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-rw-r--r--src/path/splinefit/bezier-fit.cpp90
-rw-r--r--src/path/splinefit/bezier-fit.h20
-rw-r--r--src/path/splinefit/splinefit.c1427
-rw-r--r--src/path/splinefit/splinefit.h78
-rw-r--r--src/path/splinefit/splinefont.c1174
-rw-r--r--src/path/splinefit/splinefont.h191
-rw-r--r--src/path/splinefit/splinerefigure.c117
-rw-r--r--src/path/splinefit/splinerefigure.h9
8 files changed, 3106 insertions, 0 deletions
diff --git a/src/path/splinefit/bezier-fit.cpp b/src/path/splinefit/bezier-fit.cpp
new file mode 100644
index 0000000..111f538
--- /dev/null
+++ b/src/path/splinefit/bezier-fit.cpp
@@ -0,0 +1,90 @@
+// SPDX-License-Identifier: GPL-2.0-or-later
+
+#include <iostream>
+#include <vector>
+#include "bezier-fit.h"
+#include <2geom/bezier-utils.h>
+#include <2geom/point.h>
+
+extern "C" {
+ #include "splinefit.h"
+ #include "splinefont.h"
+}
+
+int bezier_fit(Geom::Point bezier[4], const std::vector<InputPoint>& data) {
+
+ if (data.size() <= 2) return 0;
+
+ int order2 = false; // not 2nd order, so cubic
+ // "Fitting cubic Bézier curves"
+ // https://raphlinus.github.io/curves/2021/03/11/bezier-fitting.html
+ mergetype mt = mt_levien;
+ auto len = data.size();
+
+ std::vector<FitPoint> fit;
+ for (int i = 0; i < len; ++i) {
+ fit.push_back({});
+ auto& fp = fit.back();
+ fp.p.x = data[i].x();
+ fp.p.y = data[i].y();
+ fp.t = data[i].t;
+ fp.ut.x = fp.ut.y = 0;
+ }
+
+ // transform data into spline set format
+
+ auto input = (SplineSet*)chunkalloc(sizeof(SplineSet));
+
+ for (int i = 0; i < len; ++i) {
+ auto& d = data[i];
+ auto sp = SplinePointCreate(d.x(), d.y());
+ if (d.have_slope) {
+ sp->nextcp.x = d.front.x();
+ sp->nextcp.y = d.front.y();
+ sp->nonextcp = false;
+ sp->prevcp.x = d.back.x();
+ sp->prevcp.y = d.back.y();
+ sp->noprevcp = false;
+ }
+
+ if (i == 0) {
+ input->first = input->last = sp;
+ }
+ else {
+ SplineMake(input->last, sp, order2);
+ input->last = sp;
+ }
+ }
+
+ Spline* spline = ApproximateSplineFromPointsSlopes(input->first, input->last, fit.data(), fit.size(), order2, mt);
+ bool ok = spline != nullptr;
+
+ if (!spline) {
+ std::vector<Geom::Point> inp;
+ inp.reserve(data.size());
+ for (auto& pt : data) {
+ inp.push_back(pt);
+ }
+ ok = bezier_fit_cubic(bezier, inp.data(), inp.size(), 0.5) > 0;
+ }
+
+ if (spline) {
+ bezier[0].x() = spline->from->me.x;
+ bezier[0].y() = spline->from->me.y;
+
+ bezier[1].x() = spline->from->nextcp.x;
+ bezier[1].y() = spline->from->nextcp.y;
+
+ bezier[2].x() = spline->to->prevcp.x;
+ bezier[2].y() = spline->to->prevcp.y;
+
+ bezier[3].x() = spline->to->me.x;
+ bezier[3].y() = spline->to->me.y;
+ }
+
+ SplinePointListFree(input);
+ //TODO: verify that all C structs are freed up
+ // SplineFree(spline);
+
+ return ok;
+}
diff --git a/src/path/splinefit/bezier-fit.h b/src/path/splinefit/bezier-fit.h
new file mode 100644
index 0000000..7eb0440
--- /dev/null
+++ b/src/path/splinefit/bezier-fit.h
@@ -0,0 +1,20 @@
+// SPDX-License-Identifier: GPL-2.0-or-later
+
+#include <vector>
+#include "2geom/point.h"
+
+struct InputPoint : Geom::Point {
+ InputPoint() {}
+ InputPoint(const Geom::Point& pt) : Point(pt) {}
+ InputPoint(const Geom::Point& pt, double t) : Point(pt), t(t) {}
+ InputPoint(const Geom::Point& pt, const Geom::Point& front, const Geom::Point& back, double t)
+ : Point(pt), front(front), back(back), t(t), have_slope(true) {}
+
+ Geom::Point front;
+ Geom::Point back;
+ double t = 0;
+ bool have_slope = false;
+};
+
+// Fit cubic Bezier to input points; use slope of the first and last points to find a fit
+int bezier_fit(Geom::Point bezier[4], const std::vector<InputPoint>& data);
diff --git a/src/path/splinefit/splinefit.c b/src/path/splinefit/splinefit.c
new file mode 100644
index 0000000..9e3039a
--- /dev/null
+++ b/src/path/splinefit/splinefit.c
@@ -0,0 +1,1427 @@
+// SPDX-License-Identifier: GPL-2.0-or-later
+/* -*- coding: utf-8 -*- */
+/* Copyright (C) 2000-2012 by George Williams, 2021 by Linus Romer */
+/*
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions are met:
+
+ * Redistributions of source code must retain the above copyright notice, this
+ * list of conditions and the following disclaimer.
+
+ * Redistributions in binary form must reproduce the above copyright notice,
+ * this list of conditions and the following disclaimer in the documentation
+ * and/or other materials provided with the distribution.
+
+ * The name of the author may not be used to endorse or promote products
+ * derived from this software without specific prior written permission.
+
+ * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED
+ * WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
+ * MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO
+ * EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+ * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+ * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS;
+ * OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,
+ * WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR
+ * OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF
+ * ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+ */
+
+/* Incorporated into Inkscape sources; splinefit.c as of 2023-02-15.
+
+ * Note: The only changes to this file in Inkscape codebase are modification of includes
+ * and adding this note. Formatting is intact to facilitate future merges.
+ */
+
+#include <memory.h>
+#include <stdbool.h>
+#include <stdint.h>
+#include "splinefit.h"
+
+#include <math.h>
+
+static Spline *IsLinearApprox(SplinePoint *from, SplinePoint *to,
+ FitPoint *mid, int cnt, int order2) {
+ bigreal vx, vy, slope;
+ int i;
+
+ vx = to->me.x-from->me.x; vy = to->me.y-from->me.y;
+ if ( vx==0 && vy==0 ) {
+ for ( i=0; i<cnt; ++i )
+ if ( mid[i].p.x != from->me.x || mid[i].p.y != from->me.y )
+return( NULL );
+ } else if ( fabs(vx)>fabs(vy) ) {
+ slope = vy/vx;
+ for ( i=0; i<cnt; ++i )
+ if ( !RealWithin(mid[i].p.y,from->me.y+slope*(mid[i].p.x-from->me.x),.7) )
+return( NULL );
+ } else {
+ slope = vx/vy;
+ for ( i=0; i<cnt; ++i )
+ if ( !RealWithin(mid[i].p.x,from->me.x+slope*(mid[i].p.y-from->me.y),.7) )
+return( NULL );
+ }
+ from->nextcp = from->me;
+ to->prevcp = to->me;
+return( SplineMake(from,to,order2) );
+}
+
+/* This routine should almost never be called now. It uses a flawed algorithm */
+/* which won't produce the best results. It gets called only when the better */
+/* approach doesn't work (singular matrices, etc.) */
+/* Old comment, back when I was confused... */
+/* Least squares tells us that:
+ | S(xi*ti^3) | | S(ti^6) S(ti^5) S(ti^4) S(ti^3) | | a |
+ | S(xi*ti^2) | = | S(ti^5) S(ti^4) S(ti^3) S(ti^2) | * | b |
+ | S(xi*ti) | | S(ti^4) S(ti^3) S(ti^2) S(ti) | | c |
+ | S(xi) | | S(ti^3) S(ti^2) S(ti) n | | d |
+ and the definition of a spline tells us:
+ | x1 | = | 1 1 1 1 | * (a b c d)
+ | x0 | = | 0 0 0 1 | * (a b c d)
+So we're a bit over specified. Let's use the last two lines of least squares
+and the 2 from the spline defn. So d==x0. Now we've got three unknowns
+and only three equations...
+
+For order2 splines we've got
+ | S(xi*ti^2) | | S(ti^4) S(ti^3) S(ti^2) | | b |
+ | S(xi*ti) | = | S(ti^3) S(ti^2) S(ti) | * | c |
+ | S(xi) | | S(ti^2) S(ti) n | | d |
+ and the definition of a spline tells us:
+ | x1 | = | 1 1 1 | * (b c d)
+ | x0 | = | 0 0 1 | * (b c d)
+=>
+ d = x0
+ b+c = x1-x0
+ S(ti^2)*b + S(ti)*c = S(xi)-n*x0
+ S(ti^2)*b + S(ti)*(x1-x0-b) = S(xi)-n*x0
+ [ S(ti^2)-S(ti) ]*b = S(xi)-S(ti)*(x1-x0) - n*x0
+*/
+static int _ApproximateSplineFromPoints(SplinePoint *from, SplinePoint *to,
+ FitPoint *mid, int cnt, BasePoint *nextcp, BasePoint *prevcp,
+ int order2) {
+ bigreal tt, ttn;
+ int i, j, ret;
+ bigreal vx[3], vy[3], m[3][3];
+ bigreal ts[7], xts[4], yts[4];
+ BasePoint nres, pres;
+ int nrescnt=0, prescnt=0;
+ bigreal nmin, nmax, pmin, pmax, test, ptest;
+ bigreal bx, by, cx, cy;
+
+ memset(&nres,0,sizeof(nres)); memset(&pres,0,sizeof(pres));
+
+ /* Add the initial and end points */
+ ts[0] = 2; for ( i=1; i<7; ++i ) ts[i] = 1;
+ xts[0] = from->me.x+to->me.x; yts[0] = from->me.y+to->me.y;
+ xts[3] = xts[2] = xts[1] = to->me.x; yts[3] = yts[2] = yts[1] = to->me.y;
+ nmin = pmin = 0; nmax = pmax = (to->me.x-from->me.x)*(to->me.x-from->me.x)+(to->me.y-from->me.y)*(to->me.y-from->me.y);
+ for ( i=0; i<cnt; ++i ) {
+ xts[0] += mid[i].p.x;
+ yts[0] += mid[i].p.y;
+ ++ts[0];
+ tt = mid[i].t;
+ xts[1] += tt*mid[i].p.x;
+ yts[1] += tt*mid[i].p.y;
+ ts[1] += tt;
+ ts[2] += (ttn=tt*tt);
+ xts[2] += ttn*mid[i].p.x;
+ yts[2] += ttn*mid[i].p.y;
+ ts[3] += (ttn*=tt);
+ xts[3] += ttn*mid[i].p.x;
+ yts[3] += ttn*mid[i].p.y;
+ ts[4] += (ttn*=tt);
+ ts[5] += (ttn*=tt);
+ ts[6] += (ttn*=tt);
+
+ test = (mid[i].p.x-from->me.x)*(to->me.x-from->me.x) + (mid[i].p.y-from->me.y)*(to->me.y-from->me.y);
+ if ( test<nmin ) nmin=test;
+ if ( test>nmax ) nmax=test;
+ test = (mid[i].p.x-to->me.x)*(from->me.x-to->me.x) + (mid[i].p.y-to->me.y)*(from->me.y-to->me.y);
+ if ( test<pmin ) pmin=test;
+ if ( test>pmax ) pmax=test;
+ }
+ pmin *= 1.2; pmax *= 1.2; nmin *= 1.2; nmax *= 1.2;
+
+ for ( j=0; j<3; ++j ) {
+ if ( order2 ) {
+ if ( RealNear(ts[j+2],ts[j+1]) )
+ continue;
+ /* This produces really bad results!!!! But I don't see what I can do to improve it */
+ bx = (xts[j]-ts[j+1]*(to->me.x-from->me.x) - ts[j]*from->me.x) / (ts[j+2]-ts[j+1]);
+ by = (yts[j]-ts[j+1]*(to->me.y-from->me.y) - ts[j]*from->me.y) / (ts[j+2]-ts[j+1]);
+ cx = to->me.x-from->me.x-bx;
+ cy = to->me.y-from->me.y-by;
+
+ nextcp->x = from->me.x + cx/2;
+ nextcp->y = from->me.y + cy/2;
+ *prevcp = *nextcp;
+ } else {
+ vx[0] = xts[j+1]-ts[j+1]*from->me.x;
+ vx[1] = xts[j]-ts[j]*from->me.x;
+ vx[2] = to->me.x-from->me.x; /* always use the defn of spline */
+
+ vy[0] = yts[j+1]-ts[j+1]*from->me.y;
+ vy[1] = yts[j]-ts[j]*from->me.y;
+ vy[2] = to->me.y-from->me.y;
+
+ m[0][0] = ts[j+4]; m[0][1] = ts[j+3]; m[0][2] = ts[j+2];
+ m[1][0] = ts[j+3]; m[1][1] = ts[j+2]; m[1][2] = ts[j+1];
+ m[2][0] = 1; m[2][1] = 1; m[2][2] = 1;
+
+ /* Remove a terms from rows 0 and 1 */
+ vx[0] -= ts[j+4]*vx[2];
+ vy[0] -= ts[j+4]*vy[2];
+ m[0][0] = 0; m[0][1] -= ts[j+4]; m[0][2] -= ts[j+4];
+ vx[1] -= ts[j+3]*vx[2];
+ vy[1] -= ts[j+3]*vy[2];
+ m[1][0] = 0; m[1][1] -= ts[j+3]; m[1][2] -= ts[j+3];
+
+ if ( fabs(m[1][1])<fabs(m[0][1]) ) {
+ bigreal temp;
+ temp = vx[1]; vx[1] = vx[0]; vx[0] = temp;
+ temp = vy[1]; vy[1] = vy[0]; vy[0] = temp;
+ temp = m[1][1]; m[1][1] = m[0][1]; m[0][1] = temp;
+ temp = m[1][2]; m[1][2] = m[0][2]; m[0][2] = temp;
+ }
+ /* remove b terms from rows 0 and 2 (first normalize row 1 so m[1][1] is 1*/
+ vx[1] /= m[1][1];
+ vy[1] /= m[1][1];
+ m[1][2] /= m[1][1];
+ m[1][1] = 1;
+ vx[0] -= m[0][1]*vx[1];
+ vy[0] -= m[0][1]*vy[1];
+ m[0][2] -= m[0][1]*m[1][2]; m[0][1] = 0;
+ vx[2] -= m[2][1]*vx[1];
+ vy[2] -= m[2][1]*vy[1];
+ m[2][2] -= m[2][1]*m[1][2]; m[2][1] = 0;
+
+ vx[0] /= m[0][2]; /* This is cx */
+ vy[0] /= m[0][2]; /* This is cy */
+ /*m[0][2] = 1;*/
+
+ vx[1] -= m[1][2]*vx[0]; /* This is bx */
+ vy[1] -= m[1][2]*vy[0]; /* This is by */
+ /* m[1][2] = 0; */
+ vx[2] -= m[2][2]*vx[0]; /* This is ax */
+ vy[2] -= m[2][2]*vy[0]; /* This is ay */
+ /* m[2][2] = 0; */
+
+ nextcp->x = from->me.x + vx[0]/3;
+ nextcp->y = from->me.y + vy[0]/3;
+ prevcp->x = nextcp->x + (vx[0]+vx[1])/3;
+ prevcp->y = nextcp->y + (vy[0]+vy[1])/3;
+ }
+
+ test = (nextcp->x-from->me.x)*(to->me.x-from->me.x) +
+ (nextcp->y-from->me.y)*(to->me.y-from->me.y);
+ ptest = (prevcp->x-to->me.x)*(from->me.x-to->me.x) +
+ (prevcp->y-to->me.y)*(from->me.y-to->me.y);
+ if ( order2 &&
+ (test<nmin || test>nmax || ptest<pmin || ptest>pmax))
+ continue;
+ if ( test>=nmin && test<=nmax ) {
+ nres.x += nextcp->x; nres.y += nextcp->y;
+ ++nrescnt;
+ }
+ if ( test>=pmin && test<=pmax ) {
+ pres.x += prevcp->x; pres.y += prevcp->y;
+ ++prescnt;
+ }
+ if ( nrescnt==1 && prescnt==1 )
+ break;
+ }
+
+ ret = 0;
+ if ( nrescnt>0 ) {
+ ret |= 1;
+ nextcp->x = nres.x/nrescnt;
+ nextcp->y = nres.y/nrescnt;
+ } else
+ *nextcp = from->nextcp;
+ if ( prescnt>0 ) {
+ ret |= 2;
+ prevcp->x = pres.x/prescnt;
+ prevcp->y = pres.y/prescnt;
+ } else
+ *prevcp = to->prevcp;
+ if ( order2 && ret!=3 ) {
+ nextcp->x = (nextcp->x + prevcp->x)/2;
+ nextcp->y = (nextcp->y + prevcp->y)/2;
+ }
+ if ( order2 )
+ *prevcp = *nextcp;
+return( ret );
+}
+
+static void TestForLinear(SplinePoint *from,SplinePoint *to) {
+ BasePoint off, cpoff, cpoff2;
+ bigreal len, co, co2;
+
+ /* Did we make a line? */
+ off.x = to->me.x-from->me.x; off.y = to->me.y-from->me.y;
+ len = sqrt(off.x*off.x + off.y*off.y);
+ if ( len!=0 ) {
+ off.x /= len; off.y /= len;
+ cpoff.x = from->nextcp.x-from->me.x; cpoff.y = from->nextcp.y-from->me.y;
+ len = sqrt(cpoff.x*cpoff.x + cpoff.y*cpoff.y);
+ if ( len!=0 ) {
+ cpoff.x /= len; cpoff.y /= len;
+ }
+ cpoff2.x = to->prevcp.x-from->me.x; cpoff2.y = to->prevcp.y-from->me.y;
+ len = sqrt(cpoff2.x*cpoff2.x + cpoff2.y*cpoff2.y);
+ if ( len!=0 ) {
+ cpoff2.x /= len; cpoff2.y /= len;
+ }
+ co = cpoff.x*off.y - cpoff.y*off.x; co2 = cpoff2.x*off.y - cpoff2.y*off.x;
+ if ( co<.05 && co>-.05 && co2<.05 && co2>-.05 ) {
+ from->nextcp = from->me;
+ to->prevcp = to->me;
+ } else {
+ Spline temp;
+ memset(&temp,0,sizeof(temp));
+ temp.from = from; temp.to = to;
+ SplineRefigure(&temp);
+ if ( SplineIsLinear(&temp)) {
+ from->nextcp = from->me;
+ to->prevcp = to->me;
+ }
+ }
+ }
+}
+
+/* Find a spline which best approximates the list of intermediate points we */
+/* are given. No attempt is made to use fixed slope angles */
+/* given a set of points (x,y,t) */
+/* find the bezier spline which best fits those points */
+
+/* OK, we know the end points, so all we really need are the control points */
+ /* For cubics.... */
+/* Pf = point from */
+/* CPf = control point, from nextcp */
+/* CPt = control point, to prevcp */
+/* Pt = point to */
+/* S(t) = Pf + 3*(CPf-Pf)*t + 3*(CPt-2*CPf+Pf)*t^2 + (Pt-3*CPt+3*CPf-Pf)*t^3 */
+/* S(t) = Pf - 3*Pf*t + 3*Pf*t^2 - Pf*t^3 + Pt*t^3 + */
+/* 3*(t-2*t^2+t^3)*CPf + */
+/* 3*(t^2-t^3)*CPt */
+/* We want to minimize Σ [S(ti)-Pi]^2 */
+/* There are four variables CPf.x, CPf.y, CPt.x, CPt.y */
+/* When we take the derivative of the error term above with each of these */
+/* variables, we find that the two coordinates are separate. So I shall only */
+/* work through the equations once, leaving off the coordinate */
+/* d error/dCPf = Σ 2*3*(t-2*t^2+t^3) * [S(ti)-Pi] = 0 */
+/* d error/dCPt = Σ 2*3*(t^2-t^3) * [S(ti)-Pi] = 0 */
+ /* For quadratics.... */
+/* CP = control point, there's only one */
+/* S(t) = Pf + 2*(CP-Pf)*t + (Pt-2*CP+Pf)*t^2 */
+/* S(t) = Pf - 2*Pf*t + Pf*t^2 + Pt*t^2 + */
+/* 2*(t-2*t^2)*CP */
+/* We want to minimize Σ [S(ti)-Pi]^2 */
+/* There are two variables CP.x, CP.y */
+/* d error/dCP = Σ 2*2*(t-2*t^2) * [S(ti)-Pi] = 0 */
+/* Σ (t-2*t^2) * [Pf - 2*Pf*t + Pf*t^2 + Pt*t^2 - Pi + */
+/* 2*(t-2*t^2)*CP] = 0 */
+/* CP * (Σ 2*(t-2*t^2)*(t-2*t^2)) = Σ (t-2*t^2) * [Pf - 2*Pf*t + Pf*t^2 + Pt*t^2 - Pi] */
+
+/* Σ (t-2*t^2) * [Pf - 2*Pf*t + Pf*t^2 + Pt*t^2 - Pi] */
+/* CP = ----------------------------------------------------- */
+/* Σ 2*(t-2*t^2)*(t-2*t^2) */
+Spline *ApproximateSplineFromPoints(SplinePoint *from, SplinePoint *to,
+ FitPoint *mid, int cnt, int order2) {
+ int ret;
+ Spline *spline;
+ BasePoint nextcp, prevcp;
+ int i;
+
+ if ( order2 ) {
+ bigreal xconst, yconst, term /* Same for x and y */;
+ xconst = yconst = term = 0;
+ for ( i=0; i<cnt; ++i ) {
+ bigreal t = mid[i].t, t2 = t*t;
+ bigreal tfactor = (t-2*t2);
+ term += 2*tfactor*tfactor;
+ xconst += tfactor*(from->me.x*(1-2*t+t2) + to->me.x*t2 - mid[i].p.x);
+ yconst += tfactor*(from->me.y*(1-2*t+t2) + to->me.y*t2 - mid[i].p.y);
+ }
+ if ( term!=0 ) {
+ BasePoint cp;
+ cp.x = xconst/term; cp.y = yconst/term;
+ from->nextcp = to->prevcp = cp;
+return( SplineMake2(from,to));
+ }
+ } else {
+ bigreal xconst[2], yconst[2], f_term[2], t_term[2] /* Same for x and y */;
+ bigreal tfactor[2], determinant;
+ xconst[0] = xconst[1] = yconst[0] = yconst[1] =
+ f_term[0] = f_term[1] = t_term[0] = t_term[1] = 0;
+ for ( i=0; i<cnt; ++i ) {
+ bigreal t = mid[i].t, t2 = t*t, t3=t*t2;
+ bigreal xc = (from->me.x*(1-3*t+3*t2-t3) + to->me.x*t3 - mid[i].p.x);
+ bigreal yc = (from->me.y*(1-3*t+3*t2-t3) + to->me.y*t3 - mid[i].p.y);
+ tfactor[0] = (t-2*t2+t3); tfactor[1]=(t2-t3);
+ xconst[0] += tfactor[0]*xc;
+ xconst[1] += tfactor[1]*xc;
+ yconst[0] += tfactor[0]*yc;
+ yconst[1] += tfactor[1]*yc;
+ f_term[0] += 3*tfactor[0]*tfactor[0];
+ f_term[1] += 3*tfactor[0]*tfactor[1];
+ /*t_term[0] += 3*tfactor[1]*tfactor[0];*/
+ t_term[1] += 3*tfactor[1]*tfactor[1];
+ }
+ t_term[0] = f_term[1];
+ determinant = f_term[1]*t_term[0] - f_term[0]*t_term[1];
+ if ( determinant!=0 ) {
+ to->prevcp.x = -(xconst[0]*f_term[1]-xconst[1]*f_term[0])/determinant;
+ to->prevcp.y = -(yconst[0]*f_term[1]-yconst[1]*f_term[0])/determinant;
+ if ( f_term[0]!=0 ) {
+ from->nextcp.x = (-xconst[0]-t_term[0]*to->prevcp.x)/f_term[0];
+ from->nextcp.y = (-yconst[0]-t_term[0]*to->prevcp.y)/f_term[0];
+ } else {
+ from->nextcp.x = (-xconst[1]-t_term[1]*to->prevcp.x)/f_term[1];
+ from->nextcp.y = (-yconst[1]-t_term[1]*to->prevcp.y)/f_term[1];
+ }
+return( SplineMake3(from,to));
+ }
+ }
+
+ if ( (spline = IsLinearApprox(from,to,mid,cnt,order2))!=NULL )
+return( spline );
+
+ ret = _ApproximateSplineFromPoints(from,to,mid,cnt,&nextcp,&prevcp,order2);
+
+ if ( ret&1 ) {
+ from->nextcp = nextcp;
+ } else {
+ from->nextcp = from->me;
+ }
+ if ( ret&2 ) {
+ to->prevcp = prevcp;
+ } else {
+ to->prevcp = to->me;
+ }
+ TestForLinear(from,to);
+ spline = SplineMake(from,to,order2);
+return( spline );
+}
+
+static bigreal ClosestSplineSolve(Spline1D *sp,bigreal sought,bigreal close_to_t) {
+ /* We want to find t so that spline(t) = sought */
+ /* find the value which is closest to close_to_t */
+ /* on error return closetot */
+ extended ts[3];
+ int i;
+ bigreal t, best, test;
+
+ _CubicSolve(sp,sought,ts);
+ best = 9e20; t= close_to_t;
+ for ( i=0; i<3; ++i ) if ( ts[i]>-.0001 && ts[i]<1.0001 ) {
+ if ( (test=ts[i]-close_to_t)<0 ) test = -test;
+ if ( test<best ) {
+ best = test;
+ t = ts[i];
+ }
+ }
+
+return( t );
+}
+
+struct dotbounds {
+ BasePoint unit;
+ BasePoint base;
+ bigreal len;
+ bigreal min,max; /* If min<0 || max>len the spline extends beyond its endpoints */
+};
+
+static bigreal SigmaDeltas(Spline *spline, FitPoint *mid, int cnt, DBounds *b, struct dotbounds *db) {
+ int i;
+ bigreal xdiff, ydiff, sum, temp, t;
+ SplinePoint *to = spline->to, *from = spline->from;
+ extended ts[2], x,y;
+ struct dotbounds db2;
+ bigreal dot;
+ int near_vert, near_horiz;
+
+ if ( (xdiff = to->me.x-from->me.x)<0 ) xdiff = -xdiff;
+ if ( (ydiff = to->me.y-from->me.y)<0 ) ydiff = -ydiff;
+ near_vert = ydiff>2*xdiff;
+ near_horiz = xdiff>2*ydiff;
+
+ sum = 0;
+ for ( i=0; i<cnt; ++i ) {
+ if ( near_vert ) {
+ t = ClosestSplineSolve(&spline->splines[1],mid[i].p.y,mid[i].t);
+ } else if ( near_horiz ) {
+ t = ClosestSplineSolve(&spline->splines[0],mid[i].p.x,mid[i].t);
+ } else {
+ t = (ClosestSplineSolve(&spline->splines[1],mid[i].p.y,mid[i].t) +
+ ClosestSplineSolve(&spline->splines[0],mid[i].p.x,mid[i].t))/2;
+ }
+ temp = mid[i].p.x - ( ((spline->splines[0].a*t+spline->splines[0].b)*t+spline->splines[0].c)*t + spline->splines[0].d );
+ sum += temp*temp;
+ temp = mid[i].p.y - ( ((spline->splines[1].a*t+spline->splines[1].b)*t+spline->splines[1].c)*t + spline->splines[1].d );
+ sum += temp*temp;
+ }
+
+ /* Ok, we've got distances from a set of points on the old spline to the */
+ /* new one. Let's do the reverse: find the extrema of the new and see how*/
+ /* close they get to the bounding box of the old */
+ /* And get really unhappy if the spline extends beyond the end-points */
+ db2.min = 0; db2.max = db->len;
+ SplineFindExtrema(&spline->splines[0],&ts[0],&ts[1]);
+ for ( i=0; i<2; ++i ) {
+ if ( ts[i]!=-1 ) {
+ x = ((spline->splines[0].a*ts[i]+spline->splines[0].b)*ts[i]+spline->splines[0].c)*ts[i] + spline->splines[0].d;
+ y = ((spline->splines[1].a*ts[i]+spline->splines[1].b)*ts[i]+spline->splines[1].c)*ts[i] + spline->splines[1].d;
+ if ( x<b->minx )
+ sum += (x-b->minx)*(x-b->minx);
+ else if ( x>b->maxx )
+ sum += (x-b->maxx)*(x-b->maxx);
+ dot = (x-db->base.x)*db->unit.x + (y-db->base.y)*db->unit.y;
+ if ( dot<db2.min ) db2.min = dot;
+ if ( dot>db2.max ) db2.max = dot;
+ }
+ }
+ SplineFindExtrema(&spline->splines[1],&ts[0],&ts[1]);
+ for ( i=0; i<2; ++i ) {
+ if ( ts[i]!=-1 ) {
+ x = ((spline->splines[0].a*ts[i]+spline->splines[0].b)*ts[i]+spline->splines[0].c)*ts[i] + spline->splines[0].d;
+ y = ((spline->splines[1].a*ts[i]+spline->splines[1].b)*ts[i]+spline->splines[1].c)*ts[i] + spline->splines[1].d;
+ if ( y<b->miny )
+ sum += (y-b->miny)*(y-b->miny);
+ else if ( y>b->maxy )
+ sum += (y-b->maxy)*(y-b->maxy);
+ dot = (x-db->base.x)*db->unit.x + (y-db->base.y)*db->unit.y;
+ if ( dot<db2.min ) db2.min = dot;
+ if ( dot>db2.max ) db2.max = dot;
+ }
+ }
+
+ /* Big penalty for going beyond the range of the desired spline */
+ if ( db->min==0 && db2.min<0 )
+ sum += 10000 + db2.min*db2.min;
+ else if ( db2.min<db->min )
+ sum += 100 + (db2.min-db->min)*(db2.min-db->min);
+ if ( db->max==db->len && db2.max>db->len )
+ sum += 10000 + (db2.max-db->max)*(db2.max-db->max);
+ else if ( db2.max>db->max )
+ sum += 100 + (db2.max-db->max)*(db2.max-db->max);
+
+return( sum );
+}
+
+static void ApproxBounds(DBounds *b, FitPoint *mid, int cnt, struct dotbounds *db) {
+ int i;
+ bigreal dot;
+
+ b->minx = b->maxx = mid[0].p.x;
+ b->miny = b->maxy = mid[0].p.y;
+ db->min = 0; db->max = db->len;
+ for ( i=1; i<cnt; ++i ) {
+ if ( mid[i].p.x>b->maxx ) b->maxx = mid[i].p.x;
+ if ( mid[i].p.x<b->minx ) b->minx = mid[i].p.x;
+ if ( mid[i].p.y>b->maxy ) b->maxy = mid[i].p.y;
+ if ( mid[i].p.y<b->miny ) b->miny = mid[i].p.y;
+ dot = (mid[i].p.x-db->base.x)*db->unit.x + (mid[i].p.y-db->base.y)*db->unit.y;
+ if ( dot<db->min ) db->min = dot;
+ if ( dot>db->max ) db->max = dot;
+ }
+}
+
+/* pf == point from (start point) */
+/* Δf == slope from (cp(from) - from) */
+/* pt == point to (end point, t==1) */
+/* Δt == slope to (cp(to) - to) */
+
+/* A spline from pf to pt with slope vectors rf*Δf, rt*Δt is: */
+/* p(t) = pf + [ 3*rf*Δf ]*t + 3*[pt-pf+rt*Δt-2*rf*Δf] *t^2 + */
+/* [2*pf-2*pt+3*rf*Δf-3*rt*Δt]*t^3 */
+
+/* So I want */
+/* d Σ (p(t(i))-p(i))^2/ d rf == 0 */
+/* d Σ (p(t(i))-p(i))^2/ d rt == 0 */
+/* now... */
+/* d Σ (p(t(i))-p(i))^2/ d rf == 0 */
+/* => Σ 3*t*Δf*(1-2*t+t^2)*
+ * [pf-pi+ 3*(pt-pf)*t^2 + 2*(pf-pt)*t^3] +
+ * 3*[t - 2*t^2 + t^3]*Δf*rf +
+ * 3*[t^2-t^3]*Δt*rt */
+/* and... */
+/* d Σ (p(t(i))-p(i))^2/ d rt == 0 */
+/* => Σ 3*t^2*Δt*(1-t)*
+ * [pf-pi+ 3*(pt-pf)*t^2 + 2*(pf-pt)*t^3] +
+ * 3*[t - 2*t^2 + t^3]*Δf*rf +
+ * 3*[t^2-t^3]*Δt*rt */
+
+/* Now for a long time I looked at that and saw four equations and two unknowns*/
+/* That was I was trying to solve for x and y separately, and that doesn't work. */
+/* There are really just two equations and each sums over both x and y components */
+
+/* Old comment: */
+/* I used to do a least squares approach adding two more to the above set of equations */
+/* which held the slopes constant. But that didn't work very well. So instead*/
+/* Then I tried doing the approximation, and then forcing the control points */
+/* to be in line (with the original slopes), getting a better approximation */
+/* to "t" for each data point and then calculating an error array, approximating*/
+/* it, and using that to fix up the final result */
+/* Then I tried checking various possible cp lengths in the desired directions*/
+/* finding the best one or two, and doing a 2D binary search using that as a */
+/* starting point. */
+/* And sometimes a least squares approach will give us the right answer, so */
+/* try that too. */
+/* This still isn't as good as I'd like it... But I haven't been able to */
+/* improve it further yet */
+/* The mergetype mt is either of: */
+/* mt_matrix; original, fast, all-purpose (relies on matrix calculations) */
+/* mt_levien; by Raph Levien (implemented by Linus Romer), fast, accurate, use only if mid is on spline */
+/* mt_bruteforce; slow, all-purpose, normally more accurate than mt_matrix.*/
+/* The mt_levien algorithm is explained here: */
+/* raphlinus.github.io/curves/2021/03/11/bezier-fitting.html */
+/* The notation used here is a bit different: Instead of theta1, theta2, */
+/* delta1, delta2, momentx, area we use alpha,beta,a,b,m,f: */
+/* Here is to complete math that we are using: */
+/* Signed area of the cubic bezier spline a .. controls b and c .. d to the x-axis */
+/* f = ((xb-xa)*(10*ya+6*yb+3*yc+yd)+(xc-xb)*(4*ya+6*yb+6*yc+4*yd)+(xd-xc)*(ya+3*yb+6*yc+10*yd))/20; */
+/* simplified for the normed case */
+/* f = 3/20*(2*a*sin(alpha)+2*b*sin(beta)-a*b*sin(alpha+beta)); */
+/* solved for b */
+/* b = (20*f-6*a*sin(alpha))/(6*sin(beta)-3*a*sin(alpha+beta)). */
+/* Signed area of the cubic bezier spline a .. controls b and c .. d to the x-axis */
+/* from point a up to the bezier point at time t */
+/* f(t) = ((((1-t)*xa+xb*t)-xa)*(10*ya+6*((1-t)*ya+yb*t)+3*((1-t)^2*ya+2*(1-t)*t*yb+t^2*yc) */
+/* +((1-t)^3*ya+3*(1-t)^2*t*yb+3*(1-t)*t^2*yc+t^3*yd))+(((1-t)^2*xa+2*(1-t)*t*xb+t^2*xc) */
+/* -((1-t)*xa+xb*t))*(4*ya+6*((1-t)*ya+yb*t)+6*((1-t)^2*ya+2*(1-t)*t*yb+t^2*yc) */
+/* +4*((1-t)^3*ya+3*(1-t)^2*t*yb+3*(1-t)*t^2*yc+t^3*yd))+(((1-t)^3*xa+3*(1-t)^2*t*xb */
+/* +3*(1-t)*t^2*xc+t^3*xd)-((1-t)^2*xa+2*(1-t)*t*xb+t^2*xc))*(ya+3*((1-t)*ya+yb*t) */
+/* +6*((1-t)^2*ya+2*(1-t)*t*yb+t^2*yc)+10*((1-t)^3*ya+3*(1-t)^2*t*yb+3*(1-t)*t^2*yc+t^3*yd)))/20; */
+/* simplified for the normed case: */
+/* f(t) = -(3*(30*a*b*sin(beta-alpha)*t^6+15*b^2*sin(2*beta)*t^6-20*b*sin(beta)*t^6 */
+/* -15*a^2*sin(2*alpha)*t^6+20*a*sin(alpha)*t^6+6*a*b*sin(beta+alpha)*t^5 */
+/* -90*a*b*sin(beta-alpha)*t^5-30*b^2*sin(2*beta)*t^5+48*b*sin(beta)*t^5 */
+/* +60*a^2*sin(2*alpha)*t^5-72*a*sin(alpha)*t^5-15*a*b*sin(beta+alpha)*t^4 */
+/* +90*a*b*sin(beta-alpha)*t^4+15*b^2*sin(2*beta)*t^4-30*b*sin(beta)*t^4 */
+/* -90*a^2*sin(2*alpha)*t^4+90*a*sin(alpha)*t^4+10*a*b*sin(beta+alpha)*t^3 */
+/* -30*a*b*sin(beta-alpha)*t^3+60*a^2*sin(2*alpha)*t^3-40*a*sin(alpha)*t^3 */
+/* -15*a^2*sin(2*alpha)*t^2))/20. */
+/* First moment about y-axis = \int x dA = \int x dA/dt dt for a cubic bezier */
+/* path a .. controls b and c .. d */
+/* m = (280*xd^2*yd-105*xc*xd*yd-30*xb*xd*yd-5*xa*xd*yd-45*xc^2*yd-45*xb*xc*yd */
+/* -12*xa*xc*yd-18*xb^2*yd-15*xa*xb*yd-5*xa^2*yd+105*xd^2*yc+45*xc*xd*yc */
+/* -3*xa*xd*yc-27*xb*xc*yc-18*xa*xc*yc-27*xb^2*yc-45*xa*xb*yc-30*xa^2*yc */
+/* +30*xd^2*yb+45*xc*xd*yb+18*xb*xd*yb+3*xa*xd*yb+27*xc^2*yb+27*xb*xc*yb */
+/* -45*xa*xb*yb-105*xa^2*yb+5*xd^2*ya+15*xc*xd*ya+12*xb*xd*ya+5*xa*xd*ya */
+/* +18*xc^2*ya+45*xb*xc*ya+30*xa*xc*ya+45*xb^2*ya+105*xa*xb*ya-280*xa^2*ya)/840; */
+/* simplified for the normed case */
+/* m = (9*a*cos(alpha)*b^2*cos(beta)*sin(beta)-15*b^2*cos(beta)*sin(beta) */
+/* -9*a^2*cos(alpha)^2*b*sin(beta)-9*a*cos(alpha)*b*sin(beta)+50*b*sin(beta) */
+/* +9*a*sin(alpha)*b^2*cos(beta)^2-9*a^2*cos(alpha)*sin(alpha)*b*cos(beta) */
+/* -33*a*sin(alpha)*b*cos(beta)+15*a^2*cos(alpha)*sin(alpha)+34*a*sin(alpha))/280; */
+/* normed case combined with the formula for b depending on the area (see above): */
+/* m = (34*a*sin(alpha)+50*(20*f-6*a*sin(alpha))/(6*sin(beta)-3*a*sin(beta+alpha))*sin(beta) */
+/* +15*a^2*sin(alpha)*cos(alpha)-15*(20*f-6*a*sin(alpha))/(6*sin(beta) */
+/* -3*a*sin(beta+alpha))^2*sin(beta)*cos(beta)-a*(20*f-6*a*sin(alpha))/(6*sin(beta) */
+/* -3*a*sin(beta+alpha))*(33*sin(alpha)*cos(beta)+9*cos(alpha)*sin(beta)) */
+/* -9*a^2*(20*f-6*a*sin(alpha))/(6*sin(beta)-3*a*sin(beta+alpha))*sin(alpha+beta)*cos(alpha) */
+/* +9*a*(20*f-6*a*sin(alpha))/(6*sin(beta)-3*a*sin(beta+alpha))^2*sin(alpha+beta)*cos(beta))/280; */
+/* and reduced to a quartic equation with sa = sin(alpha), sb = sin(beta), ca = cos(alpha), cb = cos(beta) */
+/* 0 = -9*ca*(((2*sb*cb*ca+sa*(2*cb*cb-1))*ca-2*sb*cb)*ca-cb*cb*sa) * a^4 */
+/* + 12*((((cb*(30*f*cb-sb)-15*f)*ca+2*sa-cb*sa*(cb+30*f*sb))*ca+cb*(sb-15*f*cb))*ca-sa*cb*cb) * a^3 */
+/* + 12*((((70*m+15*f)*sb^2+cb*(9*sb-70*cb*m-5*cb*f))*ca-5*sa*sb*(3*sb-4*cb*(7*m+f)))*ca-cb*(9*sb-70*cb*m-5*cb*f)) * a^2 */
+/* + 16*(((12*sa-5*ca*(42*m-17*f))*sb-70*cb*(3*m-f)*sa-75*ca*cb*f*f)*sb-75*cb^2*f^2*sa) * a */
+/* + 80*sb*(42*sb*m-25*f*(sb-cb*f)); */
+/* this quartic equation reduces to a quadratic for the special case beta = pi - alpha or beta = -alpha */
+/* 0 = -9*ca*sa^2 * a^3 */
+/* + 6*sa*(4*sa+5*ca*f) * a^2 */
+/* + 10*((42*m-25*f)*sa-25*ca*f^2). */
+/* The derivative of the first moment (not the quartic) = 0 results in a quartic as well: */
+/* 0 = -9*ca*sa*sab^3 * a^4 */
+/* -3*sab^2*(9*ca*sa*sb-(17*sa+30*ca*f)*sab+15*cb*sa^2) * a^3 */
+/* +18*sab*sb*(21*ca*sa*sb-(17*sa+30*ca*f)*sab+15*cb*sa^2) * a^2 */
+/* -4*(144*ca*sa*sb^3+((-78*sa-135*ca*f)*sab+108*cb*sa^2)*sb^2+(-125*f*sab^2-45*cb*f*sa*sab)*sb+150*cb*f^2*sab^2) * a */
+/* +8*sb*((24*sa+45*ca*f)*sb^2+(15*cb*f*sa-125*f*sab)*sb+100*cb*f^2*sab) */
+/* this quartic equation reduces to a linear for the special case beta = pi - alpha or beta = -alpha */
+/* 0 = -3*ca*sa * a */
+/* +4*sa+5*ca*f */
+#define TRY_CNT 2
+#define DECIMATION 5
+Spline *ApproximateSplineFromPointsSlopes(SplinePoint *from, SplinePoint *to,
+ FitPoint *mid, int cnt, int order2, enum mergetype mt) {
+ BasePoint tounit, fromunit, ftunit;
+ bigreal flen,tlen,ftlen,dot;
+ Spline *spline, temp;
+ BasePoint nextcp;
+ int bettern, betterp;
+ bigreal offn, offp, incrn, incrp, trylen;
+ int nocnt = 0, totcnt;
+ bigreal curdiff, bestdiff[TRY_CNT];
+ int i,j,besti[TRY_CNT],bestj[TRY_CNT],k,l;
+ bigreal fdiff, tdiff, fmax, tmax, fdotft, tdotft;
+ DBounds b;
+ struct dotbounds db;
+ bigreal offn_, offp_, finaldiff;
+ bigreal pt_pf_x, pt_pf_y, determinant;
+ bigreal consts[2], rt_terms[2], rf_terms[2];
+
+ /* If all the selected points are at the same spot, and one of the */
+ /* end-points is also at that spot, then just copy the control point */
+ /* But our caller seems to have done that for us */
+
+ /* If the two end-points are corner points then allow the slope to vary */
+ /* Or if one end-point is a tangent but the point defining the tangent's */
+ /* slope is being removed then allow the slope to vary */
+ /* Except if the slope is horizontal or vertical then keep it fixed */
+ if ( ( !from->nonextcp && ( from->nextcp.x==from->me.x || from->nextcp.y==from->me.y)) ||
+ (!to->noprevcp && ( to->prevcp.x==to->me.x || to->prevcp.y==to->me.y)) )
+ /* Preserve the slope */;
+ else if ( ((from->pointtype == pt_corner && from->nonextcp) ||
+ (from->pointtype == pt_tangent &&
+ ((from->nonextcp && from->noprevcp) || !from->noprevcp))) &&
+ ((to->pointtype == pt_corner && to->noprevcp) ||
+ (to->pointtype == pt_tangent &&
+ ((to->nonextcp && to->noprevcp) || !to->nonextcp))) ) {
+ from->pointtype = to->pointtype = pt_corner;
+return( ApproximateSplineFromPoints(from,to,mid,cnt,order2) );
+ }
+
+ /* If we are going to honour the slopes of a quadratic spline, there is */
+ /* only one possibility */
+ if ( order2 ) {
+ if ( from->nonextcp )
+ from->nextcp = from->next->to->me;
+ if ( to->noprevcp )
+ to->prevcp = to->prev->from->me;
+ fromunit.x = from->nextcp.x-from->me.x; fromunit.y = from->nextcp.y-from->me.y;
+ tounit.x = to->prevcp.x-to->me.x; tounit.y = to->prevcp.y-to->me.y;
+
+ if ( !IntersectLines(&nextcp,&from->nextcp,&from->me,&to->prevcp,&to->me) ||
+ (nextcp.x-from->me.x)*fromunit.x + (nextcp.y-from->me.y)*fromunit.y < 0 ||
+ (nextcp.x-to->me.x)*tounit.x + (nextcp.y-to->me.y)*tounit.y < 0 ) {
+ /* If the slopes don't intersect then use a line */
+ /* (or if the intersection is patently absurd) */
+ from->nextcp = from->me;
+ to->prevcp = to->me;
+ TestForLinear(from,to);
+ } else {
+ from->nextcp = to->prevcp = nextcp;
+ }
+return( SplineMake2(from,to));
+ }
+ /* From here down we are only working with cubic splines */
+
+ if ( cnt==0 ) {
+ /* Just use what we've got, no data to improve it */
+ /* But we do sometimes get some cps which are too big */
+ bigreal len = sqrt((to->me.x-from->me.x)*(to->me.x-from->me.x) + (to->me.y-from->me.y)*(to->me.y-from->me.y));
+ if ( len==0 ) {
+ from->nextcp = from->me;
+ to->prevcp = to->me;
+ } else {
+ BasePoint noff, poff;
+ bigreal nlen, plen;
+ noff.x = from->nextcp.x-from->me.x; noff.y = from->nextcp.y-from->me.y;
+ poff.x = to->me.x-to->prevcp.x; poff.y = to->me.y-to->prevcp.y;
+ nlen = sqrt(noff.x*noff.x + noff.y+noff.y);
+ plen = sqrt(poff.x*poff.x + poff.y+poff.y);
+ if ( nlen>len/3 ) {
+ noff.x *= len/3/nlen; noff.y *= len/3/nlen;
+ from->nextcp.x = from->me.x + noff.x;
+ from->nextcp.y = from->me.y + noff.y;
+ }
+ if ( plen>len/3 ) {
+ poff.x *= len/3/plen; poff.y *= len/3/plen;
+ to->prevcp.x = to->me.x + poff.x;
+ to->prevcp.y = to->me.y + poff.y;
+ }
+ }
+return( SplineMake3(from,to));
+ }
+
+ if ( to->prev!=NULL && (( to->noprevcp && to->nonextcp ) || to->prev->knownlinear )) {
+ tounit.x = to->prev->from->me.x-to->me.x; tounit.y = to->prev->from->me.y-to->me.y;
+// g_message("1 tu: %f %f, %d %d %d", tounit.x, tounit.y, to->noprevcp, to->nonextcp, to->prev->knownlinear);
+ } else if ( !to->noprevcp || to->pointtype == pt_corner ) {
+ tounit.x = to->prevcp.x-to->me.x; tounit.y = to->prevcp.y-to->me.y;
+// g_message("2 tu: %f %f", tounit.x, tounit.y);
+ } else {
+ tounit.x = to->me.x-to->nextcp.x; tounit.y = to->me.y-to->nextcp.y;
+// g_message("3 tu: %f %f", tounit.x, tounit.y);
+ }
+ tlen = sqrt(tounit.x*tounit.x + tounit.y*tounit.y);
+ if ( from->next!=NULL && (( from->noprevcp && from->nonextcp ) || from->next->knownlinear) ) {
+ fromunit.x = from->next->to->me.x-from->me.x; fromunit.y = from->next->to->me.y-from->me.y;
+// g_message("1 fu");
+ } else if ( !from->nonextcp || from->pointtype == pt_corner ) {
+ fromunit.x = from->nextcp.x-from->me.x; fromunit.y = from->nextcp.y-from->me.y;
+// g_message("2 fu");
+ } else {
+ fromunit.x = from->me.x-from->prevcp.x; fromunit.y = from->me.y-from->prevcp.y;
+// g_message("3 fu");
+ }
+ flen = sqrt(fromunit.x*fromunit.x + fromunit.y*fromunit.y);
+// g_message("tu: %f %f, fu: %f, %f", tounit.x, tounit.y, fromunit.x, fromunit.y);
+ if ( (tlen==0 || flen==0) && (from->next==NULL || to->prev==NULL) ) {
+ memset(&temp,0,sizeof(temp));
+ temp.from = from; temp.to = to;
+ SplineRefigure(&temp);
+ from->next = to->prev = NULL;
+ }
+ if ( tlen==0 ) {
+ if ( to->prev!=NULL ) {
+ temp = *to->prev;
+ }
+ if ( (to->pointtype==pt_curve || to->pointtype==pt_hvcurve) &&
+ to->next && !to->nonextcp ) {
+ tounit.x = to->me.x-to->nextcp.x; tounit.y = to->me.y-to->nextcp.y;
+ } else {
+ tounit.x = -( (3*temp.splines[0].a*.9999+2*temp.splines[0].b)*.9999+temp.splines[0].c );
+ tounit.y = -( (3*temp.splines[1].a*.9999+2*temp.splines[1].b)*.9999+temp.splines[1].c );
+ }
+ tlen = sqrt(tounit.x*tounit.x + tounit.y*tounit.y);
+ }
+ tounit.x /= tlen; tounit.y /= tlen;
+
+ if ( flen==0 ) {
+ if ( from->next!=NULL ) {
+ temp = *from->next;
+ }
+ if ( (from->pointtype==pt_curve || from->pointtype==pt_hvcurve) &&
+ from->prev && !from->noprevcp ) {
+ fromunit.x = from->me.x-from->prevcp.x; fromunit.y = from->me.y-from->prevcp.y;
+ } else {
+ fromunit.x = ( (3*temp.splines[0].a*.0001+2*temp.splines[0].b)*.0001+temp.splines[0].c );
+ fromunit.y = ( (3*temp.splines[1].a*.0001+2*temp.splines[1].b)*.0001+temp.splines[1].c );
+ }
+ flen = sqrt(fromunit.x*fromunit.x + fromunit.y*fromunit.y);
+ }
+ fromunit.x /= flen; fromunit.y /= flen;
+
+ ftunit.x = (to->me.x-from->me.x); ftunit.y = (to->me.y-from->me.y);
+ ftlen = sqrt(ftunit.x*ftunit.x + ftunit.y*ftunit.y);
+ if ( ftlen!=0 ) {
+ ftunit.x /= ftlen; ftunit.y /= ftlen;
+ }
+
+ if ( (dot=fromunit.x*tounit.y - fromunit.y*tounit.x)<.0001 && dot>-.0001 &&
+ (dot=ftunit.x*tounit.y - ftunit.y*tounit.x)<.0001 && dot>-.0001 ) {
+ /* It's a line. Slopes are parallel, and parallel to vector between (from,to) */
+ from->nextcp = from->me; to->prevcp = to->me;
+return( SplineMake3(from,to));
+ }
+ /* This is the generic case, where a generic part is approximated by a cubic */
+ /* bezier spline. */
+ if ( ( ftlen == 0 ) && ( mt != mt_matrix ) )
+ mt = mt_matrix;
+ if ( mt == mt_levien ) {
+ bigreal f,m,xa,ya,xb,yb,xc,yc,xd,yd,sasa,sab;
+ int numberOfSolutions;
+ SplinePoint *frompoint,*topoint;
+ f = 0; /* area */
+ m = 0; /* first area moment about y (along x) */
+ frompoint = from;
+ if ( from->next==NULL )
+ topoint=to;
+ else
+ topoint=from->next->to;
+ for ( ; ; frompoint = topoint->next->from, topoint = topoint->next->to ) {
+ /* normalizing transformation (chord to x axis and length 1) */
+ xa = ((frompoint->me.x-from->me.x)*ftunit.x+(frompoint->me.y-from->me.y)*ftunit.y)/ftlen;
+ ya = (-(frompoint->me.x-from->me.x)*ftunit.y+(frompoint->me.y-from->me.y)*ftunit.x)/ftlen;
+ xb = ((frompoint->nextcp.x-from->me.x)*ftunit.x+(frompoint->nextcp.y-from->me.y)*ftunit.y)/ftlen;
+ yb = (-(frompoint->nextcp.x-from->me.x)*ftunit.y+(frompoint->nextcp.y-from->me.y)*ftunit.x)/ftlen;
+ xc = ((topoint->prevcp.x-from->me.x)*ftunit.x+(topoint->prevcp.y-from->me.y)*ftunit.y)/ftlen;
+ yc = (-(topoint->prevcp.x-from->me.x)*ftunit.y+(topoint->prevcp.y-from->me.y)*ftunit.x)/ftlen;
+ xd = ((topoint->me.x-from->me.x)*ftunit.x+(topoint->me.y-from->me.y)*ftunit.y)/ftlen;
+ yd = (-(topoint->me.x-from->me.x)*ftunit.y+(topoint->me.y-from->me.y)*ftunit.x)/ftlen;
+ f += ((xb-xa)*(10*ya+6*yb+3*yc+yd)+(xc-xb)*(4*ya+6*yb+6*yc+4*yd)+(xd-xc)*(ya+3*yb+6*yc+10*yd))/20;
+ m += (280*xd*xd*yd-105*xc*xd*yd-30*xb*xd*yd-5*xa*xd*yd-45*xc*xc*yd-45*xb*xc*yd-12*xa*xc*yd-18*xb*xb*yd
+ -15*xa*xb*yd-5*xa*xa*yd+105*xd*xd*yc+45*xc*xd*yc-3*xa*xd*yc-27*xb*xc*yc-18*xa*xc*yc-27*xb*xb*yc
+ -45*xa*xb*yc-30*xa*xa*yc+30*xd*xd*yb+45*xc*xd*yb+18*xb*xd*yb+3*xa*xd*yb+27*xc*xc*yb+27*xb*xc*yb
+ -45*xa*xb*yb-105*xa*xa*yb+5*xd*xd*ya+15*xc*xd*ya+12*xb*xd*ya+5*xa*xd*ya+18*xc*xc*ya+45*xb*xc*ya
+ +30*xa*xc*ya+45*xb*xb*ya+105*xa*xb*ya-280*xa*xa*ya)/840;
+ if ( topoint==to )
+ break;
+ }
+ BasePoint aunit = (BasePoint) { BPDot(ftunit, fromunit), BPCross(ftunit, fromunit) }; /* normed direction at "from" */
+ BasePoint bunit = (BasePoint) { BPDot(BPRev(ftunit), tounit),BPCross(ftunit, tounit) }; /* normed direction at "to" */
+ if ( aunit.y < 0 ) { /* normalize aunit.y to >= 0: */
+ aunit.y = -aunit.y;
+ bunit.y = -bunit.y;
+ m = -m;
+ f = -f;
+ }
+ /* calculate the Tunni point (where the tangents at "from" and "to" intersect) */
+ bigreal aMax = 100; /* maximal value that the handle a can reach up to the Tunni point, 100 is really long */
+ bigreal bMax = 100; /* maximal value that the handle b can reach up to the Tunni point, 100 is really long */
+ sab = aunit.y*bunit.x+aunit.x*bunit.y;
+ if (sab != 0) { /* if handles not parallel */
+ aMax = bunit.y/sab;
+ bMax = aunit.y/sab;
+ if ( aMax < 0 ) {
+ aMax = 100;
+ }
+ if ( bMax < 0 ) {
+ bMax = 100;
+ }
+ }
+ /* start approximation by solving the quartic equation */
+ sasa = aunit.y*aunit.y; /* reducing the multiplications */
+ Quartic quad;
+ if ( (aunit.x == -bunit.x && aunit.y == bunit.y) || (aunit.x == bunit.x && aunit.y == -bunit.y) ) { /* handles are parallel */
+ quad.a = 0;
+ quad.b = 0;
+ quad.c = -9*aunit.x*sasa;
+ quad.d = 6*aunit.y*(4*aunit.y+5*aunit.x*f);
+ quad.e = 10*((42*m-25*f)*aunit.y-25*aunit.x*f*f);
+ } else { /* generic situation */
+ quad.a = -9*aunit.x*(((2*bunit.y*bunit.x*aunit.x+aunit.y
+ *(2*bunit.x*bunit.x-1))*aunit.x-2*bunit.y*bunit.x)
+ *aunit.x-bunit.x*bunit.x*aunit.y);
+ quad.b = 12*((((bunit.x*(30*f*bunit.x-bunit.y)-15*f)
+ *aunit.x+2*aunit.y-bunit.x*aunit.y*(bunit.x+30*f*bunit.y))
+ *aunit.x+bunit.x*(bunit.y-15*f*bunit.x))
+ *aunit.x-aunit.y*bunit.x*bunit.x);
+ quad.c = 12*((((70*m+15*f)*bunit.y*bunit.y+bunit.x
+ *(9*bunit.y-70*bunit.x*m-5*bunit.x*f))
+ *aunit.x-5*aunit.y*bunit.y*(3*bunit.y-4*bunit.x
+ *(7*m+f)))*aunit.x-bunit.x*(9*bunit.y-70*bunit.x*m-5*bunit.x*f));
+ quad.d = 16*(((12*aunit.y-5*aunit.x*(42*m-17*f))*bunit.y
+ -70*bunit.x*(3*m-f)*aunit.y-75*aunit.x*bunit.x*f*f)
+ *bunit.y-75*bunit.x*bunit.x*f*f*aunit.y);
+ quad.e = 80*bunit.y*(42*bunit.y*m-25*f*(bunit.y-bunit.x*f));
+ }
+ extended solutions[4] = {-999999,-999999,-999999,-999999};
+ _QuarticSolve(&quad,solutions);
+ extended abSolutions[10][2]; /* there are at most 4+4+1+1=10 solutions of pairs of a and b (quartic=0,derivative=0,b=0.01,a=0.01) */
+ numberOfSolutions = 0;
+ extended a,b;
+ for( int i = 0; i < 4; i++ ){
+ a = solutions[i];
+ if ( a >= 0 && a < aMax ) {
+ b = (20*f-6*a*aunit.y)/(3*(2*bunit.y-a*sab));
+ if ( b >= 0 && b < bMax ) {
+ abSolutions[numberOfSolutions][0] = a;
+ abSolutions[numberOfSolutions++][1] = b;
+ }
+ }
+ }
+ /* and now again for the derivative (of m not of the upper quartic): */
+ if ( (aunit.x == -bunit.x && aunit.y == bunit.y) || (aunit.x == bunit.x && aunit.y == -bunit.y) ) { /* handles are parallel */
+ quad.a = 0;
+ quad.b = 0;
+ quad.c = 0;
+ quad.d = -3*aunit.x*aunit.y;
+ quad.e = 4*aunit.y+5*aunit.x*f;
+ } else { /* generic situation */
+ bigreal sbsb = bunit.y*bunit.y;
+ bigreal sabsab = sab*sab;
+ quad.a = -9*aunit.x*aunit.y*sabsab*sab;
+ quad.b = -3*sabsab*(9*aunit.x*aunit.y*bunit.y-(17*aunit.y
+ +30*aunit.x*f)*sab+15*bunit.x*sasa);
+ quad.c = 18*sab*bunit.y*(21*aunit.x*aunit.y*bunit.y-(17*aunit.y
+ +30*aunit.x*f)*sab+15*bunit.x*sasa);
+ quad.d = -4*(144*aunit.x*aunit.y*sbsb*bunit.y+((-78*aunit.y-
+ 135*aunit.x*f)*sab+108*bunit.x*sasa)*sbsb+(-125*f*sabsab
+ -45*bunit.x*f*aunit.y*sab)*bunit.y+150*bunit.x*f*f*sabsab);
+ quad.e = 8*bunit.y*((24*aunit.y+45*aunit.x*f)*sbsb
+ +(15*bunit.x*f*aunit.y-125*f*sab)*bunit.y+100*bunit.x*f*f*sab);
+ }
+ for( int i = 0; i < 4; i++ ) /* overwriting (reusing) */
+ solutions[i] = -999999;
+ _QuarticSolve(&quad,solutions);
+ for( int i = 0; i < 4; i++ ){
+ a = solutions[i];
+ if ( a >= 0 && a < aMax ) {
+ b = (20*f-6*a*aunit.y)/(3*(2*bunit.y-a*sab));
+ if ( b >= 0 && b < bMax ) {
+ abSolutions[numberOfSolutions][0] = a;
+ abSolutions[numberOfSolutions++][1] = b;
+ }
+ }
+ }
+ /* Add the solution of b = 0.01 (approximately 0 but above because of direction). */
+#if 0 /* those solutions lead to subpar approximations */
+ /* This solution is not part of the original algorithm by Raph Levien. */
+ a = (2000*f-6*bunit.y)/(600*aunit.y-3*sab);
+ if ( a >= 0 && a < aMax ) {
+ abSolutions[numberOfSolutions][0] = a;
+ abSolutions[numberOfSolutions++][1] = 0.01;
+ }
+ /* Add the solution of a = 0.01 (approximately 0 but above because of direction). */
+ /* This solution is not part of the original algorithm by Raph Levien. */
+ b = (2000*f-6*aunit.y)/(600*bunit.y-3*sab);
+ if ( b >= 0 && b < bMax ) {
+ abSolutions[numberOfSolutions][0] = 0.01;
+ abSolutions[numberOfSolutions++][1] = b;
+ }
+ if ( numberOfSolutions == 0) { /* add solutions that extend up to the Tunni point */
+ /* try solution with a = aMax and b area-equal*/
+ b = (20*f-6*aMax*aunit.y)/(3*(2*bunit.y-aMax*sab));
+ if ( b >= 0 && b < bMax ) {
+ abSolutions[numberOfSolutions][0] = aMax;
+ abSolutions[numberOfSolutions++][1] = b;
+ }
+ /* try solution with b = bMax and a area-equal*/
+ a = (20*f-6*bMax*bunit.y)/(3*(2*aunit.y-bMax*sab));
+ if ( a >= 0 && a < aMax ) {
+ abSolutions[numberOfSolutions][0] = a;
+ abSolutions[numberOfSolutions++][1] = bMax;
+ }
+ }
+#endif
+ if ( numberOfSolutions == 0) {
+ /* no solutions found, quit */
+return NULL;
+
+ /* solution with a = aMax and b = bMax*/
+ abSolutions[numberOfSolutions][0] = aMax;
+ abSolutions[numberOfSolutions++][1] = bMax;
+ }
+ if ( numberOfSolutions == 1) {
+ from->nextcp.x = from->me.x+ftlen*fromunit.x*abSolutions[0][0];
+ from->nextcp.y = from->me.y+ftlen*fromunit.y*abSolutions[0][0];
+ to->prevcp.x = to->me.x+ftlen*tounit.x*abSolutions[0][1];
+ to->prevcp.y = to->me.y+ftlen*tounit.y*abSolutions[0][1];
+ } else { /* compare L2 errors to choose the best solution */
+ bigreal bestError = 1e30;
+ bigreal t,error,errorsum,dist;
+ BasePoint prevcp,nextcp,coeff1,coeff2,coeff3;
+ int last_best_j;
+ for (int k=0; k<numberOfSolutions; k++) {
+ nextcp.x = from->me.x+ftlen*fromunit.x*abSolutions[k][0];
+ nextcp.y = from->me.y+ftlen*fromunit.y*abSolutions[k][0];
+ prevcp.x = to->me.x+ftlen*tounit.x*abSolutions[k][1];
+ prevcp.y = to->me.y+ftlen*tounit.y*abSolutions[k][1];
+ /* Calculate the error of the cubic bezier path from,nextcp,prevcp,to: */
+ /* In order to do that we calculate 99 points on the bezier path. */
+ coeff3.x = -from->me.x+3*nextcp.x-3*prevcp.x+to->me.x;
+ coeff3.y = -from->me.y+3*nextcp.y-3*prevcp.y+to->me.y;
+ coeff2.x = 3*from->me.x-6*nextcp.x+3*prevcp.x;
+ coeff2.y = 3*from->me.y-6*nextcp.y+3*prevcp.y;
+ coeff1.x = -3*from->me.x+3*nextcp.x;
+ coeff1.y = -3*from->me.y+3*nextcp.y;
+ BasePoint approx[99];
+ for (int i=0; i<99; i++) {
+ t = (i+1)/100.0;
+ approx[i].x = from->me.x+t*(coeff1.x+t*(coeff2.x+t*coeff3.x));
+ approx[i].y = from->me.y+t*(coeff1.y+t*(coeff2.y+t*coeff3.y));
+ }
+ /* Now we calculate the error by determining the minimal quadratic distance to the mid points. */
+ errorsum = 0.0;
+ last_best_j = 0;
+ for (int i=0; i<cnt; i++) { /* Going through the mid points */
+ error = 1e30;
+ /* For this mid point, find the distance to the closest one of the */
+ /* 99 points on the approximate cubic bezier. */
+ /* To not favour approximations which trace the original multiple times */
+ /* by going back and forth, only consider monotonic mappings. */
+ /* I.e., start from the point that was closest to the previous mid point: */
+ for (int j=last_best_j; j<99; j++) {
+ dist = (mid[i].p.x-approx[j].x)*(mid[i].p.x-approx[j].x)
+ +(mid[i].p.y-approx[j].y)*(mid[i].p.y-approx[j].y);
+ if (dist < error) {
+ error = dist;
+ last_best_j = j;
+ }
+ }
+ errorsum += error;
+ if (errorsum > bestError)
+ break;
+ }
+ if (errorsum < bestError) {
+ bestError = errorsum;
+ from->nextcp = nextcp;
+ to->prevcp = prevcp;
+ }
+ }
+ }
+ return( SplineMake3(from,to));
+ } else if ( mt == mt_bruteforce ) {
+ bigreal best_error = 1e30;
+ bigreal t,error,errorsum,dist;
+ BasePoint prevcp,coeff1,coeff2,coeff3;
+ bigreal best_fromhandle = 0.0;
+ bigreal best_tohandle = 0.0;
+ BasePoint approx[99]; /* The 99 points on the approximate cubic bezier */
+ /* We make 2 runs: The first run to narrow the variation range, the second run to finetune */
+ /* The optimal length of the two handles are determined by brute force. */
+ for (int run=0; run<2; ++run) {
+ for (int fromhandle=((run==0)?1:-29); fromhandle<=((run==0)?60:29); ++fromhandle) {
+ for (int tohandle=((run==0)?1:-29); tohandle<=((run==0)?60:29); ++tohandle) {
+ nextcp.x = from->me.x+ftlen*fromunit.x*( (run==0)?fromhandle:best_fromhandle+fromhandle/30.0 )/60.0;
+ nextcp.y = from->me.y+ftlen*fromunit.y*( (run==0)?fromhandle:best_fromhandle+fromhandle/30.0 )/60.0;
+ prevcp.x = to->me.x+ftlen*tounit.x*( (run==0)?tohandle:best_tohandle+tohandle/30.0 )/60.0;
+ prevcp.y = to->me.y+ftlen*tounit.y*( (run==0)?tohandle:best_tohandle+tohandle/30.0 )/60.0;
+ /* Calculate the error of the cubic bezier path from,nextcp,prevcp,to: */
+ /* In order to do that we calculate 99 points on the bezier path. */
+ coeff3.x = -from->me.x+3*nextcp.x-3*prevcp.x+to->me.x;
+ coeff3.y = -from->me.y+3*nextcp.y-3*prevcp.y+to->me.y;
+ coeff2.x = 3*from->me.x-6*nextcp.x+3*prevcp.x;
+ coeff2.y = 3*from->me.y-6*nextcp.y+3*prevcp.y;
+ coeff1.x = -3*from->me.x+3*nextcp.x;
+ coeff1.y = -3*from->me.y+3*nextcp.y;
+ for (int i=0; i<99; ++i) {
+ t = (i+1)/100.0;
+ approx[i].x = from->me.x+t*(coeff1.x+t*(coeff2.x+t*coeff3.x));
+ approx[i].y = from->me.y+t*(coeff1.y+t*(coeff2.y+t*coeff3.y));
+ }
+ /* Now we calculate the error by determining the minimal quadratic distance to the mid points. */
+ errorsum = 0.0;
+ for (int i=0; i<cnt; ++i) { /* Going through the mid points */
+ error = (mid[i].p.x-approx[0].x)*(mid[i].p.x-approx[0].x)
+ +(mid[i].p.y-approx[0].y)*(mid[i].p.y-approx[0].y);
+ /* Above we have just initialized the error and */
+ /* now we are going through the remaining 98 of */
+ /* 99 points on the approximate cubic bezier: */
+ for (int j=1; j<99; ++j) {
+ dist = (mid[i].p.x-approx[j].x)*(mid[i].p.x-approx[j].x)
+ +(mid[i].p.y-approx[j].y)*(mid[i].p.y-approx[j].y);
+ if (dist < error)
+ error = dist;
+ }
+ errorsum += error;
+ if (errorsum > best_error)
+ break;
+ }
+ if (errorsum < best_error) {
+ best_error = errorsum;
+ if (run == 0) {
+ best_fromhandle = fromhandle;
+ best_tohandle = tohandle;
+ }
+ from->nextcp = nextcp;
+ to->prevcp = prevcp;
+ }
+ }
+ }
+ }
+ return( SplineMake3(from,to));
+ }
+ else { /* mergetype mt_matrix (original algorithm) */
+ pt_pf_x = to->me.x - from->me.x;
+ pt_pf_y = to->me.y - from->me.y;
+ consts[0] = consts[1] = rt_terms[0] = rt_terms[1] = rf_terms[0] = rf_terms[1] = 0;
+ for ( i=0; i<cnt; ++i ) {
+ bigreal t = mid[i].t, t2 = t*t, t3=t2*t;
+ bigreal factor_from = t-2*t2+t3;
+ bigreal factor_to = t2-t3;
+ bigreal const_x = from->me.x-mid[i].p.x + 3*pt_pf_x*t2 - 2*pt_pf_x*t3;
+ bigreal const_y = from->me.y-mid[i].p.y + 3*pt_pf_y*t2 - 2*pt_pf_y*t3;
+ bigreal temp1 = 3*(t-2*t2+t3);
+ bigreal rf_term_x = temp1*fromunit.x;
+ bigreal rf_term_y = temp1*fromunit.y;
+ bigreal temp2 = 3*(t2-t3);
+ bigreal rt_term_x = -temp2*tounit.x;
+ bigreal rt_term_y = -temp2*tounit.y;
+
+ consts[0] += factor_from*( fromunit.x*const_x + fromunit.y*const_y );
+ consts[1] += factor_to *( -tounit.x*const_x + -tounit.y*const_y);
+ rf_terms[0] += factor_from*( fromunit.x*rf_term_x + fromunit.y*rf_term_y);
+ rf_terms[1] += factor_to*( -tounit.x*rf_term_x + -tounit.y*rf_term_y);
+ rt_terms[0] += factor_from*( fromunit.x*rt_term_x + fromunit.y*rt_term_y);
+ rt_terms[1] += factor_to*( -tounit.x*rt_term_x + -tounit.y*rt_term_y);
+ }
+
+ /* I've only seen singular matrices (determinant==0) when cnt==1 */
+ /* but even with cnt==1 the determinant is usually non-0 (16 times out of 17)*/
+ determinant = (rt_terms[0]*rf_terms[1]-rt_terms[1]*rf_terms[0]);
+ if ( determinant!=0 ) {
+ bigreal rt, rf;
+ rt = (consts[1]*rf_terms[0]-consts[0]*rf_terms[1])/determinant;
+ if ( rf_terms[0]!=0 )
+ rf = -(consts[0]+rt*rt_terms[0])/rf_terms[0];
+ else /* if ( rf_terms[1]!=0 ) This can't happen, otherwise the determinant would be 0 */
+ rf = -(consts[1]+rt*rt_terms[1])/rf_terms[1];
+ /* If we get bad values (ones that point diametrically opposed to what*/
+ /* we need), then fix that factor at 0, and see what we get for the */
+ /* other */
+ if ( rf>=0 && rt>0 && rf_terms[0]!=0 &&
+ (rf = -consts[0]/rf_terms[0])>0 ) {
+ rt = 0;
+ } else if ( rf<0 && rt<=0 && rt_terms[1]!=0 &&
+ (rt = -consts[1]/rt_terms[1])<0 ) {
+ rf = 0;
+ }
+ if ( rt<=0 && rf>=0 ) {
+ from->nextcp.x = from->me.x + rf*fromunit.x;
+ from->nextcp.y = from->me.y + rf*fromunit.y;
+ to->prevcp.x = to->me.x - rt*tounit.x;
+ to->prevcp.y = to->me.y - rt*tounit.y;
+return( SplineMake3(from,to));
+ }
+ }
+
+ trylen = (to->me.x-from->me.x)*fromunit.x + (to->me.y-from->me.y)*fromunit.y;
+ if ( trylen>flen ) flen = trylen;
+
+ trylen = (from->me.x-to->me.x)*tounit.x + (from->me.y-to->me.y)*tounit.y;
+ if ( trylen>tlen ) tlen = trylen;
+
+ for ( i=0; i<cnt; ++i ) {
+ trylen = (mid[i].p.x-from->me.x)*fromunit.x + (mid[i].p.y-from->me.y)*fromunit.y;
+ if ( trylen>flen ) flen = trylen;
+ trylen = (mid[i].p.x-to->me.x)*tounit.x + (mid[i].p.y-to->me.y)*tounit.y;
+ if ( trylen>tlen ) tlen = trylen;
+ }
+
+ fdotft = fromunit.x*ftunit.x + fromunit.y*ftunit.y;
+ fmax = fdotft>0 ? ftlen/fdotft : 1e10;
+ tdotft = -tounit.x*ftunit.x - tounit.y*ftunit.y;
+ tmax = tdotft>0 ? ftlen/tdotft : 1e10;
+ /* At fmax, tmax the control points will stretch beyond the other endpoint*/
+ /* when projected along the line between the two endpoints */
+
+ db.base = from->me;
+ db.unit = ftunit;
+ db.len = ftlen;
+ ApproxBounds(&b,mid,cnt,&db);
+
+ for ( k=0; k<TRY_CNT; ++k ) {
+ bestdiff[k] = 1e20;
+ besti[k] = -1; bestj[k] = -1;
+ }
+ fdiff = flen/DECIMATION;
+ tdiff = tlen/DECIMATION;
+ from->nextcp = from->me;
+ memset(&temp,0,sizeof(Spline));
+ temp.from = from; temp.to = to;
+ for ( i=1; i<DECIMATION; ++i ) {
+ from->nextcp.x += fdiff*fromunit.x; from->nextcp.y += fdiff*fromunit.y;
+ to->prevcp = to->me;
+ for ( j=1; j<DECIMATION; ++j ) {
+ to->prevcp.x += tdiff*tounit.x; to->prevcp.y += tdiff*tounit.y;
+ SplineRefigure(&temp);
+ curdiff = SigmaDeltas(&temp,mid,cnt,&b,&db);
+ for ( k=0; k<TRY_CNT; ++k ) {
+ if ( curdiff<bestdiff[k] ) {
+ for ( l=TRY_CNT-1; l>k; --l ) {
+ bestdiff[l] = bestdiff[l-1];
+ besti[l] = besti[l-1];
+ bestj[l] = bestj[l-1];
+ }
+ bestdiff[k] = curdiff;
+ besti[k] = i; bestj[k]=j;
+ break;
+ }
+ }
+ }
+ }
+
+ finaldiff = 1e20;
+ offn_ = offp_ = -1;
+ spline = SplineMake(from,to,false);
+ for ( k=-1; k<TRY_CNT; ++k ) {
+ if ( k<0 ) {
+ BasePoint nextcp, prevcp;
+ bigreal temp1, temp2;
+ int ret = _ApproximateSplineFromPoints(from,to,mid,cnt,&nextcp,&prevcp,false);
+ /* sometimes least squares gives us the right answer */
+ if ( !(ret&1) || !(ret&2))
+ continue;
+ temp1 = (prevcp.x-to->me.x)*tounit.x + (prevcp.y-to->me.y)*tounit.y;
+ temp2 = (nextcp.x-from->me.x)*fromunit.x + (nextcp.y-from->me.y)*fromunit.y;
+ if ( temp1<=0 || temp2<=0 ) /* A nice solution... but the control points are diametrically opposed to what they should be */
+ continue;
+ tlen = temp1; flen = temp2;
+ } else {
+ if ( bestj[k]<0 || besti[k]<0 )
+ continue;
+ tlen = bestj[k]*tdiff; flen = besti[k]*fdiff;
+ }
+ to->prevcp.x = to->me.x + tlen*tounit.x; to->prevcp.y = to->me.y + tlen*tounit.y;
+ from->nextcp.x = from->me.x + flen*fromunit.x; from->nextcp.y = from->me.y + flen*fromunit.y;
+ SplineRefigure(spline);
+
+ bettern = betterp = false;
+ incrn = tdiff/2.0; incrp = fdiff/2.0;
+ offn = flen; offp = tlen;
+ nocnt = 0;
+ curdiff = SigmaDeltas(spline,mid,cnt,&b,&db);
+ totcnt = 0;
+ for (;;) {
+ bigreal fadiff, fsdiff;
+ bigreal tadiff, tsdiff;
+
+ from->nextcp.x = from->me.x + (offn+incrn)*fromunit.x; from->nextcp.y = from->me.y + (offn+incrn)*fromunit.y;
+ to->prevcp.x = to->me.x + offp*tounit.x; to->prevcp.y = to->me.y + offp*tounit.y;
+ SplineRefigure(spline);
+ fadiff = SigmaDeltas(spline,mid,cnt,&b,&db);
+ from->nextcp.x = from->me.x + (offn-incrn)*fromunit.x; from->nextcp.y = from->me.y + (offn-incrn)*fromunit.y;
+ SplineRefigure(spline);
+ fsdiff = SigmaDeltas(spline,mid,cnt,&b,&db);
+ from->nextcp.x = from->me.x + offn*fromunit.x; from->nextcp.y = from->me.y + offn*fromunit.y;
+ if ( offn-incrn<=0 )
+ fsdiff += 1e10;
+
+ to->prevcp.x = to->me.x + (offp+incrp)*tounit.x; to->prevcp.y = to->me.y + (offp+incrp)*tounit.y;
+ SplineRefigure(spline);
+ tadiff = SigmaDeltas(spline,mid,cnt,&b,&db);
+ to->prevcp.x = to->me.x + (offp-incrp)*tounit.x; to->prevcp.y = to->me.y + (offp-incrp)*tounit.y;
+ SplineRefigure(spline);
+ tsdiff = SigmaDeltas(spline,mid,cnt,&b,&db);
+ to->prevcp.x = to->me.x + offp*tounit.x; to->prevcp.y = to->me.y + offp*tounit.y;
+ if ( offp-incrp<=0 )
+ tsdiff += 1e10;
+
+ if ( offn>=incrn && fsdiff<curdiff &&
+ (fsdiff<fadiff && fsdiff<tsdiff && fsdiff<tadiff)) {
+ offn -= incrn;
+ if ( bettern>0 )
+ incrn /= 2;
+ bettern = -1;
+ nocnt = 0;
+ curdiff = fsdiff;
+ } else if ( offn+incrn<fmax && fadiff<curdiff &&
+ (fadiff<=fsdiff && fadiff<tsdiff && fadiff<tadiff)) {
+ offn += incrn;
+ if ( bettern<0 )
+ incrn /= 2;
+ bettern = 1;
+ nocnt = 0;
+ curdiff = fadiff;
+ } else if ( offp>=incrp && tsdiff<curdiff &&
+ (tsdiff<=fsdiff && tsdiff<=fadiff && tsdiff<tadiff)) {
+ offp -= incrp;
+ if ( betterp>0 )
+ incrp /= 2;
+ betterp = -1;
+ nocnt = 0;
+ curdiff = tsdiff;
+ } else if ( offp+incrp<tmax && tadiff<curdiff &&
+ (tadiff<=fsdiff && tadiff<=fadiff && tadiff<=tsdiff)) {
+ offp += incrp;
+ if ( betterp<0 )
+ incrp /= 2;
+ betterp = 1;
+ nocnt = 0;
+ curdiff = tadiff;
+ } else {
+ if ( ++nocnt > 6 )
+ break;
+ incrn /= 2;
+ incrp /= 2;
+ }
+ if ( curdiff<1 )
+ break;
+ if ( incrp<tdiff/1024 || incrn<fdiff/1024 )
+ break;
+ if ( ++totcnt>200 )
+ break;
+ if ( offn<0 || offp<0 ) {
+ IError("Approximation got inverse control points");
+ break;
+ }
+ }
+ if ( curdiff<finaldiff ) {
+ finaldiff = curdiff;
+ offn_ = offn;
+ offp_ = offp;
+ }
+ }
+
+ to->noprevcp = offp_==0;
+ from->nonextcp = offn_==0;
+ to->prevcp.x = to->me.x + offp_*tounit.x; to->prevcp.y = to->me.y + offp_*tounit.y;
+ from->nextcp.x = from->me.x + offn_*fromunit.x; from->nextcp.y = from->me.y + offn_*fromunit.y;
+ /* I used to check for a spline very close to linear (and if so, make it */
+ /* linear). But in when stroking a path with an elliptical pen we transform*/
+ /* the coordinate system and our normal definitions of "close to linear" */
+ /* don't apply */
+ /*TestForLinear(from,to);*/
+ SplineRefigure(spline);
+
+return( spline );
+ }
+}
+#undef TRY_CNT
+#undef DECIMATION
+
+SplinePoint *_ApproximateSplineSetFromGen(SplinePoint *from, SplinePoint *to,
+ bigreal start_t, bigreal end_t,
+ bigreal toler, int toler_is_sumsq,
+ GenPointsP genp, void *tok,
+ int order2, int depth) {
+ int cnt, i, maxerri=0, created = false;
+ bigreal errsum=0, maxerr=0, d, mid_t;
+ FitPoint *fp;
+ SplinePoint *mid, *r;
+
+ cnt = (*genp)(tok, start_t, end_t, &fp);
+ if ( cnt < 2 )
+ return NULL;
+
+ // Rescale zero to one
+ for ( i=1; i<(cnt-1); ++i )
+ fp[i].t = (fp[i].t-fp[0].t)/(fp[cnt-1].t-fp[0].t);
+ fp[0].t = 0.0;
+ fp[cnt-1].t = 1.0;
+
+ from->nextcp.x = from->me.x + fp[0].ut.x;
+ from->nextcp.y = from->me.y + fp[0].ut.y;
+ from->nonextcp = false;
+ if ( to!=NULL )
+ to->me = fp[cnt-1].p;
+ else {
+ to = SplinePointCreate(fp[cnt-1].p.x, fp[cnt-1].p.y);
+ created = true;
+ }
+ to->prevcp.x = to->me.x - fp[cnt-1].ut.x;
+ to->prevcp.y = to->me.y - fp[cnt-1].ut.y;
+ to->noprevcp = false;
+ ApproximateSplineFromPointsSlopes(from,to,fp+1,cnt-2,order2,mt_matrix);
+
+ for ( i=0; i<cnt; ++i ) {
+ d = SplineMinDistanceToPoint(from->next, &fp[i].p);
+ errsum += d*d;
+ if ( d>maxerr ) {
+ maxerr = d;
+ maxerri = i;
+ }
+ }
+ // printf(" Error sum %lf, max error %lf at depth %d\n", errsum, maxerr, depth);
+
+ if ( (toler_is_sumsq ? errsum : maxerr) > toler && depth < 6 ) {
+ mid_t = fp[maxerri].t * (end_t-start_t) + start_t;
+ free(fp);
+ SplineFree(from->next);
+ from->next = NULL;
+ to->prev = NULL;
+ mid = _ApproximateSplineSetFromGen(from, NULL, start_t, mid_t, toler,
+ toler_is_sumsq, genp, tok, order2,
+ depth+1);
+ if ( mid ) {
+ r = _ApproximateSplineSetFromGen(mid, to, mid_t, end_t, toler,
+ toler_is_sumsq, genp, tok,
+ order2, depth+1);
+ if ( r )
+ return r;
+ else {
+ if ( created )
+ SplinePointFree(to);
+ else
+ to->prev = NULL;
+ SplinePointFree(mid);
+ SplineFree(from->next);
+ from->next = NULL;
+ return NULL;
+ }
+ } else {
+ if ( created )
+ SplinePointFree(to);
+ return NULL;
+ }
+ } else if ( (toler_is_sumsq ? errsum : maxerr) > toler ) {
+ TRACE("%s %lf exceeds %lf at maximum depth %d\n",
+ toler_is_sumsq ? "Sum of squared errors" : "Maximum error length",
+ toler_is_sumsq ? errsum : maxerr, toler, depth);
+ }
+ free(fp);
+ return to;
+}
+
+SplinePoint *ApproximateSplineSetFromGen(SplinePoint *from, SplinePoint *to,
+ bigreal start_t, bigreal end_t,
+ bigreal toler, int toler_is_sumsq,
+ GenPointsP genp, void *tok,
+ int order2) {
+ return _ApproximateSplineSetFromGen(from, to, start_t, end_t, toler,
+ toler_is_sumsq, genp, tok, order2, 0);
+}
diff --git a/src/path/splinefit/splinefit.h b/src/path/splinefit/splinefit.h
new file mode 100644
index 0000000..22e21fc
--- /dev/null
+++ b/src/path/splinefit/splinefit.h
@@ -0,0 +1,78 @@
+// SPDX-License-Identifier: GPL-2.0-or-later
+/* Copyright (C) 2000-2012 by George Williams, 2019 by Skef Iterum, 2021 by Linus Romer */
+/*
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions are met:
+
+ * Redistributions of source code must retain the above copyright notice, this
+ * list of conditions and the following disclaimer.
+
+ * Redistributions in binary form must reproduce the above copyright notice,
+ * this list of conditions and the following disclaimer in the documentation
+ * and/or other materials provided with the distribution.
+
+ * The name of the author may not be used to endorse or promote products
+ * derived from this software without specific prior written permission.
+
+ * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED
+ * WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
+ * MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO
+ * EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+ * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+ * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS;
+ * OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,
+ * WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR
+ * OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF
+ * ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+ */
+
+#ifndef FONTFORGE_SPLINEFIT_H
+#define FONTFORGE_SPLINEFIT_H
+
+// #include <fontforge-config.h>
+#include "splinefont.h"
+
+typedef struct fitpoint {
+ BasePoint p;
+ BasePoint ut;
+ bigreal t;
+} FitPoint;
+
+#define FITPOINT_EMPTY { {0.0, 0.0}, {0.0, 0.0}, 0.0 }
+
+enum mergetype { mt_matrix, mt_levien, mt_bruteforce };
+
+extern Spline *ApproximateSplineFromPoints(SplinePoint *from, SplinePoint *to,
+ FitPoint *mid, int cnt, int order2);
+extern Spline *ApproximateSplineFromPointsSlopes(SplinePoint *from, SplinePoint *to,
+ FitPoint *mid, int cnt, int order2, enum mergetype mt);
+
+/* ApproximateSplineSetFromGen() fits a one or more splines to data
+ * generated by calls to genp, within the tolerance toler. The data
+ * are treated as noiseless although some noise may be harmless. When
+ * an interval cannot be fit with a single spline it is divided at
+ * the point of largest error, where the position and slope are fixed
+ * according to the FitPoint entries.
+ *
+ * A GenPointsP function should allocate a free-able array of FitPoints,
+ * set *fpp to point to it, and return the number of points allocated.
+ * The points should correspond to values chosen between the interval
+ * t_start and t_end, but just what this interval corresponds to depends
+ * on vinfo and is therefore opaque to the tracing system. It is up to the
+ * implementation how many points to return on the interval and how they are
+ * "t-spaced".
+ *
+ * The function can also return 0 points, in which case *fpp should be set
+ * to NULL. This will cancel the tracing. This semantic is useful for
+ * communicating information found during tracing back to the caller
+ * of ApproximateSplineSetFromGen(), presumably via the vinfo structure,
+ * so that it can retrace with a different choice of interval.
+ */
+typedef int (*GenPointsP)(void *vinfo, bigreal t_start, bigreal t_end, FitPoint **fpp);
+
+extern SplinePoint *ApproximateSplineSetFromGen(SplinePoint *from, SplinePoint *to,
+ bigreal start_t, bigreal end_t,
+ bigreal toler, int toler_is_sumsq,
+ GenPointsP genp, void *tok, int order2);
+
+#endif /* FONTFORGE_SPLINEFIT_H */
diff --git a/src/path/splinefit/splinefont.c b/src/path/splinefit/splinefont.c
new file mode 100644
index 0000000..4681628
--- /dev/null
+++ b/src/path/splinefit/splinefont.c
@@ -0,0 +1,1174 @@
+// SPDX-License-Identifier: GPL-2.0-or-later
+
+#include <math.h>
+#include <stdbool.h>
+#include <stdint.h>
+#include "splinefont.h"
+#include "splinefit.h"
+
+#define FONTFORGE_CONFIG_USE_DOUBLE 1
+
+bigreal BPDot(BasePoint v1, BasePoint v2) {
+ return v1.x * v2.x + v1.y * v2.y;
+}
+
+bigreal BPCross(BasePoint v1, BasePoint v2) {
+ return v1.x * v2.y - v1.y * v2.x;
+}
+
+BasePoint BPRev(BasePoint v) {
+ return (BasePoint) { -v.x, -v.y };
+}
+
+int RealWithin(real a,real b,real fudge) {
+ return( b>=a-fudge && b<=a+fudge );
+}
+
+BOOL RealNear(real a,real b) {
+ real d = a-b;
+#ifdef FONTFORGE_CONFIG_USE_DOUBLE
+ // These tighter equals-zero tests are retained for code tuned when
+ // passing zero as a constant
+ if ( a==0 )
+ return b>-1e-8 && b<1e-8;
+ if ( b==0 )
+ return a>-1e-8 && a<1e-8;
+
+ return d>-1e-6 && d<1e-6;
+#else /* For floats */
+ return d>-1e-5 && d<1e-5
+#endif
+}
+
+int RealApprox(real a,real b) {
+
+ if ( a==0 ) {
+ if ( b<.0001 && b>-.0001 )
+return( true );
+ } else if ( b==0 ) {
+ if ( a<.0001 && a>-.0001 )
+return( true );
+ } else {
+ a /= b;
+ if ( a>=.95 && a<=1.05 )
+return( true );
+ }
+return( false );
+}
+
+void LineListFree(LineList *ll) {
+ LineList *next;
+
+ while ( ll!=NULL ) {
+ next = ll->next;
+ chunkfree(ll,sizeof(LineList));
+ ll = next;
+ }
+}
+
+void LinearApproxFree(LinearApprox *la) {
+ LinearApprox *next;
+
+ while ( la!=NULL ) {
+ next = la->next;
+ LineListFree(la->lines);
+ chunkfree(la,sizeof(LinearApprox));
+ la = next;
+ }
+}
+
+void SplineFree(Spline *spline) {
+ LinearApproxFree(spline->approx);
+ chunkfree(spline,sizeof(Spline));
+}
+
+SplinePoint *SplinePointCreate(real x, real y) {
+ SplinePoint *sp;
+ if ( (sp=chunkalloc(sizeof(SplinePoint)))!=NULL ) {
+ sp->me.x = x; sp->me.y = y;
+ sp->nextcp = sp->prevcp = sp->me;
+ sp->nonextcp = sp->noprevcp = true;
+ sp->nextcpdef = sp->prevcpdef = false;
+ sp->ttfindex = sp->nextcpindex = 0xfffe;
+ sp->name = NULL;
+ }
+ return( sp );
+}
+
+void SplinePointsFree(SplinePointList *spl) {
+ Spline *first, *spline, *next;
+ int nonext;
+
+ if ( spl==NULL )
+ return;
+ if ( spl->first!=NULL ) {
+ nonext = spl->first->next==NULL; // If there is no spline, we set a flag.
+ first = NULL;
+ // We start on the first spline if it exists.
+ for ( spline = spl->first->next; spline!=NULL && spline!=first; spline = next ) {
+ next = spline->to->next; // Cache the location of the next spline.
+ SplinePointFree(spline->to); // Free the destination point.
+ SplineFree(spline); // Free the spline.
+ if ( first==NULL ) first = spline; // We want to avoid repeating the circuit.
+ }
+ // If the path is open or has no splines, free the starting point.
+ if ( spl->last!=spl->first || nonext )
+ SplinePointFree(spl->first);
+ }
+}
+
+void SplinePointListFree(SplinePointList *spl) {
+
+ if ( spl==NULL ) return;
+ SplinePointsFree(spl);
+ // free(spl->spiros);
+ free(spl->contour_name);
+ chunkfree(spl,sizeof(SplinePointList));
+}
+
+void SplineRefigure2(Spline *spline) {
+ SplinePoint *from = spline->from, *to = spline->to;
+ Spline1D *xsp = &spline->splines[0], *ysp = &spline->splines[1];
+ Spline old;
+
+#ifdef DEBUG
+ if ( RealNear(from->me.x,to->me.x) && RealNear(from->me.y,to->me.y))
+ IError("Zero length spline created");
+#endif
+ if ( spline->acceptableextrema )
+ old = *spline;
+
+ if ( ( from->nextcp.x==from->me.x && from->nextcp.y==from->me.y && from->nextcpindex>=0xfffe )
+ || ( to->prevcp.x==to->me.x && to->prevcp.y==to->me.y && from->nextcpindex>=0xfffe ) ) {
+ from->nonextcp = to->noprevcp = true;
+ from->nextcp = from->me;
+ to->prevcp = to->me;
+ } else {
+ from->nonextcp = to->noprevcp = false;
+ if ( from->nextcp.x==from->me.x && from->nextcp.y==from->me.y )
+ to->prevcp = from->me;
+ else if ( to->prevcp.x==to->me.x && to->prevcp.y==to->me.y )
+ from->nextcp = to->me;
+ }
+
+ if ( from->nonextcp && to->noprevcp )
+ /* Ok */;
+ else if ( from->nextcp.x!=to->prevcp.x || from->nextcp.y!=to->prevcp.y ) {
+ if ( RealNear(from->nextcp.x,to->prevcp.x) &&
+ RealNear(from->nextcp.y,to->prevcp.y)) {
+ from->nextcp.x = to->prevcp.x = (from->nextcp.x+to->prevcp.x)/2;
+ from->nextcp.y = to->prevcp.y = (from->nextcp.y+to->prevcp.y)/2;
+ } else {
+ IError("Invalid 2nd order spline in SplineRefigure2" );
+#ifndef GWW_TEST
+ /* I don't want these to go away when I'm debugging. I want to */
+ /* know how I got them */
+ from->nextcp.x = to->prevcp.x = (from->nextcp.x+to->prevcp.x)/2;
+ from->nextcp.y = to->prevcp.y = (from->nextcp.y+to->prevcp.y)/2;
+#endif
+ }
+ }
+
+ xsp->d = from->me.x; ysp->d = from->me.y;
+ if ( from->nonextcp && to->noprevcp ) {
+ spline->islinear = true;
+ xsp->c = to->me.x-from->me.x;
+ ysp->c = to->me.y-from->me.y;
+ xsp->a = xsp->b = 0;
+ ysp->a = ysp->b = 0;
+ } else {
+ /* from p. 393 (Operator Details, curveto) PostScript Lang. Ref. Man. (Red book) */
+ xsp->c = 2*(from->nextcp.x-from->me.x);
+ ysp->c = 2*(from->nextcp.y-from->me.y);
+ xsp->b = to->me.x-from->me.x-xsp->c;
+ ysp->b = to->me.y-from->me.y-ysp->c;
+ xsp->a = 0;
+ ysp->a = 0;
+ if ( RealNear(xsp->c,0)) xsp->c=0;
+ if ( RealNear(ysp->c,0)) ysp->c=0;
+ if ( RealNear(xsp->b,0)) xsp->b=0;
+ if ( RealNear(ysp->b,0)) ysp->b=0;
+ spline->islinear = false;
+ if ( ysp->b==0 && xsp->b==0 )
+ spline->islinear = true; /* This seems extremely unlikely... */
+ if ( from->nextcpselected || to->prevcpselected ) {
+ // The convention for tracking selection of quadratic control
+ // points is to use nextcpselected except at the tail of the
+ // list, where it's prevcpselected on the first point.
+ from->nextcpselected = true;
+ to->prevcpselected = false;
+ }
+ }
+ if ( isnan(ysp->b) || isnan(xsp->b) )
+ IError("NaN value in spline creation");
+ LinearApproxFree(spline->approx);
+ spline->approx = NULL;
+ spline->knowncurved = false;
+ spline->knownlinear = spline->islinear;
+ SplineIsLinear(spline);
+ spline->isquadratic = !spline->knownlinear;
+ spline->order2 = true;
+
+ if ( spline->acceptableextrema ) {
+ /* I don't check "d", because changes to that reflect simple */
+ /* translations which will not affect the shape of the spline */
+ /* (I don't check "a" because it is always 0 in a quadratic spline) */
+ if ( !RealNear(old.splines[0].b,spline->splines[0].b) ||
+ !RealNear(old.splines[0].c,spline->splines[0].c) ||
+ !RealNear(old.splines[1].b,spline->splines[1].b) ||
+ !RealNear(old.splines[1].c,spline->splines[1].c) )
+ spline->acceptableextrema = false;
+ }
+}
+
+Spline *SplineMake(SplinePoint *from, SplinePoint *to, int order2) {
+ if (order2 > 0)
+return( SplineMake2(from,to));
+ else
+return( SplineMake3(from,to));
+}
+
+Spline *SplineMake2(SplinePoint *from, SplinePoint *to) {
+ Spline *spline = chunkalloc(sizeof(Spline));
+
+ spline->from = from; spline->to = to;
+ from->next = to->prev = spline;
+ spline->order2 = true;
+ SplineRefigure2(spline);
+return( spline );
+}
+
+Spline *SplineMake3(SplinePoint *from, SplinePoint *to) {
+ Spline *spline = chunkalloc(sizeof(Spline));
+
+ spline->from = from; spline->to = to;
+ from->next = to->prev = spline;
+ SplineRefigure3(spline);
+return( spline );
+}
+
+void SplinePointFree(SplinePoint *sp) {
+ // chunkfree(sp->hintmask,sizeof(HintMask));
+ free(sp->name);
+ chunkfree(sp,sizeof(SplinePoint));
+}
+
+void SplineRefigure(Spline *spline) {
+ if ( spline==NULL )
+return;
+ if ( spline->order2 )
+ SplineRefigure2(spline);
+ else
+ SplineRefigure3(spline);
+}
+
+# define RE_NearZero .00000001
+# define RE_Factor (1024.0*1024.0*1024.0*1024.0*1024.0*2.0) /* 52 bits => divide by 2^51 */
+
+int Within16RoundingErrors(bigreal v1, bigreal v2) {
+ bigreal temp=v1*v2;
+ bigreal re;
+
+ if ( temp<0 ) /* Ok, if the two values are on different sides of 0 there */
+return( false ); /* is no way they can be within a rounding error of each other */
+ else if ( temp==0 ) {
+ if ( v1==0 )
+return( v2<RE_NearZero && v2>-RE_NearZero );
+ else
+return( v1<RE_NearZero && v1>-RE_NearZero );
+ } else if ( v1>0 ) {
+ if ( v1>v2 ) { /* Rounding error from the biggest absolute value */
+ re = v1/ (RE_Factor/16);
+return( v1-v2 < re );
+ } else {
+ re = v2/ (RE_Factor/16);
+return( v2-v1 < re );
+ }
+ } else {
+ if ( v1<v2 ) {
+ re = v1/ (RE_Factor/16); /* This will be a negative number */
+return( v1-v2 > re );
+ } else {
+ re = v2/ (RE_Factor/16);
+return( v2-v1 > re );
+ }
+ }
+}
+
+/* An IEEE double has 52 bits of precision. So one unit of rounding error will be */
+/* the number divided by 2^51 */
+# define D_RE_Factor (1024.0*1024.0*1024.0*1024.0*1024.0*2.0)
+/* But that's not going to work near 0, so, since the t values we care about */
+/* are [0,1], let's use 1.0/D_RE_Factor */
+
+double CheckExtremaForSingleBitErrors(const Spline1D *sp, double t, double othert) {
+ double u1, um1;
+ double slope, slope1, slopem1;
+ int err;
+ double diff, factor;
+
+ if ( t<0 || t>1 )
+return( t );
+
+ factor = t*0x40000/D_RE_Factor;
+ if ( (diff = t-othert)<0 ) diff= -diff;
+ if ( factor>diff/4 && diff!=0 ) /* This little check is to insure we don't skip beyond the well of this extremum into the next */
+ factor = diff/4;
+
+ slope = (3*(double) sp->a*t+2*sp->b)*t+sp->c;
+ if ( slope<0 ) slope = -slope;
+
+ for ( err = 0x40000; err!=0; err>>=1 ) {
+ u1 = t+factor;
+ slope1 = (3*(double) sp->a*u1+2*sp->b)*u1+sp->c;
+ if ( slope1<0 ) slope1 = -slope1;
+
+ um1 = t-factor;
+ slopem1 = (3*(double) sp->a*um1+2*sp->b)*um1+sp->c;
+ if ( slopem1<0 ) slopem1 = -slopem1;
+
+ if ( slope1<slope && slope1<=slopem1 && u1<=1.0 ) {
+ t = u1;
+ } else if ( slopem1<slope && slopem1<=slope1 && um1>=0.0 ) {
+ t = um1;
+ }
+ factor /= 2.0;
+ }
+ /* that seems as good as it gets */
+
+return( t );
+}
+
+void SplineFindExtrema(const Spline1D *sp, extended *_t1, extended *_t2 ) {
+ extended t1= -1, t2= -1;
+ extended b2_fourac;
+
+ /* Find the extreme points on the curve */
+ /* Set to -1 if there are none or if they are outside the range [0,1] */
+ /* Order them so that t1<t2 */
+ /* If only one valid extremum it will be t1 */
+ /* (Does not check the end points unless they have derivative==0) */
+ /* (Does not check to see if d/dt==0 points are inflection points (rather than extrema) */
+ if ( sp->a!=0 ) {
+ /* cubic, possibly 2 extrema (possibly none) */
+ b2_fourac = 4*(extended) sp->b*sp->b - 12*(extended) sp->a*sp->c;
+ if ( b2_fourac>=0 ) {
+ b2_fourac = sqrt(b2_fourac);
+ t1 = (-2*sp->b - b2_fourac) / (6*sp->a);
+ t2 = (-2*sp->b + b2_fourac) / (6*sp->a);
+ t1 = CheckExtremaForSingleBitErrors(sp,t1,t2);
+ t2 = CheckExtremaForSingleBitErrors(sp,t2,t1);
+ if ( t1>t2 ) { extended temp = t1; t1 = t2; t2 = temp; }
+ else if ( t1==t2 ) t2 = -1;
+ if ( RealNear(t1,0)) t1=0; else if ( RealNear(t1,1)) t1=1;
+ if ( RealNear(t2,0)) t2=0; else if ( RealNear(t2,1)) t2=1;
+ if ( t2<=0 || t2>=1 ) t2 = -1;
+ if ( t1<=0 || t1>=1 ) { t1 = t2; t2 = -1; }
+ }
+ } else if ( sp->b!=0 ) {
+ /* Quadratic, at most one extremum */
+ t1 = -sp->c/(2.0*(extended) sp->b);
+ if ( t1<=0 || t1>=1 ) t1 = -1;
+ } else /*if ( sp->c!=0 )*/ {
+ /* linear, no extrema */
+ }
+ *_t1 = t1; *_t2 = t2;
+}
+
+int IntersectLines(BasePoint *inter,
+ BasePoint *line1_1, BasePoint *line1_2,
+ BasePoint *line2_1, BasePoint *line2_2) {
+ // A lot of functions call this with the same address as an input and the output.
+ // In order to avoid unexpected behavior, we delay writing to the output until the end.
+ bigreal s1, s2;
+ BasePoint _output;
+ BasePoint * output = &_output;
+ if ( line1_1->x == line1_2->x ) {
+ // Line 1 is vertical.
+ output->x = line1_1->x;
+ if ( line2_1->x == line2_2->x ) {
+ // Line 2 is vertical.
+ if ( line2_1->x!=line1_1->x )
+ return( false ); /* Parallel vertical lines */
+ output->y = (line1_1->y+line2_1->y)/2;
+ } else {
+ output->y = line2_1->y + (output->x-line2_1->x) * (line2_2->y - line2_1->y)/(line2_2->x - line2_1->x);
+ }
+ *inter = *output;
+ return( true );
+ } else if ( line2_1->x == line2_2->x ) {
+ // Line 2 is vertical, but we know that line 1 is not.
+ output->x = line2_1->x;
+ output->y = line1_1->y + (output->x-line1_1->x) * (line1_2->y - line1_1->y)/(line1_2->x - line1_1->x);
+ *inter = *output;
+ return( true );
+ } else {
+ // Both lines are oblique.
+ s1 = (line1_2->y - line1_1->y)/(line1_2->x - line1_1->x);
+ s2 = (line2_2->y - line2_1->y)/(line2_2->x - line2_1->x);
+ if ( RealNear(s1,s2)) {
+ if ( !RealNear(line1_1->y + (line2_1->x-line1_1->x) * s1,line2_1->y))
+ return( false );
+ output->x = (line1_2->x+line2_2->x)/2;
+ output->y = (line1_2->y+line2_2->y)/2;
+ } else {
+ output->x = (s1*line1_1->x - s2*line2_1->x - line1_1->y + line2_1->y)/(s1-s2);
+ output->y = line1_1->y + (output->x-line1_1->x) * s1;
+ }
+ *inter = *output;
+ return( true );
+ }
+}
+
+static int MinMaxWithin(Spline *spline) {
+ extended dx, dy;
+ int which;
+ extended t1, t2;
+ extended w;
+ /* We know that this "spline" is basically one dimensional. As long as its*/
+ /* extrema are between the start and end points on that line then we can */
+ /* treat it as a line. If the extrema are way outside the line segment */
+ /* then it's a line that backtracks on itself */
+
+ if ( (dx = spline->to->me.x - spline->from->me.x)<0 ) dx = -dx;
+ if ( (dy = spline->to->me.y - spline->from->me.y)<0 ) dy = -dy;
+ which = dx<dy;
+ SplineFindExtrema(&spline->splines[which],&t1,&t2);
+ if ( t1==-1 )
+return( true );
+ w = ((spline->splines[which].a*t1 + spline->splines[which].b)*t1
+ + spline->splines[which].c)*t1 + spline->splines[which].d;
+ if ( RealNear(w, (&spline->to->me.x)[which]) || RealNear(w, (&spline->from->me.x)[which]) )
+ /* Close enough */;
+ else if ( (w<(&spline->to->me.x)[which] && w<(&spline->from->me.x)[which]) ||
+ (w>(&spline->to->me.x)[which] && w>(&spline->from->me.x)[which]) )
+return( false ); /* Outside */
+
+ w = ((spline->splines[which].a*t2 + spline->splines[which].b)*t2
+ + spline->splines[which].c)*t2 + spline->splines[which].d;
+ if ( RealNear(w, (&spline->to->me.x)[which]) || RealNear(w, (&spline->from->me.x)[which]) )
+ /* Close enough */;
+ else if ( (w<(&spline->to->me.x)[which] && w<(&spline->from->me.x)[which]) ||
+ (w>(&spline->to->me.x)[which] && w>(&spline->from->me.x)[which]) )
+return( false ); /* Outside */
+
+return( true );
+}
+
+int SplineIsLinear(Spline *spline) {
+ bigreal t1,t2, t3,t4;
+ int ret;
+
+ if ( spline->knownlinear )
+return( true );
+ if ( spline->knowncurved )
+return( false );
+
+ if ( spline->splines[0].a==0 && spline->splines[0].b==0 &&
+ spline->splines[1].a==0 && spline->splines[1].b==0 )
+return( true );
+
+ /* Something is linear if the control points lie on the line between the */
+ /* two base points */
+
+ /* Vertical lines */
+ if ( RealNear(spline->from->me.x,spline->to->me.x) ) {
+ ret = RealNear(spline->from->me.x,spline->from->nextcp.x) &&
+ RealNear(spline->from->me.x,spline->to->prevcp.x);
+ if ( ret && ! ((spline->from->nextcp.y >= spline->from->me.y &&
+ spline->from->nextcp.y <= spline->to->me.y &&
+ spline->to->prevcp.y >= spline->from->me.y &&
+ spline->to->prevcp.y <= spline->to->me.y ) ||
+ (spline->from->nextcp.y <= spline->from->me.y &&
+ spline->from->nextcp.y >= spline->to->me.y &&
+ spline->to->prevcp.y <= spline->from->me.y &&
+ spline->to->prevcp.y >= spline->to->me.y )) )
+ ret = MinMaxWithin(spline);
+ /* Horizontal lines */
+ } else if ( RealNear(spline->from->me.y,spline->to->me.y) ) {
+ ret = RealNear(spline->from->me.y,spline->from->nextcp.y) &&
+ RealNear(spline->from->me.y,spline->to->prevcp.y);
+ if ( ret && ! ((spline->from->nextcp.x >= spline->from->me.x &&
+ spline->from->nextcp.x <= spline->to->me.x &&
+ spline->to->prevcp.x >= spline->from->me.x &&
+ spline->to->prevcp.x <= spline->to->me.x) ||
+ (spline->from->nextcp.x <= spline->from->me.x &&
+ spline->from->nextcp.x >= spline->to->me.x &&
+ spline->to->prevcp.x <= spline->from->me.x &&
+ spline->to->prevcp.x >= spline->to->me.x)) )
+ ret = MinMaxWithin(spline);
+ } else {
+ ret = true;
+ t1 = (spline->from->nextcp.y-spline->from->me.y)/(spline->to->me.y-spline->from->me.y);
+ t2 = (spline->from->nextcp.x-spline->from->me.x)/(spline->to->me.x-spline->from->me.x);
+ t3 = (spline->to->me.y-spline->to->prevcp.y)/(spline->to->me.y-spline->from->me.y);
+ t4 = (spline->to->me.x-spline->to->prevcp.x)/(spline->to->me.x-spline->from->me.x);
+ ret = (Within16RoundingErrors(t1,t2) || (RealApprox(t1,0) && RealApprox(t2,0))) &&
+ (Within16RoundingErrors(t3,t4) || (RealApprox(t3,0) && RealApprox(t4,0)));
+ if ( ret ) {
+ if ( t1<0 || t2<0 || t3<0 || t4<0 ||
+ t1>1 || t2>1 || t3>1 || t4>1 )
+ ret = MinMaxWithin(spline);
+ }
+ }
+ spline->knowncurved = !ret;
+ spline->knownlinear = ret;
+ if ( ret ) {
+ /* A few places that if the spline is knownlinear then its splines[?] */
+ /* are linear. So give the linear version and not that suggested by */
+ /* the control points */
+ spline->splines[0].a = spline->splines[0].b = 0;
+ spline->splines[0].d = spline->from->me.x;
+ spline->splines[0].c = spline->to->me.x-spline->from->me.x;
+ spline->splines[1].a = spline->splines[1].b = 0;
+ spline->splines[1].d = spline->from->me.y;
+ spline->splines[1].c = spline->to->me.y-spline->from->me.y;
+ }
+return( ret );
+}
+
+static bigreal FindZero5(bigreal w[7],bigreal tlow, bigreal thigh) {
+ /* Somewhere between tlow and thigh there is a value of t where w(t)==0 */
+ /* It is conceiveable that there might be 3 such ts if there are some high frequency effects */
+ /* but I ignore that for now */
+ bigreal t, test;
+ int bot_negative;
+
+ t = tlow;
+ test = ((((w[5]*t+w[4])*t+w[3])*t+w[2])*t+w[1])*t + w[0];
+ bot_negative = test<0;
+
+ for (;;) {
+ t = (thigh+tlow)/2;
+ if ( thigh==t || tlow==t )
+return( t ); /* close as we can get */
+ test = ((((w[5]*t+w[4])*t+w[3])*t+w[2])*t+w[1])*t + w[0];
+ if ( test==0 )
+return( t );
+ if ( bot_negative ) {
+ if ( test<0 )
+ tlow = t;
+ else
+ thigh = t;
+ } else {
+ if ( test<0 )
+ thigh = t;
+ else
+ tlow = t;
+ }
+ }
+}
+
+static bigreal FindZero3(bigreal w[7],bigreal tlow, bigreal thigh) {
+ /* Somewhere between tlow and thigh there is a value of t where w(t)==0 */
+ /* It is conceiveable that there might be 3 such ts if there are some high frequency effects */
+ /* but I ignore that for now */
+ bigreal t, test;
+ int bot_negative;
+
+ t = tlow;
+ test = ((w[3]*t+w[2])*t+w[1])*t + w[0];
+ bot_negative = test<0;
+
+ for (;;) {
+ t = (thigh+tlow)/2;
+ if ( thigh==t || tlow==t )
+return( t ); /* close as we can get */
+ test = ((w[3]*t+w[2])*t+w[1])*t + w[0];
+ if ( test==0 )
+return( t );
+ if ( bot_negative ) {
+ if ( test<0 )
+ tlow = t;
+ else
+ thigh = t;
+ } else {
+ if ( test<0 )
+ thigh = t;
+ else
+ tlow = t;
+ }
+ }
+}
+
+bigreal SplineMinDistanceToPoint(Spline *s, BasePoint *p) {
+ /* So to find the minimum distance we want the sqrt( (sx(t)-px)^2 + (sy(t)-py)^2 ) */
+ /* Same minima as (sx(t)-px)^2 + (sy(t)-py)^2, which is easier to deal with */
+ bigreal w[7];
+ Spline1D *x = &s->splines[0], *y = &s->splines[1];
+ bigreal off[2], best;
+
+ off[0] = (x->d-p->x); off[1] = (y->d-p->y);
+
+ w[6] = (x->a*x->a) + (y->a*y->a);
+ w[5] = 2*(x->a*x->b + y->a*y->b);
+ w[4] = (x->b*x->b) + 2*(x->a*x->c) + (y->b*y->b) + 2*(y->a*y->c);
+ w[3] = 2* (x->b*x->c + x->a*off[0] + y->b*y->c + y->a*off[1]);
+ w[2] = (x->c*x->c) + 2*(x->b*off[0]) + (y->c*y->c) + 2*y->b*off[1];
+ w[1] = 2*(x->c*off[0] + y->c*off[1]);
+ w[0] = off[0]*off[0] + off[1]*off[1];
+
+ /* Take derivative */
+ w[0] = w[1];
+ w[1] = 2*w[2];
+ w[2] = 3*w[3];
+ w[3] = 4*w[4];
+ w[4] = 5*w[5];
+ w[5] = 6*w[6];
+ w[6] = 0;
+
+ if ( w[5]!=0 ) {
+ bigreal tzeros[8], t, incr, test, lasttest, zerot;
+ int i, zcnt=0;
+ /* Well, we've got a 5th degree poly and no way to play cute tricks. */
+ /* brute force it */
+ incr = 1.0/1024;
+ lasttest = w[0];
+ for ( t = incr; t<=1.0; t += incr ) {
+ test = ((((w[5]*t+w[4])*t+w[3])*t+w[2])*t+w[1])*t + w[0];
+ if ( test==0 )
+ tzeros[zcnt++] = t;
+ else {
+ if ( lasttest!=0 && (test>0) != (lasttest>0) ) {
+ zerot = FindZero5(w,t-incr,t);
+ if ( zerot>0 )
+ tzeros[zcnt++] = zerot;
+ }
+ }
+ lasttest = test;
+ }
+ best = off[0]*off[0] + off[1]*off[1]; /* t==0 */
+ test = (x->a+x->b+x->c+off[0])*(x->a+x->b+x->c+off[0]) +
+ (y->a+y->b+y->c+off[1])*(y->a+y->b+y->c+off[1]); /* t==1 */
+ if ( best>test ) best = test;
+ for ( i=0; i<zcnt; ++i ) {
+ bigreal tx, ty;
+ tx = ((x->a*tzeros[i]+x->b)*tzeros[i]+x->c)*tzeros[i] + off[0];
+ ty = ((y->a*tzeros[i]+y->b)*tzeros[i]+y->c)*tzeros[i] + off[1];
+ test = tx*tx + ty*ty;
+ if ( best>test ) best = test;
+ }
+return( sqrt(best));
+ } else if ( w[4]==0 && w[3]!=0 ) {
+ /* Started with a quadratic -- now, find 0s of a cubic */
+ /* We could find the extrema, so we have a bunch of monotonics */
+ /* Or we could brute force it as above */
+ bigreal tzeros[8], test, zerot;
+ bigreal quad[3], disc, e[5], t1, t2;
+ int i, zcnt=0, ecnt;
+
+ quad[2] = 3*w[3]; quad[1] = 2*w[2]; quad[0] = w[1];
+ disc = (-quad[1]*quad[1] - 4*quad[2]*quad[0]);
+ e[0] = 0;
+ if ( disc<0 ) {
+ e[1] = 1.0;
+ ecnt = 2;
+ } else
+ disc = sqrt(disc);
+ t1 = (-w[1] - disc) / (2*w[2]);
+ t2 = (-w[1] + disc) / (2*w[2]);
+ if ( t1>t2 ) {
+ bigreal temp = t1;
+ t1 = t2;
+ t2 = temp;
+ }
+ ecnt=1;
+ if ( t1>0 && t1<1 )
+ e[ecnt++] = t1;
+ if ( t2>0 && t2<1 && t1!=t2 )
+ e[ecnt++] = t2;
+ e[ecnt++] = 1.0;
+ for ( i=1; i<ecnt; ++i ) {
+ zerot = FindZero3(w,e[i-1],e[i]);
+ if ( zerot>0 )
+ tzeros[zcnt++] = zerot;
+ }
+ best = off[0]*off[0] + off[1]*off[1]; /* t==0 */
+ test = (x->b+x->c+off[0])*(x->b+x->c+off[0]) +
+ (y->b+y->c+off[1])*(y->b+y->c+off[1]); /* t==1 */
+ if ( best>test ) best = test;
+ for ( i=0; i<zcnt; ++i ) {
+ bigreal tx, ty;
+ tx = (x->b*tzeros[i]+x->c)*tzeros[i] + off[0];
+ ty = (y->b*tzeros[i]+y->c)*tzeros[i] + off[1];
+ test = tx*tx + ty*ty;
+ if ( best>test ) best = test;
+ }
+return( sqrt(best));
+ } else if ( w[2]==0 && w[1]!=0 ) {
+ /* Started with a line */
+ bigreal t = -w[0]/w[1], test, best;
+ best = off[0]*off[0] + off[1]*off[1]; /* t==0 */
+ test = (x->c+off[0])*(x->c+off[0]) + (y->c+off[1])*(y->c+off[1]); /* t==1 */
+ if ( best>test ) best = test;
+ if ( t>0 && t<1 ) {
+ test = (x->c*t+off[0])*(x->c*t+off[0]) + (y->c*t+off[1])*(y->c*t+off[1]);
+ if ( best>test ) best = test;
+ }
+return(sqrt(best));
+ } else if ( w[4]!=0 && w[3]!=0 && w[2]!=0 && w[1]!=0 ) {
+ IError( "Impossible condition in SplineMinDistanceToPoint");
+ } else {
+ /* It's a point, minimum distance is the only distance */
+return( sqrt(off[0]*off[0] + off[1]*off[1]) );
+ }
+return( -1 );
+}
+
+/* This returns all real solutions, even those out of bounds */
+/* I use -999999 as an error flag, since we're really only interested in */
+/* solns near 0 and 1 that should be ok. -1 is perhaps a little too close */
+/* Sigh. When solutions are near 0, the rounding errors are appalling. */
+int _CubicSolve(const Spline1D *sp,bigreal sought, extended ts[3]) {
+ extended d, xN, yN, delta2, temp, delta, h, t2, t3, theta;
+ extended sa=sp->a, sb=sp->b, sc=sp->c, sd=sp->d-sought;
+ int i=0;
+
+ ts[0] = ts[1] = ts[2] = -999999;
+ if ( sd==0 && sa!=0 ) {
+ /* one of the roots is 0, the other two are the soln of a quadratic */
+ ts[0] = 0;
+ if ( sc==0 ) {
+ ts[1] = -sb/(extended) sa; /* two zero roots */
+ } else {
+ temp = sb*(extended) sb-4*(extended) sa*sc;
+ if ( RealNear(temp,0))
+ ts[1] = -sb/(2*(extended) sa);
+ else if ( temp>=0 ) {
+ temp = sqrt(temp);
+ ts[1] = (-sb+temp)/(2*(extended) sa);
+ ts[2] = (-sb-temp)/(2*(extended) sa);
+ }
+ }
+ } else if ( sa!=0 ) {
+ /* http://www.m-a.org.uk/eb/mg/mg077ch.pdf */
+ /* this nifty solution to the cubic neatly avoids complex arithmetic */
+ xN = -sb/(3*(extended) sa);
+ yN = ((sa*xN + sb)*xN+sc)*xN + sd;
+
+ delta2 = (sb*(extended) sb-3*(extended) sa*sc)/(9*(extended) sa*sa);
+ /*if ( RealWithin(delta2,0,.00000001) ) delta2 = 0;*/
+
+ /* the descriminant is yN^2-h^2, but delta might be <0 so avoid using h */
+ d = yN*yN - 4*sa*sa*delta2*delta2*delta2;
+ if ( ((yN>.01 || yN<-.01) && RealNear(d/yN,0)) || ((yN<=.01 && yN>=-.01) && RealNear(d,0)) )
+ d = 0;
+ if ( d>0 ) {
+ temp = sqrt(d);
+ t2 = (-yN-temp)/(2*sa);
+ t2 = (t2==0) ? 0 : (t2<0) ? -pow(-t2,1./3.) : pow(t2,1./3.);
+ t3 = (-yN+temp)/(2*sa);
+ t3 = t3==0 ? 0 : (t3<0) ? -pow(-t3,1./3.) : pow(t3,1./3.);
+ ts[0] = xN + t2 + t3;
+ } else if ( d<0 ) {
+ if ( delta2>=0 ) {
+ delta = sqrt(delta2);
+ h = 2*sa*delta2*delta;
+ temp = -yN/h;
+ if ( temp>=-1.0001 && temp<=1.0001 ) {
+ if ( temp<-1 ) temp = -1; else if ( temp>1 ) temp = 1;
+ theta = acos(temp)/3;
+ ts[i++] = xN+2*delta*cos(theta);
+ ts[i++] = xN+2*delta*cos(2.0943951+theta); /* 2*pi/3 */
+ ts[i++] = xN+2*delta*cos(4.1887902+theta); /* 4*pi/3 */
+ }
+ }
+ } else if ( /* d==0 && */ delta2!=0 ) {
+ delta = yN/(2*sa);
+ delta = delta==0 ? 0 : delta>0 ? pow(delta,1./3.) : -pow(-delta,1./3.);
+ ts[i++] = xN + delta; /* this root twice, but that's irrelevant to me */
+ ts[i++] = xN - 2*delta;
+ } else if ( /* d==0 && */ delta2==0 ) {
+ if ( xN>=-0.0001 && xN<=1.0001 ) ts[0] = xN;
+ }
+ } else if ( sb!=0 ) {
+ extended d = sc*(extended) sc-4*(extended) sb*sd;
+ if ( d<0 && RealNear(d,0)) d=0;
+ if ( d<0 )
+return(false); /* All roots imaginary */
+ d = sqrt(d);
+ ts[0] = (-sc-d)/(2*(extended) sb);
+ ts[1] = (-sc+d)/(2*(extended) sb);
+ } else if ( sc!=0 ) {
+ ts[0] = -sd/(extended) sc;
+ } else {
+ /* If it's a point then either everything is a solution, or nothing */
+ }
+return( ts[0]!=-999999 );
+}
+
+int _QuarticSolve(Quartic *q,extended ts[4]) {
+ extended extrema[5];
+ Spline1D sp;
+ int ecnt = 0, i, zcnt;
+
+ /* Two special cases */
+ if ( q->a==0 ) { /* It's really a cubic */
+ sp.a = q->b;
+ sp.b = q->c;
+ sp.c = q->d;
+ sp.d = q->e;
+ ts[3] = -999999;
+return( _CubicSolve(&sp,0,ts));
+ } else if ( q->e==0 ) { /* we can factor out a zero root */
+ sp.a = q->a;
+ sp.b = q->b;
+ sp.c = q->c;
+ sp.d = q->d;
+ ts[0] = 0;
+return( _CubicSolve(&sp,0,ts+1)+1);
+ }
+
+ sp.a = 4*q->a;
+ sp.b = 3*q->b;
+ sp.c = 2*q->c;
+ sp.d = q->d;
+ if ( _CubicSolve(&sp,0,extrema)) {
+ ecnt = 1;
+ if ( extrema[1]!=-999999 ) {
+ ecnt = 2;
+ if ( extrema[1]<extrema[0] ) {
+ extended temp = extrema[1]; extrema[1] = extrema[0]; extrema[0]=temp;
+ }
+ if ( extrema[2]!=-999999 ) {
+ ecnt = 3;
+ if ( extrema[2]<extrema[0] ) {
+ extended temp = extrema[2]; extrema[2] = extrema[0]; extrema[0]=temp;
+ }
+ if ( extrema[2]<extrema[1] ) {
+ extended temp = extrema[2]; extrema[2] = extrema[1]; extrema[1]=temp;
+ }
+ }
+ }
+ }
+ for ( i=ecnt-1; i>=0 ; --i )
+ extrema[i+1] = extrema[i];
+ /* Upper and lower bounds within which we'll search */
+ extrema[0] = -999;
+ extrema[ecnt+1] = 999;
+ ecnt += 2;
+ /* divide into monotonic sections & use binary search to find zeroes */
+ for ( i=zcnt=0; i<ecnt-1; ++i ) {
+ extended top, bottom, val;
+ extended topt, bottomt, t;
+ topt = extrema[i+1];
+ bottomt = extrema[i];
+ top = (((q->a*topt+q->b)*topt+q->c)*topt+q->d)*topt+q->e;
+ bottom = (((q->a*bottomt+q->b)*bottomt+q->c)*bottomt+q->d)*bottomt+q->e;
+ if ( top<bottom ) {
+ extended temp = top; top = bottom; bottom = temp;
+ temp = topt; topt = bottomt; bottomt = temp;
+ }
+ if ( bottom>.001 ) /* this monotonic is all above 0 */
+ continue;
+ if ( top<-.001 ) /* this monotonic is all below 0 */
+ continue;
+ if ( bottom>0 ) {
+ ts[zcnt++] = bottomt;
+ continue;
+ }
+ if ( top<0 ) {
+ ts[zcnt++] = topt;
+ continue;
+ }
+ for (;;) {
+ t = (topt+bottomt)/2;
+ if ( isnan(t) ) {
+ break;
+ } else if ( t==topt || t==bottomt ) {
+ ts[zcnt++] = t;
+ break;
+ }
+
+ val = (((q->a*t+q->b)*t+q->c)*t+q->d)*t+q->e;
+ if ( val>-.0001 && val<.0001 ) {
+ ts[zcnt++] = t;
+ break;
+ } else if ( val>0 ) {
+ top = val;
+ topt = t;
+ } else {
+ bottom = val;
+ bottomt = t;
+ }
+ }
+ }
+ for ( i=zcnt; i<4; ++i )
+ ts[i] = -999999;
+return( zcnt );
+}
+
+/* calculating the actual length of a spline is hard, this gives a very */
+/* rough (but quick) approximation */
+static bigreal SplineLenApprox(Spline *spline) {
+ bigreal len, slen, temp;
+
+ if ( (temp = spline->to->me.x-spline->from->me.x)<0 ) temp = -temp;
+ len = temp;
+ if ( (temp = spline->to->me.y-spline->from->me.y)<0 ) temp = -temp;
+ len += temp;
+ if ( !spline->to->noprevcp || !spline->from->nonextcp ) {
+ if ( (temp = spline->from->nextcp.x-spline->from->me.x)<0 ) temp = -temp;
+ slen = temp;
+ if ( (temp = spline->from->nextcp.y-spline->from->me.y)<0 ) temp = -temp;
+ slen += temp;
+ if ( (temp = spline->to->prevcp.x-spline->from->nextcp.x)<0 ) temp = -temp;
+ slen += temp;
+ if ( (temp = spline->to->prevcp.y-spline->from->nextcp.y)<0 ) temp = -temp;
+ slen += temp;
+ if ( (temp = spline->to->me.x-spline->to->prevcp.x)<0 ) temp = -temp;
+ slen += temp;
+ if ( (temp = spline->to->me.y-spline->to->prevcp.y)<0 ) temp = -temp;
+ slen += temp;
+ len = (len + slen)/2;
+ }
+return( len );
+}
+
+FitPoint *SplinesFigureFPsBetween(SplinePoint *from, SplinePoint *to,
+ int *tot) {
+ int cnt, i, j, pcnt;
+ bigreal len, slen, lbase;
+ SplinePoint *np;
+ FitPoint *fp;
+ bigreal _lens[10], *lens = _lens;
+ int _cnts[10], *cnts = _cnts;
+ /* I used just to give every spline 10 points. But that gave much more */
+ /* weight to small splines than to big ones */
+
+ cnt = 0;
+ for ( np = from->next->to; ; np = np->next->to ) {
+ ++cnt;
+ if ( np==to )
+ break;
+ }
+ if ( cnt>10 ) {
+ lens = malloc(cnt*sizeof(bigreal));
+ cnts = malloc(cnt*sizeof(int));
+ }
+ cnt = 0; len = 0;
+ for ( np = from->next->to; ; np = np->next->to ) {
+ lens[cnt] = SplineLenApprox(np->prev);
+ len += lens[cnt];
+ ++cnt;
+ if ( np==to )
+ break;
+ }
+ if ( len!=0 ) {
+ pcnt = 0;
+ for ( i=0; i<cnt; ++i ) {
+ int pnts = rint( (10*cnt*lens[i])/len );
+ if ( pnts<2 ) pnts = 2;
+ cnts[i] = pnts;
+ pcnt += pnts;
+ }
+ } else
+ pcnt = 2*cnt;
+
+ fp = malloc((pcnt+1)*sizeof(FitPoint)); i = 0;
+ if ( len==0 ) {
+ for ( i=0; i<=pcnt; ++i ) {
+ fp[i].t = i/(pcnt);
+ fp[i].p.x = from->me.x;
+ fp[i].p.y = from->me.y;
+ }
+ } else {
+ lbase = 0;
+ for ( i=cnt=0, np = from->next->to; ; np = np->next->to, ++cnt ) {
+ slen = SplineLenApprox(np->prev);
+ for ( j=0; j<cnts[cnt]; ++j ) {
+ bigreal t = j/(bigreal) cnts[cnt];
+ fp[i].t = (lbase+ t*slen)/len;
+ fp[i].p.x = ((np->prev->splines[0].a*t+np->prev->splines[0].b)*t+np->prev->splines[0].c)*t + np->prev->splines[0].d;
+ fp[i++].p.y = ((np->prev->splines[1].a*t+np->prev->splines[1].b)*t+np->prev->splines[1].c)*t + np->prev->splines[1].d;
+ }
+ lbase += slen;
+ if ( np==to )
+ break;
+ }
+ }
+ if ( cnts!=_cnts ) free(cnts);
+ if ( lens!=_lens ) free(lens);
+
+ *tot = i;
+
+return( fp );
+}
+
+static int SplinePointCategory(SplinePoint *sp) {
+ enum pointtype pt;
+
+ pt = pt_corner;
+ if ( sp->next==NULL && sp->prev==NULL )
+ ;
+ else if ( (sp->next!=NULL && sp->next->to->me.x==sp->me.x && sp->next->to->me.y==sp->me.y) ||
+ (sp->prev!=NULL && sp->prev->from->me.x==sp->me.x && sp->prev->from->me.y==sp->me.y ))
+ ;
+ else if ( sp->next==NULL ) {
+ pt = sp->noprevcp ? pt_corner : pt_curve;
+ } else if ( sp->prev==NULL ) {
+ pt = sp->nonextcp ? pt_corner : pt_curve;
+ } else if ( sp->nonextcp && sp->noprevcp ) {
+ ;
+ } else {
+ BasePoint ndir, ncdir, ncunit, pdir, pcdir, pcunit;
+ bigreal nlen, nclen, plen, pclen;
+ bigreal cross, bounds;
+
+ ncdir.x = sp->nextcp.x - sp->me.x; ncdir.y = sp->nextcp.y - sp->me.y;
+ pcdir.x = sp->prevcp.x - sp->me.x; pcdir.y = sp->prevcp.y - sp->me.y;
+ ndir.x = ndir.y = pdir.x = pdir.y = 0;
+ if ( sp->next!=NULL ) {
+ ndir.x = sp->next->to->me.x - sp->me.x; ndir.y = sp->next->to->me.y - sp->me.y;
+ }
+ if ( sp->prev!=NULL ) {
+ pdir.x = sp->prev->from->me.x - sp->me.x; pdir.y = sp->prev->from->me.y - sp->me.y;
+ }
+ nclen = sqrt(ncdir.x*ncdir.x + ncdir.y*ncdir.y);
+ pclen = sqrt(pcdir.x*pcdir.x + pcdir.y*pcdir.y);
+ nlen = sqrt(ndir.x*ndir.x + ndir.y*ndir.y);
+ plen = sqrt(pdir.x*pdir.x + pdir.y*pdir.y);
+ ncunit = ncdir; pcunit = pcdir;
+ if ( nclen!=0 ) { ncunit.x /= nclen; ncunit.y /= nclen; }
+ if ( pclen!=0 ) { pcunit.x /= pclen; pcunit.y /= pclen; }
+ if ( nlen!=0 ) { ndir.x /= nlen; ndir.y /= nlen; }
+ if ( plen!=0 ) { pdir.x /= plen; pdir.y /= plen; }
+
+ /* find out which side has the shorter control vector. Cross that vector */
+ /* with the normal of the unit vector on the other side. If the */
+ /* result is less than 1 em-unit then we've got colinear control points */
+ /* (within the resolution of the integer grid) */
+ /* Not quite... they could point in the same direction */
+ if ( sp->pointtype==pt_curve )
+ bounds = 4.0;
+ else
+ bounds = 1.0;
+ if ( nclen!=0 && pclen!=0 &&
+ ((nclen>=pclen && (cross = pcdir.x*ncunit.y - pcdir.y*ncunit.x)<bounds && cross>-bounds ) ||
+ (pclen>nclen && (cross = ncdir.x*pcunit.y - ncdir.y*pcunit.x)<bounds && cross>-bounds )) &&
+ ncdir.x*pcdir.x + ncdir.y*pcdir.y < 0 )
+ pt = pt_curve;
+ /* Cross product of control point with unit vector normal to line in */
+ /* opposite direction should be less than an em-unit for a tangent */
+ else if ( ( nclen==0 && pclen!=0
+ && (cross = pcdir.x*ndir.y-pcdir.y*ndir.x)<bounds
+ && cross>-bounds && (pcdir.x*ndir.x+pcdir.y*ndir.y)<0 )
+ ||
+ ( pclen==0 && nclen!=0
+ && (cross = ncdir.x*pdir.y-ncdir.y*pdir.x)<bounds
+ && cross>-bounds && (ncdir.x*pdir.x+ncdir.y*pdir.y)<0 ) )
+ pt = pt_tangent;
+
+ if (pt == pt_curve &&
+ ((sp->nextcp.x==sp->me.x && sp->prevcp.x==sp->me.x && sp->nextcp.y!=sp->me.y) ||
+ (sp->nextcp.y==sp->me.y && sp->prevcp.y==sp->me.y && sp->nextcp.x!=sp->me.x)))
+ pt = pt_hvcurve;
+ }
+ return pt;
+}
+
+static enum pointtype SplinePointDowngrade(int current, int geom) {
+ enum pointtype np = current;
+
+ if ( current==pt_curve && geom!=pt_curve ) {
+ if ( geom==pt_hvcurve )
+ np = pt_curve;
+ else
+ np = pt_corner;
+ } else if ( current==pt_hvcurve && geom!=pt_hvcurve ) {
+ if ( geom==pt_curve )
+ np = pt_curve;
+ else
+ np = pt_corner;
+ } else if ( current==pt_tangent && geom!=pt_tangent ) {
+ np = pt_corner;
+ }
+
+ return np;
+}
+
+// Assumes flag combinations are already verified. Only returns false
+// when called with check_compat
+int _SplinePointCategorize(SplinePoint *sp, int flags) {
+ enum pointtype geom, dg, cur;
+
+ if ( flags & pconvert_flag_none )
+ // No points selected for conversion -- keep type as is
+ return true;
+ if ( flags & pconvert_flag_smooth && sp->pointtype == pt_corner )
+ // Convert only "smooth" points, not corners
+ return true;
+
+ geom = SplinePointCategory(sp);
+ dg = SplinePointDowngrade(sp->pointtype, geom);
+
+ if ( flags & pconvert_flag_incompat && sp->pointtype == dg )
+ // Only convert points incompatible with current type
+ return true;
+
+ if ( flags & pconvert_flag_by_geom ) {
+ if ( ! ( flags & pconvert_flag_hvcurve ) && geom == pt_hvcurve )
+ sp->pointtype = pt_curve;
+ else
+ sp->pointtype = geom;
+ } else if ( flags & pconvert_flag_downgrade ) {
+ sp->pointtype = dg;
+ } else if ( flags & pconvert_flag_force_type ) {
+ if ( sp->pointtype != dg ) {
+ cur = sp->pointtype;
+ sp->pointtype = dg;
+ /* SPChangePointType(sp,cur); */
+ }
+ } else if ( flags & pconvert_flag_check_compat ) {
+ if ( sp->pointtype != dg )
+ return false;
+ }
+ return true;
+}
+
+void SplinePointCategorize(SplinePoint *sp) {
+ _SplinePointCategorize(sp, pconvert_flag_all|pconvert_flag_by_geom);
+}
+
+static void SplinePointReCategorize(SplinePoint *sp,int oldpt) {
+ SplinePointCategorize(sp);
+ if ( sp->pointtype!=oldpt ) {
+ if ( sp->pointtype==pt_curve && oldpt==pt_hvcurve &&
+ ((sp->nextcp.x == sp->me.x && sp->nextcp.y != sp->me.y ) ||
+ (sp->nextcp.y == sp->me.y && sp->nextcp.x != sp->me.x )))
+ sp->pointtype = pt_hvcurve;
+ }
+}
+
+void SplinesRemoveBetween(SplinePoint *from, SplinePoint *to, int type) {
+ int tot;
+ FitPoint *fp;
+ SplinePoint *np, oldfrom;
+ int oldfpt = from->pointtype, oldtpt = to->pointtype;
+ Spline *sp;
+ int order2 = from->next->order2;
+
+ oldfrom = *from;
+ fp = SplinesFigureFPsBetween(from,to,&tot);
+
+ if ( type==1 )
+ ApproximateSplineFromPointsSlopes(from,to,fp,tot-1,order2,mt_levien);
+ else
+ ApproximateSplineFromPoints(from,to,fp,tot-1,order2);
+
+ /* Have to do the frees after the approximation because the approx */
+ /* uses the splines to determine slopes */
+ for ( sp = oldfrom.next; ; ) {
+ np = sp->to;
+ SplineFree(sp);
+ if ( np==to )
+ break;
+ sp = np->next;
+ // SplinePointMDFree(sc,np);
+ }
+
+ free(fp);
+
+ SplinePointReCategorize(from,oldfpt);
+ SplinePointReCategorize(to,oldtpt);
+}
diff --git a/src/path/splinefit/splinefont.h b/src/path/splinefit/splinefont.h
new file mode 100644
index 0000000..83db934
--- /dev/null
+++ b/src/path/splinefit/splinefont.h
@@ -0,0 +1,191 @@
+// SPDX-License-Identifier: GPL-2.0-or-later
+
+#ifndef _SEEN_SPLINEFONT_H_
+#define _SEEN_SPLINEFONT_H_
+
+#include <glib.h>
+
+typedef double real;
+typedef double bigreal;
+typedef double extended;
+typedef int BOOL;
+
+#define chunkalloc(size) calloc(1,size)
+#define chunkfree(item,size) free(item)
+
+typedef struct basepoint {
+ real x;
+ real y;
+} BasePoint;
+
+typedef struct ipoint {
+ int x;
+ int y;
+} IPoint;
+
+enum pointtype { pt_curve, pt_corner, pt_tangent, pt_hvcurve };
+typedef struct splinepoint {
+ BasePoint me;
+ BasePoint nextcp; /* control point */
+ BasePoint prevcp; /* control point */
+ unsigned int nonextcp:1;
+ unsigned int noprevcp:1;
+ unsigned int nextcpdef:1;
+ unsigned int prevcpdef:1;
+ unsigned int selected:1; /* for UI */
+ unsigned int nextcpselected: 2; /* Is the next BCP selected */
+ unsigned int prevcpselected: 2; /* Is the prev BCP selected */
+ unsigned int pointtype:2;
+ unsigned int isintersection: 1;
+ unsigned int flexy: 1; /* When "freetype_markup" is on in charview.c:DrawPoint */
+ unsigned int flexx: 1; /* flexy means select nextcp, and flexx means draw circle around nextcp */
+ unsigned int roundx: 1; /* For true type hinting */
+ unsigned int roundy: 1; /* For true type hinting */
+ unsigned int dontinterpolate: 1; /* in ttf, don't imply point by interpolating between cps */
+ unsigned int ticked: 1;
+ unsigned int watched: 1;
+ /* 1 bits left... */
+ uint16_t ptindex; /* Temporary value used by metafont routine */
+ uint16_t ttfindex; /* Truetype point index */
+ /* Special values 0xffff => point implied by averaging control points */
+ /* 0xfffe => point created with no real number yet */
+ /* (or perhaps point in context where no number is possible as in a glyph with points & refs) */
+ uint16_t nextcpindex; /* Truetype point index */
+ struct spline *next;
+ struct spline *prev;
+ /* Inkscape: not used; HintMask *hintmask; */
+ char* name;
+} SplinePoint;
+
+
+typedef struct spline1d {
+ real a, b, c, d;
+} Spline1D;
+
+typedef struct spline {
+ unsigned int islinear: 1; /* No control points */
+ unsigned int isquadratic: 1; /* probably read in from ttf */
+ unsigned int isticked: 1;
+ unsigned int isneeded: 1; /* Used in remove overlap */
+ unsigned int isunneeded: 1; /* Used in remove overlap */
+ unsigned int exclude: 1; /* Used in remove overlap variant: exclude */
+ unsigned int ishorvert: 1;
+ unsigned int knowncurved: 1; /* We know that it curves */
+ unsigned int knownlinear: 1; /* it might have control points, but still traces out a line */
+ /* If neither knownlinear nor curved then we haven't checked */
+ unsigned int order2: 1; /* It's a bezier curve with only one cp */
+ unsigned int touched: 1;
+ unsigned int leftedge: 1;
+ unsigned int rightedge: 1;
+ unsigned int acceptableextrema: 1; /* This spline has extrema, but we don't care */
+ SplinePoint *from;
+ SplinePoint *to;
+ Spline1D splines[2]; /* splines[0] is the x spline, splines[1] is y */
+ struct linearapprox *approx;
+ /* Possible optimizations:
+ Precalculate bounding box
+ Precalculate min/max/ points of inflection
+ */
+} Spline;
+
+typedef struct splinepointlist {
+ SplinePoint *first, *last;
+ struct splinepointlist *next;
+ /* Not used: spiro_cp *spiros; */
+ uint16_t spiro_cnt, spiro_max;
+ /* These could be bit fields, but bytes are easier to access and we */
+ /* don't need the space (yet) */
+ uint8_t ticked;
+ uint8_t beziers_need_optimizer; /* If the spiros have changed in spiro mode, then reverting to bezier mode might, someday, run a simplifier */
+ uint8_t is_clip_path; /* In type3/svg fonts */
+ int start_offset; // This indicates which point is the canonical first for purposes of outputting to U. F. O..
+ char *contour_name;
+} SplinePointList, SplineSet;
+
+typedef struct dbounds {
+ real minx, maxx;
+ real miny, maxy;
+} DBounds;
+
+typedef struct quartic {
+ bigreal a,b,c,d,e;
+} Quartic;
+
+
+int RealWithin(real a,real b,real fudge);
+BOOL RealNear(real a, real b);
+
+Spline *SplineMake(SplinePoint *from, SplinePoint *to, int order2);
+Spline *SplineMake2(SplinePoint *from, SplinePoint *to);
+Spline *SplineMake3(SplinePoint *from, SplinePoint *to);
+SplinePoint *SplinePointCreate(real x, real y);
+
+void SplineRefigure3(Spline *spline);
+void SplineRefigure(Spline *spline);
+int SplineIsLinear(Spline *spline);
+void SplineFindExtrema(const Spline1D *sp, extended *_t1, extended *_t2 );
+bigreal SplineMinDistanceToPoint(Spline *s, BasePoint *p);
+
+void SplinePointFree(SplinePoint *sp);
+void SplineFree(Spline *spline);
+void SplinePointListFree(SplinePointList *spl);
+
+bigreal BPDot(BasePoint v1, BasePoint v2);
+bigreal BPCross(BasePoint v1, BasePoint v2);
+BasePoint BPRev(BasePoint v);
+
+int _CubicSolve(const Spline1D *sp,bigreal sought, extended ts[3]);
+int _QuarticSolve(Quartic *q,extended ts[4]);
+int IntersectLines(BasePoint *inter,
+ BasePoint *line1_1, BasePoint *line1_2,
+ BasePoint *line2_1, BasePoint *line2_2);
+
+#define IError(msg) g_warning(msg)
+#define TRACE g_message
+
+enum linelist_flags { cvli_onscreen=0x1, cvli_clipped=0x2 };
+
+typedef struct linelist {
+ IPoint here;
+ struct linelist *next;
+ /* The first two fields are constant for the linelist, the next ones */
+ /* refer to a particular screen. If some portion of the line from */
+ /* this point to the next one is on the screen then set cvli_onscreen */
+ /* if this point needs to be clipped then set cvli_clipped */
+ /* asend and asstart are the actual screen locations where this point */
+ /* intersects the clip edge. */
+ enum linelist_flags flags;
+ IPoint asend, asstart;
+} LineList;
+
+typedef struct linearapprox {
+ real scale;
+ unsigned int oneline: 1;
+ unsigned int onepoint: 1;
+ unsigned int any: 1; /* refers to a particular screen */
+ struct linelist *lines;
+ struct linearapprox *next;
+} LinearApprox;
+
+void LinearApproxFree(LinearApprox *la);
+
+int Within16RoundingErrors(bigreal v1, bigreal v2);
+
+enum pconvert_flags {
+ // Point selection (mutually exclusive)
+ pconvert_flag_none = 0x01,
+ pconvert_flag_all = 0x02,
+ pconvert_flag_smooth = 0x04,
+ pconvert_flag_incompat = 0x08,
+ // Conversion modes (mutually exclusive)
+ pconvert_flag_by_geom = 0x100,
+ pconvert_flag_force_type = 0x200,
+ pconvert_flag_downgrade = 0x400,
+ pconvert_flag_check_compat = 0x0800,
+ // Additional
+ pconvert_flag_hvcurve = 0x4000
+};
+
+void SplinesRemoveBetween(SplinePoint *from, SplinePoint *to, int type);
+
+#endif // _SEEN_SPLINEFONT_H_
diff --git a/src/path/splinefit/splinerefigure.c b/src/path/splinefit/splinerefigure.c
new file mode 100644
index 0000000..2f6d09b
--- /dev/null
+++ b/src/path/splinefit/splinerefigure.c
@@ -0,0 +1,117 @@
+// SPDX-License-Identifier: GPL-2.0-or-later
+/* Copyright (C) 2000-2012 by George Williams */
+/*
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions are met:
+
+ * Redistributions of source code must retain the above copyright notice, this
+ * list of conditions and the following disclaimer.
+
+ * Redistributions in binary form must reproduce the above copyright notice,
+ * this list of conditions and the following disclaimer in the documentation
+ * and/or other materials provided with the distribution.
+
+ * The name of the author may not be used to endorse or promote products
+ * derived from this software without specific prior written permission.
+
+ * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED
+ * WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
+ * MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO
+ * EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+ * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+ * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS;
+ * OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,
+ * WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR
+ * OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF
+ * ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+ */
+
+#include <stdbool.h>
+#include <stdint.h>
+
+#include "splinerefigure.h"
+
+#include "splinefont.h"
+
+#include <math.h>
+#include <stdio.h>
+#ifdef HAVE_IEEEFP_H
+# include <ieeefp.h> /* Solaris defines isnan in ieeefp rather than math.h */
+#endif
+
+#include <stdbool.h>
+
+/* The slight errors introduced by the optimizer turn out to have nasty */
+/* side effects. An error on the order of 7e-8 in splines[1].b caused */
+/* the rasterizer to have kaniptions */
+void SplineRefigure3(Spline *spline) {
+ SplinePoint *from = spline->from, *to = spline->to;
+ Spline1D *xsp = &spline->splines[0], *ysp = &spline->splines[1];
+ Spline old;
+
+ spline->isquadratic = false;
+ if ( spline->acceptableextrema )
+ old = *spline;
+ xsp->d = from->me.x; ysp->d = from->me.y;
+ // Set noprevcp and nonextcp based on point values but then make sure both
+ // have the same value
+ from->nonextcp = from->nextcp.x==from->me.x && from->nextcp.y == from->me.y;
+ to->noprevcp = to->prevcp.x==to->me.x && to->prevcp.y == to->me.y;
+ if ( !from->nonextcp || !to->noprevcp )
+ from->nonextcp = to->noprevcp = false;
+ if ( from->nonextcp && to->noprevcp ) {
+ spline->islinear = true;
+ xsp->c = to->me.x-from->me.x;
+ ysp->c = to->me.y-from->me.y;
+ xsp->a = xsp->b = 0;
+ ysp->a = ysp->b = 0;
+ } else {
+ /* from p. 393 (Operator Details, curveto) PostScript Lang. Ref. Man. (Red book) */
+ xsp->c = 3*(from->nextcp.x-from->me.x);
+ ysp->c = 3*(from->nextcp.y-from->me.y);
+ xsp->b = 3*(to->prevcp.x-from->nextcp.x)-xsp->c;
+ ysp->b = 3*(to->prevcp.y-from->nextcp.y)-ysp->c;
+ xsp->a = to->me.x-from->me.x-xsp->c-xsp->b;
+ ysp->a = to->me.y-from->me.y-ysp->c-ysp->b;
+ if ( RealNear(xsp->c,0)) xsp->c=0;
+ if ( RealNear(ysp->c,0)) ysp->c=0;
+ if ( RealNear(xsp->b,0)) xsp->b=0;
+ if ( RealNear(ysp->b,0)) ysp->b=0;
+ if ( RealNear(xsp->a,0)) xsp->a=0;
+ if ( RealNear(ysp->a,0)) ysp->a=0;
+ if ( xsp->a!=0 && ( Within16RoundingErrors(xsp->a+from->me.x,from->me.x) ||
+ Within16RoundingErrors(xsp->a+to->me.x,to->me.x)))
+ xsp->a = 0;
+ if ( ysp->a!=0 && ( Within16RoundingErrors(ysp->a+from->me.y,from->me.y) ||
+ Within16RoundingErrors(ysp->a+to->me.y,to->me.y)))
+ ysp->a = 0;
+ SplineIsLinear(spline);
+ spline->islinear = false;
+ if ( ysp->a==0 && xsp->a==0 ) {
+ if ( ysp->b==0 && xsp->b==0 )
+ spline->islinear = true; /* This seems extremely unlikely... */
+ else
+ spline->isquadratic = true; /* Only likely if we read in a TTF */
+ }
+ }
+ if ( !isfinite(ysp->a) || !isfinite(xsp->a) || !isfinite(ysp->c) || !isfinite(xsp->c) || !isfinite(ysp->d) || !isfinite(xsp->d))
+ IError("NaN value in spline creation");
+ LinearApproxFree(spline->approx);
+ spline->approx = NULL;
+ spline->knowncurved = false;
+ spline->knownlinear = spline->islinear;
+ SplineIsLinear(spline);
+ spline->order2 = false;
+
+ if ( spline->acceptableextrema ) {
+ /* I don't check "d", because changes to that reflect simple */
+ /* translations which will not affect the shape of the spline */
+ if ( !RealNear(old.splines[0].a,spline->splines[0].a) ||
+ !RealNear(old.splines[0].b,spline->splines[0].b) ||
+ !RealNear(old.splines[0].c,spline->splines[0].c) ||
+ !RealNear(old.splines[1].a,spline->splines[1].a) ||
+ !RealNear(old.splines[1].b,spline->splines[1].b) ||
+ !RealNear(old.splines[1].c,spline->splines[1].c) )
+ spline->acceptableextrema = false;
+ }
+}
diff --git a/src/path/splinefit/splinerefigure.h b/src/path/splinefit/splinerefigure.h
new file mode 100644
index 0000000..110e2f9
--- /dev/null
+++ b/src/path/splinefit/splinerefigure.h
@@ -0,0 +1,9 @@
+// SPDX-License-Identifier: GPL-2.0-or-later
+#ifndef FONTFORGE_SPLINEREFIGURE_H
+#define FONTFORGE_SPLINEREFIGURE_H
+
+#include "splinefont.h"
+
+extern void SplineRefigure3(Spline *spline);
+
+#endif /* FONTFORGE_SPLINEREFIGURE_H */
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