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+/*
+ * sbasis.cpp - S-power basis function class + supporting classes
+ *
+ * Authors:
+ * Nathan Hurst <njh@mail.csse.monash.edu.au>
+ * Michael Sloan <mgsloan@gmail.com>
+ *
+ * Copyright (C) 2006-2007 authors
+ *
+ * This library is free software; you can redistribute it and/or
+ * modify it either under the terms of the GNU Lesser General Public
+ * License version 2.1 as published by the Free Software Foundation
+ * (the "LGPL") or, at your option, under the terms of the Mozilla
+ * Public License Version 1.1 (the "MPL"). If you do not alter this
+ * notice, a recipient may use your version of this file under either
+ * the MPL or the LGPL.
+ *
+ * You should have received a copy of the LGPL along with this library
+ * in the file COPYING-LGPL-2.1; if not, write to the Free Software
+ * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * You should have received a copy of the MPL along with this library
+ * in the file COPYING-MPL-1.1
+ *
+ * The contents of this file are subject to the Mozilla Public License
+ * Version 1.1 (the "License"); you may not use this file except in
+ * compliance with the License. You may obtain a copy of the License at
+ * http://www.mozilla.org/MPL/
+ *
+ * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
+ * OF ANY KIND, either express or implied. See the LGPL or the MPL for
+ * the specific language governing rights and limitations.
+ */
+
+#include <cmath>
+
+#include <2geom/sbasis.h>
+#include <2geom/math-utils.h>
+
+namespace Geom {
+
+#ifndef M_PI
+# define M_PI 3.14159265358979323846
+#endif
+
+/** bound the error from term truncation
+ \param tail first term to chop
+ \returns the largest possible error this truncation could give
+*/
+double SBasis::tailError(unsigned tail) const {
+ Interval bs = *bounds_fast(*this, tail);
+ return std::max(fabs(bs.min()),fabs(bs.max()));
+}
+
+/** test all coefficients are finite
+*/
+bool SBasis::isFinite() const {
+ for(unsigned i = 0; i < size(); i++) {
+ if(!(*this)[i].isFinite())
+ return false;
+ }
+ return true;
+}
+
+/** Compute the value and the first n derivatives
+ \param t position to evaluate
+ \param n number of derivatives (not counting value)
+ \returns a vector with the value and the n derivative evaluations
+
+There is an elegant way to compute the value and n derivatives for a polynomial using a variant of horner's rule. Someone will someday work out how for sbasis.
+*/
+std::vector<double> SBasis::valueAndDerivatives(double t, unsigned n) const {
+ std::vector<double> ret(n+1);
+ ret[0] = valueAt(t);
+ SBasis tmp = *this;
+ for(unsigned i = 1; i < n+1; i++) {
+ tmp.derive();
+ ret[i] = tmp.valueAt(t);
+ }
+ return ret;
+}
+
+
+/** Compute the pointwise sum of a and b (Exact)
+ \param a,b sbasis functions
+ \returns sbasis equal to a+b
+
+*/
+SBasis operator+(const SBasis& a, const SBasis& b) {
+ const unsigned out_size = std::max(a.size(), b.size());
+ const unsigned min_size = std::min(a.size(), b.size());
+ SBasis result(out_size, Linear());
+
+ for(unsigned i = 0; i < min_size; i++) {
+ result[i] = a[i] + b[i];
+ }
+ for(unsigned i = min_size; i < a.size(); i++)
+ result[i] = a[i];
+ for(unsigned i = min_size; i < b.size(); i++)
+ result[i] = b[i];
+
+ assert(result.size() == out_size);
+ return result;
+}
+
+/** Compute the pointwise difference of a and b (Exact)
+ \param a,b sbasis functions
+ \returns sbasis equal to a-b
+
+*/
+SBasis operator-(const SBasis& a, const SBasis& b) {
+ const unsigned out_size = std::max(a.size(), b.size());
+ const unsigned min_size = std::min(a.size(), b.size());
+ SBasis result(out_size, Linear());
+
+ for(unsigned i = 0; i < min_size; i++) {
+ result[i] = a[i] - b[i];
+ }
+ for(unsigned i = min_size; i < a.size(); i++)
+ result[i] = a[i];
+ for(unsigned i = min_size; i < b.size(); i++)
+ result[i] = -b[i];
+
+ assert(result.size() == out_size);
+ return result;
+}
+
+/** Compute the pointwise sum of a and b and store in a (Exact)
+ \param a,b sbasis functions
+ \returns sbasis equal to a+b
+
+*/
+SBasis& operator+=(SBasis& a, const SBasis& b) {
+ const unsigned out_size = std::max(a.size(), b.size());
+ const unsigned min_size = std::min(a.size(), b.size());
+ a.resize(out_size);
+
+ for(unsigned i = 0; i < min_size; i++)
+ a[i] += b[i];
+ for(unsigned i = min_size; i < b.size(); i++)
+ a[i] = b[i];
+
+ assert(a.size() == out_size);
+ return a;
+}
+
+/** Compute the pointwise difference of a and b and store in a (Exact)
+ \param a,b sbasis functions
+ \returns sbasis equal to a-b
+
+*/
+SBasis& operator-=(SBasis& a, const SBasis& b) {
+ const unsigned out_size = std::max(a.size(), b.size());
+ const unsigned min_size = std::min(a.size(), b.size());
+ a.resize(out_size);
+
+ for(unsigned i = 0; i < min_size; i++)
+ a[i] -= b[i];
+ for(unsigned i = min_size; i < b.size(); i++)
+ a[i] = -b[i];
+
+ assert(a.size() == out_size);
+ return a;
+}
+
+/** Compute the pointwise product of a and b (Exact)
+ \param a,b sbasis functions
+ \returns sbasis equal to a*b
+
+*/
+SBasis operator*(SBasis const &a, double k) {
+ SBasis c(a.size(), Linear());
+ for(unsigned i = 0; i < a.size(); i++)
+ c[i] = a[i] * k;
+ return c;
+}
+
+/** Compute the pointwise product of a and b and store the value in a (Exact)
+ \param a,b sbasis functions
+ \returns sbasis equal to a*b
+
+*/
+SBasis& operator*=(SBasis& a, double b) {
+ if (a.isZero()) return a;
+ if (b == 0)
+ a.clear();
+ else
+ for(auto & i : a)
+ i *= b;
+ return a;
+}
+
+/** multiply a by x^sh in place (Exact)
+ \param a sbasis function
+ \param sh power
+ \returns a
+
+*/
+SBasis shift(SBasis const &a, int sh) {
+ size_t n = a.size()+sh;
+ SBasis c(n, Linear());
+ size_t m = std::max(0, sh);
+
+ for(int i = 0; i < sh; i++)
+ c[i] = Linear(0,0);
+ for(size_t i = m, j = std::max(0,-sh); i < n; i++, j++)
+ c[i] = a[j];
+ return c;
+}
+
+/** multiply a by x^sh (Exact)
+ \param a linear function
+ \param sh power
+ \returns a* x^sh
+
+*/
+SBasis shift(Linear const &a, int sh) {
+ size_t n = 1+sh;
+ SBasis c(n, Linear());
+
+ for(int i = 0; i < sh; i++)
+ c[i] = Linear(0,0);
+ if(sh >= 0)
+ c[sh] = a;
+ return c;
+}
+
+#if 0
+SBasis multiply(SBasis const &a, SBasis const &b) {
+ // c = {a0*b0 - shift(1, a.Tri*b.Tri), a1*b1 - shift(1, a.Tri*b.Tri)}
+
+ // shift(1, a.Tri*b.Tri)
+ SBasis c(a.size() + b.size(), Linear(0,0));
+ if(a.isZero() || b.isZero())
+ return c;
+ for(unsigned j = 0; j < b.size(); j++) {
+ for(unsigned i = j; i < a.size()+j; i++) {
+ double tri = b[j].tri()*a[i-j].tri();
+ c[i+1/*shift*/] += Linear(-tri);
+ }
+ }
+ for(unsigned j = 0; j < b.size(); j++) {
+ for(unsigned i = j; i < a.size()+j; i++) {
+ for(unsigned dim = 0; dim < 2; dim++)
+ c[i][dim] += b[j][dim]*a[i-j][dim];
+ }
+ }
+ c.normalize();
+ //assert(!(0 == c.back()[0] && 0 == c.back()[1]));
+ return c;
+}
+#else
+
+/** Compute the pointwise product of a and b adding c (Exact)
+ \param a,b,c sbasis functions
+ \returns sbasis equal to a*b+c
+
+The added term is almost free
+*/
+SBasis multiply_add(SBasis const &a, SBasis const &b, SBasis c) {
+ if(a.isZero() || b.isZero())
+ return c;
+ c.resize(a.size() + b.size(), Linear(0,0));
+ for(unsigned j = 0; j < b.size(); j++) {
+ for(unsigned i = j; i < a.size()+j; i++) {
+ double tri = b[j].tri()*a[i-j].tri();
+ c[i+1/*shift*/] += Linear(-tri);
+ }
+ }
+ for(unsigned j = 0; j < b.size(); j++) {
+ for(unsigned i = j; i < a.size()+j; i++) {
+ for(unsigned dim = 0; dim < 2; dim++)
+ c[i][dim] += b[j][dim]*a[i-j][dim];
+ }
+ }
+ c.normalize();
+ //assert(!(0 == c.back()[0] && 0 == c.back()[1]));
+ return c;
+}
+
+/** Compute the pointwise product of a and b (Exact)
+ \param a,b sbasis functions
+ \returns sbasis equal to a*b
+
+*/
+SBasis multiply(SBasis const &a, SBasis const &b) {
+ if(a.isZero() || b.isZero()) {
+ SBasis c(1, Linear(0,0));
+ return c;
+ }
+ SBasis c(a.size() + b.size(), Linear(0,0));
+ return multiply_add(a, b, c);
+}
+#endif
+/** Compute the integral of a (Exact)
+ \param a sbasis functions
+ \returns sbasis integral(a)
+
+*/
+SBasis integral(SBasis const &c) {
+ SBasis a;
+ a.resize(c.size() + 1, Linear(0,0));
+ a[0] = Linear(0,0);
+
+ for(unsigned k = 1; k < c.size() + 1; k++) {
+ double ahat = -c[k-1].tri()/(2*k);
+ a[k][0] = a[k][1] = ahat;
+ }
+ double aTri = 0;
+ for(int k = c.size()-1; k >= 0; k--) {
+ aTri = (c[k].hat() + (k+1)*aTri/2)/(2*k+1);
+ a[k][0] -= aTri/2;
+ a[k][1] += aTri/2;
+ }
+ a.normalize();
+ return a;
+}
+
+/** Compute the derivative of a (Exact)
+ \param a sbasis functions
+ \returns sbasis da/dt
+
+*/
+SBasis derivative(SBasis const &a) {
+ SBasis c;
+ c.resize(a.size(), Linear(0,0));
+ if(a.isZero())
+ return c;
+
+ for(unsigned k = 0; k < a.size()-1; k++) {
+ double d = (2*k+1)*(a[k][1] - a[k][0]);
+
+ c[k][0] = d + (k+1)*a[k+1][0];
+ c[k][1] = d - (k+1)*a[k+1][1];
+ }
+ int k = a.size()-1;
+ double d = (2*k+1)*(a[k][1] - a[k][0]);
+ if (d == 0 && k > 0) {
+ c.pop_back();
+ } else {
+ c[k][0] = d;
+ c[k][1] = d;
+ }
+
+ return c;
+}
+
+/** Compute the derivative of this inplace (Exact)
+
+*/
+void SBasis::derive() { // in place version
+ if(isZero()) return;
+ for(unsigned k = 0; k < size()-1; k++) {
+ double d = (2*k+1)*((*this)[k][1] - (*this)[k][0]);
+
+ (*this)[k][0] = d + (k+1)*(*this)[k+1][0];
+ (*this)[k][1] = d - (k+1)*(*this)[k+1][1];
+ }
+ int k = size()-1;
+ double d = (2*k+1)*((*this)[k][1] - (*this)[k][0]);
+ if (d == 0 && k > 0) {
+ pop_back();
+ } else {
+ (*this)[k][0] = d;
+ (*this)[k][1] = d;
+ }
+}
+
+/** Compute the sqrt of a
+ \param a sbasis functions
+ \returns sbasis \f[ \sqrt{a} \f]
+
+It is recommended to use the piecewise version unless you have good reason.
+TODO: convert int k to unsigned k, and remove cast
+*/
+SBasis sqrt(SBasis const &a, int k) {
+ SBasis c;
+ if(a.isZero() || k == 0)
+ return c;
+ c.resize(k, Linear(0,0));
+ c[0] = Linear(std::sqrt(a[0][0]), std::sqrt(a[0][1]));
+ SBasis r = a - multiply(c, c); // remainder
+
+ for(unsigned i = 1; i <= (unsigned)k && i<r.size(); i++) {
+ Linear ci(r[i][0]/(2*c[0][0]), r[i][1]/(2*c[0][1]));
+ SBasis cisi = shift(ci, i);
+ r -= multiply(shift((c*2 + cisi), i), SBasis(ci));
+ r.truncate(k+1);
+ c += cisi;
+ if(r.tailError(i) == 0) // if exact
+ break;
+ }
+
+ return c;
+}
+
+/** Compute the recpirocal of a
+ \param a sbasis functions
+ \returns sbasis 1/a
+
+It is recommended to use the piecewise version unless you have good reason.
+*/
+SBasis reciprocal(Linear const &a, int k) {
+ SBasis c;
+ assert(!a.isZero());
+ c.resize(k, Linear(0,0));
+ double r_s0 = (a.tri()*a.tri())/(-a[0]*a[1]);
+ double r_s0k = 1;
+ for(unsigned i = 0; i < (unsigned)k; i++) {
+ c[i] = Linear(r_s0k/a[0], r_s0k/a[1]);
+ r_s0k *= r_s0;
+ }
+ return c;
+}
+
+/** Compute a / b to k terms
+ \param a,b sbasis functions
+ \returns sbasis a/b
+
+It is recommended to use the piecewise version unless you have good reason.
+*/
+SBasis divide(SBasis const &a, SBasis const &b, int k) {
+ SBasis c;
+ assert(!a.isZero());
+ SBasis r = a; // remainder
+
+ k++;
+ r.resize(k, Linear(0,0));
+ c.resize(k, Linear(0,0));
+
+ for(unsigned i = 0; i < (unsigned)k; i++) {
+ Linear ci(r[i][0]/b[0][0], r[i][1]/b[0][1]); //H0
+ c[i] += ci;
+ r -= shift(multiply(ci,b), i);
+ r.truncate(k+1);
+ if(r.tailError(i) == 0) // if exact
+ break;
+ }
+
+ return c;
+}
+
+/** Compute a composed with b
+ \param a,b sbasis functions
+ \returns sbasis a(b(t))
+
+ return a0 + s(a1 + s(a2 +... where s = (1-u)u; ak =(1 - u)a^0_k + ua^1_k
+*/
+SBasis compose(SBasis const &a, SBasis const &b) {
+ SBasis s = multiply((SBasis(Linear(1,1))-b), b);
+ SBasis r;
+
+ for(int i = a.size()-1; i >= 0; i--) {
+ r = multiply_add(r, s, SBasis(Linear(a[i][0])) - b*a[i][0] + b*a[i][1]);
+ }
+ return r;
+}
+
+/** Compute a composed with b to k terms
+ \param a,b sbasis functions
+ \returns sbasis a(b(t))
+
+ return a0 + s(a1 + s(a2 +... where s = (1-u)u; ak =(1 - u)a^0_k + ua^1_k
+*/
+SBasis compose(SBasis const &a, SBasis const &b, unsigned k) {
+ SBasis s = multiply((SBasis(Linear(1,1))-b), b);
+ SBasis r;
+
+ for(int i = a.size()-1; i >= 0; i--) {
+ r = multiply_add(r, s, SBasis(Linear(a[i][0])) - b*a[i][0] + b*a[i][1]);
+ }
+ r.truncate(k);
+ return r;
+}
+
+SBasis portion(const SBasis &t, double from, double to) {
+ double fv = t.valueAt(from);
+ double tv = t.valueAt(to);
+ SBasis ret = compose(t, Linear(from, to));
+ ret.at0() = fv;
+ ret.at1() = tv;
+ return ret;
+}
+
+/*
+Inversion algorithm. The notation is certainly very misleading. The
+pseudocode should say:
+
+c(v) := 0
+r(u) := r_0(u) := u
+for i:=0 to k do
+ c_i(v) := H_0(r_i(u)/(t_1)^i; u)
+ c(v) := c(v) + c_i(v)*t^i
+ r(u) := r(u) ? c_i(u)*(t(u))^i
+endfor
+*/
+
+//#define DEBUG_INVERSION 1
+
+/** find the function a^-1 such that a^-1 composed with a to k terms is the identity function
+ \param a sbasis function
+ \returns sbasis a^-1 s.t. a^-1(a(t)) = 1
+
+ The function must have 'unit range'("a00 = 0 and a01 = 1") and be monotonic.
+*/
+SBasis inverse(SBasis a, int k) {
+ assert(a.size() > 0);
+ double a0 = a[0][0];
+ if(a0 != 0) {
+ a -= a0;
+ }
+ double a1 = a[0][1];
+ assert(a1 != 0); // not invertable.
+
+ if(a1 != 1) {
+ a /= a1;
+ }
+ SBasis c(k, Linear()); // c(v) := 0
+ if(a.size() >= 2 && k == 2) {
+ c[0] = Linear(0,1);
+ Linear t1(1+a[1][0], 1-a[1][1]); // t_1
+ c[1] = Linear(-a[1][0]/t1[0], -a[1][1]/t1[1]);
+ } else if(a.size() >= 2) { // non linear
+ SBasis r = Linear(0,1); // r(u) := r_0(u) := u
+ Linear t1(1./(1+a[1][0]), 1./(1-a[1][1])); // 1./t_1
+ Linear one(1,1);
+ Linear t1i = one; // t_1^0
+ SBasis one_minus_a = SBasis(one) - a;
+ SBasis t = multiply(one_minus_a, a); // t(u)
+ SBasis ti(one); // t(u)^0
+#ifdef DEBUG_INVERSION
+ std::cout << "a=" << a << std::endl;
+ std::cout << "1-a=" << one_minus_a << std::endl;
+ std::cout << "t1=" << t1 << std::endl;
+ //assert(t1 == t[1]);
+#endif
+
+ //c.resize(k+1, Linear(0,0));
+ for(unsigned i = 0; i < (unsigned)k; i++) { // for i:=0 to k do
+#ifdef DEBUG_INVERSION
+ std::cout << "-------" << i << ": ---------" <<std::endl;
+ std::cout << "r=" << r << std::endl
+ << "c=" << c << std::endl
+ << "ti=" << ti << std::endl
+ << std::endl;
+#endif
+ if(r.size() <= i) // ensure enough space in the remainder, probably not needed
+ r.resize(i+1, Linear(0,0));
+ Linear ci(r[i][0]*t1i[0], r[i][1]*t1i[1]); // c_i(v) := H_0(r_i(u)/(t_1)^i; u)
+#ifdef DEBUG_INVERSION
+ std::cout << "t1i=" << t1i << std::endl;
+ std::cout << "ci=" << ci << std::endl;
+#endif
+ for(int dim = 0; dim < 2; dim++) // t1^-i *= 1./t1
+ t1i[dim] *= t1[dim];
+ c[i] = ci; // c(v) := c(v) + c_i(v)*t^i
+ // change from v to u parameterisation
+ SBasis civ = one_minus_a*ci[0] + a*ci[1];
+ // r(u) := r(u) - c_i(u)*(t(u))^i
+ // We can truncate this to the number of final terms, as no following terms can
+ // contribute to the result.
+ r -= multiply(civ,ti);
+ r.truncate(k);
+ if(r.tailError(i) == 0)
+ break; // yay!
+ ti = multiply(ti,t);
+ }
+#ifdef DEBUG_INVERSION
+ std::cout << "##########################" << std::endl;
+#endif
+ } else
+ c = Linear(0,1); // linear
+ c -= a0; // invert the offset
+ c /= a1; // invert the slope
+ return c;
+}
+
+/** Compute the sine of a to k terms
+ \param b linear function
+ \returns sbasis sin(a)
+
+It is recommended to use the piecewise version unless you have good reason.
+*/
+SBasis sin(Linear b, int k) {
+ SBasis s(k+2, Linear());
+ s[0] = Linear(std::sin(b[0]), std::sin(b[1]));
+ double tr = s[0].tri();
+ double t2 = b.tri();
+ s[1] = Linear(std::cos(b[0])*t2 - tr, -std::cos(b[1])*t2 + tr);
+
+ t2 *= t2;
+ for(int i = 0; i < k; i++) {
+ Linear bo(4*(i+1)*s[i+1][0] - 2*s[i+1][1],
+ -2*s[i+1][0] + 4*(i+1)*s[i+1][1]);
+ bo -= s[i]*(t2/(i+1));
+
+
+ s[i+2] = bo/double(i+2);
+ }
+
+ return s;
+}
+
+/** Compute the cosine of a
+ \param b linear function
+ \returns sbasis cos(a)
+
+It is recommended to use the piecewise version unless you have good reason.
+*/
+SBasis cos(Linear bo, int k) {
+ return sin(Linear(bo[0] + M_PI/2,
+ bo[1] + M_PI/2),
+ k);
+}
+
+/** compute fog^-1.
+ \param f,g sbasis functions
+ \returns sbasis f(g^-1(t)).
+
+("zero" = double comparison threshold. *!*we might divide by "zero"*!*)
+TODO: compute order according to tol?
+TODO: requires g(0)=0 & g(1)=1 atm... adaptation to other cases should be obvious!
+*/
+SBasis compose_inverse(SBasis const &f, SBasis const &g, unsigned order, double zero){
+ SBasis result(order, Linear(0.)); //result
+ SBasis r=f; //remainder
+ SBasis Pk=Linear(1)-g,Qk=g,sg=Pk*Qk;
+ Pk.truncate(order);
+ Qk.truncate(order);
+ Pk.resize(order,Linear(0.));
+ Qk.resize(order,Linear(0.));
+ r.resize(order,Linear(0.));
+
+ int vs = valuation(sg,zero);
+ if (vs == 0) { // to prevent infinite loop
+ return result;
+ }
+
+ for (unsigned k=0; k<order; k+=vs){
+ double p10 = Pk.at(k)[0];// we have to solve the linear system:
+ double p01 = Pk.at(k)[1];//
+ double q10 = Qk.at(k)[0];// p10*a + q10*b = r10
+ double q01 = Qk.at(k)[1];// &
+ double r10 = r.at(k)[0];// p01*a + q01*b = r01
+ double r01 = r.at(k)[1];//
+ double a,b;
+ double det = p10*q01-p01*q10;
+
+ //TODO: handle det~0!!
+ if (fabs(det)<zero){
+ a=b=0;
+ }else{
+ a=( q01*r10-q10*r01)/det;
+ b=(-p01*r10+p10*r01)/det;
+ }
+ result[k] = Linear(a,b);
+ r=r-Pk*a-Qk*b;
+
+ Pk=Pk*sg;
+ Qk=Qk*sg;
+
+ Pk.resize(order,Linear(0.)); // truncates if too high order, expands with zeros if too low
+ Qk.resize(order,Linear(0.));
+ r.resize(order,Linear(0.));
+
+ }
+ result.normalize();
+ return result;
+}
+
+}
+
+/*
+ Local Variables:
+ mode:c++
+ c-file-style:"stroustrup"
+ c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
+ indent-tabs-mode:nil
+ fill-column:99
+ End:
+*/
+// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:fileencoding=utf-8:textwidth=99 :
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