Adding upstream version 1.0.2.upstream/1.0.2upstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

1 files changed, 746 insertions, 0 deletions

diff --git a/src/2geom/convex-hull.cpp b/src/2geom/convex-hull.cpp
new file mode 100644
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+++ b/src/2geom/convex-hull.cpp
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+/** @file
+ * @brief Convex hull of a set of points
+ *//*
+ * Authors:
+ *   Nathan Hurst <njh@mail.csse.monash.edu.au>
+ *   Michael G. Sloan <mgsloan@gmail.com>
+ *   Krzysztof Kosiński <tweenk.pl@gmail.com>
+ * Copyright 2006-2015 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 <2geom/convex-hull.h>
+#include <2geom/exception.h>
+#include <algorithm>
+#include <map>
+#include <iostream>
+#include <cassert>
+#include <boost/array.hpp>
+
+/** Todo:
+    + modify graham scan to work top to bottom, rather than around angles
+    + intersection
+    + minimum distance between convex hulls
+    + maximum distance between convex hulls
+    + hausdorf metric?
+    + check all degenerate cases carefully
+    + check all algorithms meet all invariants
+    + generalise rotating caliper algorithm (iterator/circulator?)
+*/
+
+using std::vector;
+using std::map;
+using std::pair;
+using std::make_pair;
+using std::swap;
+
+namespace Geom {
+
+ConvexHull::ConvexHull(Point const &a, Point const &b)
+    : _boundary(2)
+    , _lower(0)
+{
+    _boundary[0] = a;
+    _boundary[1] = b;
+    std::sort(_boundary.begin(), _boundary.end(), Point::LexLess<X>());
+    _construct();
+}
+
+ConvexHull::ConvexHull(Point const &a, Point const &b, Point const &c)
+    : _boundary(3)
+    , _lower(0)
+{
+    _boundary[0] = a;
+    _boundary[1] = b;
+    _boundary[2] = c;
+    std::sort(_boundary.begin(), _boundary.end(), Point::LexLess<X>());
+    _construct();
+}
+
+ConvexHull::ConvexHull(Point const &a, Point const &b, Point const &c, Point const &d)
+    : _boundary(4)
+    , _lower(0)
+{
+    _boundary[0] = a;
+    _boundary[1] = b;
+    _boundary[2] = c;
+    _boundary[3] = d;
+    std::sort(_boundary.begin(), _boundary.end(), Point::LexLess<X>());
+    _construct();
+}
+
+ConvexHull::ConvexHull(std::vector<Point> const &pts)
+    : _lower(0)
+{
+    //if (pts.size() > 16) { // arbitrary threshold
+    //    _prune(pts.begin(), pts.end(), _boundary);
+    //} else {
+        _boundary = pts;
+        std::sort(_boundary.begin(), _boundary.end(), Point::LexLess<X>());
+    //}
+    _construct();
+}
+
+bool ConvexHull::_is_clockwise_turn(Point const &a, Point const &b, Point const &c)
+{
+    if (b == c) return false;
+    return cross(b-a, c-a) > 0;
+}
+
+void ConvexHull::_construct()
+{
+    // _boundary must already be sorted in LexLess<X> order
+    if (_boundary.empty()) {
+        _lower = 0;
+        return;
+    }
+    if (_boundary.size() == 1 || (_boundary.size() == 2 && _boundary[0] == _boundary[1])) {
+        _boundary.resize(1);
+        _lower = 1;
+        return;
+    }
+    if (_boundary.size() == 2) {
+        _lower = 2;
+        return;
+    }
+
+    std::size_t k = 2;
+    for (std::size_t i = 2; i < _boundary.size(); ++i) {
+        while (k >= 2 && !_is_clockwise_turn(_boundary[k-2], _boundary[k-1], _boundary[i])) {
+            --k;
+        }
+        std::swap(_boundary[k++], _boundary[i]);
+    }
+
+    _lower = k;
+    std::sort(_boundary.begin() + k, _boundary.end(), Point::LexGreater<X>());
+    _boundary.push_back(_boundary.front());
+    for (std::size_t i = _lower; i < _boundary.size(); ++i) {
+        while (k > _lower && !_is_clockwise_turn(_boundary[k-2], _boundary[k-1], _boundary[i])) {
+            --k;
+        }
+        std::swap(_boundary[k++], _boundary[i]);
+    }
+
+    _boundary.resize(k-1);
+}
+
+double ConvexHull::area() const
+{
+    if (size() <= 2) return 0;
+
+    double a = 0;
+    for (std::size_t i = 0; i < size()-1; ++i) {
+        a += cross(_boundary[i], _boundary[i+1]);
+    }
+    a += cross(_boundary.back(), _boundary.front());
+    return fabs(a * 0.5);
+}
+
+OptRect ConvexHull::bounds() const
+{
+    OptRect ret;
+    if (empty()) return ret;
+    ret = Rect(left(), top(), right(), bottom());
+    return ret;
+}
+
+Point ConvexHull::topPoint() const
+{
+    Point ret;
+    ret[Y] = std::numeric_limits<Coord>::infinity();
+
+    for (UpperIterator i = upperHull().begin(); i != upperHull().end(); ++i) {
+        if (ret[Y] >= i->y()) {
+            ret = *i;
+        } else {
+            break;
+        }
+    }
+
+    return ret;
+}
+
+Point ConvexHull::bottomPoint() const
+{
+    Point ret;
+    ret[Y] = -std::numeric_limits<Coord>::infinity();
+
+    for (LowerIterator j = lowerHull().begin(); j != lowerHull().end(); ++j) {
+        if (ret[Y] <= j->y()) {
+            ret = *j;
+        } else {
+            break;
+        }
+    }
+
+    return ret;
+}
+
+template <typename Iter, typename Lex>
+bool below_x_monotonic_polyline(Point const &p, Iter first, Iter last, Lex lex)
+{
+    typename Lex::Secondary above;
+    Iter f = std::lower_bound(first, last, p, lex);
+    if (f == last) return false;
+    if (f == first) {
+        if (p == *f) return true;
+        return false;
+    }
+
+    Point a = *(f-1), b = *f;
+    if (a[X] == b[X]) {
+        if (above(p[Y], a[Y]) || above(b[Y], p[Y])) return false;
+    } else {
+        // TODO: maybe there is a more numerically stable method
+        Coord y = lerp((p[X] - a[X]) / (b[X] - a[X]), a[Y], b[Y]);
+        if (above(p[Y], y)) return false;
+    }
+    return true;
+}
+
+bool ConvexHull::contains(Point const &p) const
+{
+    if (_boundary.empty()) return false;
+    if (_boundary.size() == 1) {
+        if (_boundary[0] == p) return true;
+        return false;
+    }
+
+    // 1. verify that the point is in the relevant X range
+    if (p[X] < _boundary[0][X] || p[X] > _boundary[_lower-1][X]) return false;
+
+    // 2. check whether it is below the upper hull
+    UpperIterator ub = upperHull().begin(), ue = upperHull().end();
+    if (!below_x_monotonic_polyline(p, ub, ue, Point::LexLess<X>())) return false;
+
+    // 3. check whether it is above the lower hull
+    LowerIterator lb = lowerHull().begin(), le = lowerHull().end();
+    if (!below_x_monotonic_polyline(p, lb, le, Point::LexGreater<X>())) return false;
+    
+    return true;
+}
+
+bool ConvexHull::contains(Rect const &r) const
+{
+    for (unsigned i = 0; i < 4; ++i) {
+        if (!contains(r.corner(i))) return false;
+    }
+    return true;
+}
+
+bool ConvexHull::contains(ConvexHull const &ch) const
+{
+    // TODO: requires interiorContains.
+    // We have to check all points of ch, and each point takes logarithmic time.
+    // If there are more points in ch that here, it is faster to make the check
+    // the other way around.
+    /*if (ch.size() > size()) {
+        for (iterator i = begin(); i != end(); ++i) {
+            if (ch.interiorContains(*i)) return false;
+        }
+        return true;
+    }*/
+
+    for (iterator i = ch.begin(); i != ch.end(); ++i) {
+        if (!contains(*i)) return false;
+    }
+    return true;
+}
+
+void ConvexHull::swap(ConvexHull &other)
+{
+    _boundary.swap(other._boundary);
+    std::swap(_lower, other._lower);
+}
+
+void ConvexHull::swap(std::vector<Point> &pts)
+{
+    _boundary.swap(pts);
+    _lower = 0;
+    std::sort(_boundary.begin(), _boundary.end(), Point::LexLess<X>());
+    _construct();
+}
+
+#if 0
+/*** SignedTriangleArea
+ * returns the area of the triangle defined by p0, p1, p2.  A clockwise triangle has positive area.
+ */
+double
+SignedTriangleArea(Point p0, Point p1, Point p2) {
+    return cross((p1 - p0), (p2 - p0));
+}
+
+class angle_cmp{
+public:
+    Point o;
+    angle_cmp(Point o) : o(o) {}
+
+#if 0
+    bool
+    operator()(Point a, Point b) {
+        // not remove this check or std::sort could crash
+        if (a == b) return false;
+        Point da = a - o;
+        Point db = b - o;
+        if (da == -db) return false;
+
+#if 1
+        double aa = da[0];
+        double ab = db[0];
+        if((da[1] == 0) && (db[1] == 0))
+            return da[0] < db[0];
+        if(da[1] == 0)
+            return true; // infinite tangent
+        if(db[1] == 0)
+            return false; // infinite tangent
+        aa = da[0] / da[1];
+        ab = db[0] / db[1];
+        if(aa > ab)
+            return true;
+#else
+        //assert((ata > atb) == (aa < ab));
+        double aa = atan2(da);
+        double ab = atan2(db);
+        if(aa < ab)
+            return true;
+#endif
+        if(aa == ab)
+            return L2sq(da) < L2sq(db);
+        return false;
+    }
+#else
+    bool operator() (Point const& a, Point const&  b)
+    {
+        // not remove this check or std::sort could generate
+        // a segmentation fault because it needs a strict '<'
+        // but due to round errors a == b doesn't mean dxy == dyx
+        if (a == b) return false;
+        Point da = a - o;
+        Point db = b - o;
+        if (da == -db) return false;
+        double dxy = da[X] * db[Y];
+        double dyx = da[Y] * db[X];
+        if (dxy > dyx) return true;
+        else if (dxy < dyx) return false;
+        return L2sq(da) < L2sq(db);
+    }
+#endif
+};
+
+//Mathematically incorrect mod, but more useful.
+int mod(int i, int l) {
+    return i >= 0 ?
+           i % l : (i % l) + l;
+}
+//OPT: usages can often be replaced by conditions
+
+/*** ConvexHull::add_point
+ * to add a point we need to find whether the new point extends the boundary, and if so, what it
+ * obscures.  Tarjan?  Jarvis?*/
+void
+ConvexHull::merge(Point p) {
+    std::vector<Point> out;
+
+    int len = boundary.size();
+
+    if(len < 2) {
+        if(boundary.empty() || boundary[0] != p)
+            boundary.push_back(p);
+        return;
+    }
+
+    bool pushed = false;
+ 
+    bool pre = is_left(p, -1);
+    for(int i = 0; i < len; i++) {
+        bool cur = is_left(p, i);
+        if(pre) {
+            if(cur) {
+                if(!pushed) {
+                    out.push_back(p);
+                    pushed = true;
+                }
+                continue;
+            }
+            else if(!pushed) {
+                out.push_back(p);
+                pushed = true;
+            }
+        }
+        out.push_back(boundary[i]);
+        pre = cur;
+    }
+
+    boundary = out;
+}
+//OPT: quickly find an obscured point and find the bounds by extending from there.  then push all points not within the bounds in order.
+  //OPT: use binary searches to find the actual starts/ends, use known rights as boundaries.  may require cooperation of find_left algo.
+
+/*** ConvexHull::is_clockwise
+ * We require that successive pairs of edges always turn right.
+ * We return false on collinear points
+ * proposed algorithm: walk successive edges and require triangle area is positive.
+ */
+bool
+ConvexHull::is_clockwise() const {
+    if(is_degenerate())
+        return true;
+    Point first = boundary[0];
+    Point second = boundary[1];
+    for(std::vector<Point>::const_iterator it(boundary.begin()+2), e(boundary.end());
+        it != e;) {
+        if(SignedTriangleArea(first, second, *it) > 0)
+            return false;
+        first = second;
+        second = *it;
+        ++it;
+    }
+    return true;
+}
+
+/*** ConvexHull::top_point_first
+ * We require that the first point in the convex hull has the least y coord, and that off all such points on the hull, it has the least x coord.
+ * proposed algorithm: track lexicographic minimum while walking the list.
+ */
+bool
+ConvexHull::top_point_first() const {
+    if(size() <= 1) return true;
+    std::vector<Point>::const_iterator pivot = boundary.begin();
+    for(std::vector<Point>::const_iterator it(boundary.begin()+1),
+            e(boundary.end());
+        it != e; it++) {
+        if((*it)[1] < (*pivot)[1])
+            pivot = it;
+        else if(((*it)[1] == (*pivot)[1]) &&
+                ((*it)[0] < (*pivot)[0]))
+            pivot = it;
+    }
+    return pivot == boundary.begin();
+}
+//OPT: since the Y values are orderly there should be something like a binary search to do this.
+
+bool
+ConvexHull::meets_invariants() const {
+    return is_clockwise() && top_point_first();
+}
+
+/*** ConvexHull::is_degenerate
+ * We allow three degenerate cases: empty, 1 point and 2 points.  In many cases these should be handled explicitly.
+ */
+bool
+ConvexHull::is_degenerate() const {
+    return boundary.size() < 3;
+}
+
+
+int sgn(double x) {
+    if(x == 0) return 0;
+    return (x<0)?-1:1;
+}
+
+bool same_side(Point L[2], Point  xs[4]) {
+    int side = 0;
+    for(int i = 0; i < 4; i++) {
+        int sn = sgn(SignedTriangleArea(L[0], L[1], xs[i]));
+        if(sn &&  !side)
+            side = sn;
+        else if(sn != side) return false;
+    }
+    return true;
+}
+
+/** find bridging pairs between two convex hulls.
+ *   this code is based on Hormoz Pirzadeh's masters thesis.  There is room for optimisation:
+ * 1. reduce recomputation
+ * 2. use more efficient angle code
+ * 3. write as iterator
+ */
+std::vector<pair<int, int> > bridges(ConvexHull a, ConvexHull b) {
+    vector<pair<int, int> > ret;
+    
+    // 1. find maximal points on a and b
+    int ai = 0, bi = 0;
+    // 2. find first copodal pair
+    double ap_angle = atan2(a[ai+1] - a[ai]);
+    double bp_angle = atan2(b[bi+1] - b[bi]);
+    Point L[2] = {a[ai], b[bi]};
+    while(ai < int(a.size()) || bi < int(b.size())) {
+        if(ap_angle == bp_angle) {
+            // In the case of parallel support lines, we must consider all four pairs of copodal points
+            {
+                assert(0); // untested
+                Point xs[4] = {a[ai-1], a[ai+1], b[bi-1], b[bi+1]};
+                if(same_side(L, xs)) ret.push_back(make_pair(ai, bi));
+                xs[2] = b[bi];
+                xs[3] = b[bi+2];
+                if(same_side(L, xs)) ret.push_back(make_pair(ai, bi));
+                xs[0] = a[ai];
+                xs[1] = a[ai+2];
+                if(same_side(L, xs)) ret.push_back(make_pair(ai, bi));
+                xs[2] = b[bi-1];
+                xs[3] = b[bi+1];
+                if(same_side(L, xs)) ret.push_back(make_pair(ai, bi));
+            }
+            ai++;
+            ap_angle += angle_between(a[ai] - a[ai-1], a[ai+1] - a[ai]);
+            L[0] = a[ai];
+            bi++;
+            bp_angle += angle_between(b[bi] - b[bi-1], b[bi+1] - b[bi]);
+            L[1] = b[bi];
+            std::cout << "parallel\n";
+        } else if(ap_angle < bp_angle) {
+            ai++;
+            ap_angle += angle_between(a[ai] - a[ai-1], a[ai+1] - a[ai]);
+            L[0] = a[ai];
+            Point xs[4] = {a[ai-1], a[ai+1], b[bi-1], b[bi+1]};
+            if(same_side(L, xs)) ret.push_back(make_pair(ai, bi));
+        } else {
+            bi++;
+            bp_angle += angle_between(b[bi] - b[bi-1], b[bi+1] - b[bi]);
+            L[1] = b[bi];
+            Point xs[4] = {a[ai-1], a[ai+1], b[bi-1], b[bi+1]};
+            if(same_side(L, xs)) ret.push_back(make_pair(ai, bi));
+        }
+    }
+    return ret;
+}
+
+unsigned find_bottom_right(ConvexHull const &a) {
+    unsigned it = 1;
+    while(it < a.boundary.size() &&
+          a.boundary[it][Y] > a.boundary[it-1][Y])
+        it++;
+    return it-1;
+}
+
+/*** ConvexHull sweepline_intersection(ConvexHull a, ConvexHull b);
+ * find the intersection between two convex hulls.  The intersection is also a convex hull.
+ * (Proof: take any two points both in a and in b.  Any point between them is in a by convexity,
+ * and in b by convexity, thus in both.  Need to prove still finite bounds.)
+ * This algorithm works by sweeping a line down both convex hulls in parallel, working out the left and right edges of the new hull.
+ */
+ConvexHull sweepline_intersection(ConvexHull const &a, ConvexHull const &b) {
+    ConvexHull ret;
+
+    unsigned al = 0;
+    unsigned bl = 0;
+
+    while(al+1 < a.boundary.size() &&
+          (a.boundary[al+1][Y] > b.boundary[bl][Y])) {
+        al++;
+    }
+    while(bl+1 < b.boundary.size() &&
+          (b.boundary[bl+1][Y] > a.boundary[al][Y])) {
+        bl++;
+    }
+    // al and bl now point to the top of the first pair of edges that overlap in y value
+    //double sweep_y = std::min(a.boundary[al][Y],
+    //                          b.boundary[bl][Y]);
+    return ret;
+}
+
+/*** ConvexHull intersection(ConvexHull a, ConvexHull b);
+ * find the intersection between two convex hulls.  The intersection is also a convex hull.
+ * (Proof: take any two points both in a and in b.  Any point between them is in a by convexity,
+ * and in b by convexity, thus in both.  Need to prove still finite bounds.)
+ */
+ConvexHull intersection(ConvexHull /*a*/, ConvexHull /*b*/) {
+    ConvexHull ret;
+    /*
+    int ai = 0, bi = 0;
+    int aj = a.boundary.size() - 1;
+    int bj = b.boundary.size() - 1;
+    */
+    /*while (true) {
+        if(a[ai]
+    }*/
+    return ret;
+}
+
+template <typename T>
+T idx_to_pair(pair<T, T> p, int idx) {
+    return idx?p.second:p.first;
+}
+
+/*** ConvexHull merge(ConvexHull a, ConvexHull b);
+ * find the smallest convex hull that surrounds a and b.
+ */
+ConvexHull merge(ConvexHull a, ConvexHull b) {
+    ConvexHull ret;
+
+    std::cout << "---\n";
+    std::vector<pair<int, int> > bpair = bridges(a, b);
+    
+    // Given our list of bridges {(pb1, qb1), ..., (pbk, qbk)}
+    // we start with the highest point in p0, q0, say it is p0.
+    // then the merged hull is p0, ..., pb1, qb1, ..., qb2, pb2, ...
+    // In other words, either of the two polygons vertices are added in order until the vertex coincides with a bridge point, at which point we swap.
+
+    unsigned state = (a[0][Y] < b[0][Y])?0:1;
+    ret.boundary.reserve(a.size() + b.size());
+    ConvexHull chs[2] = {a, b};
+    unsigned idx = 0;
+    
+    for(unsigned k = 0; k < bpair.size(); k++) {
+        unsigned limit = idx_to_pair(bpair[k], state);
+        std::cout << bpair[k].first << " , " << bpair[k].second << "; "
+                  << idx << ", " << limit << ", s: "
+                  << state
+                  << " \n";
+        while(idx <= limit) {
+            ret.boundary.push_back(chs[state][idx++]);
+        }
+        state = 1-state;
+        idx = idx_to_pair(bpair[k], state);
+    }
+    while(idx < chs[state].size()) {
+        ret.boundary.push_back(chs[state][idx++]);
+    }
+    return ret;
+}
+
+ConvexHull graham_merge(ConvexHull a, ConvexHull b) {
+    ConvexHull result;
+
+    // we can avoid the find pivot step because of top_point_first
+    if(b.boundary[0] <= a.boundary[0])
+        swap(a, b);
+
+    result.boundary = a.boundary;
+    result.boundary.insert(result.boundary.end(),
+                           b.boundary.begin(), b.boundary.end());
+
+/** if we modified graham scan to work top to bottom as proposed in lect754.pdf we could replace the
+ angle sort with a simple merge sort type algorithm. furthermore, we could do the graham scan
+ online, avoiding a bunch of memory copies.  That would probably be linear. -- njh*/
+    result.angle_sort();
+    result.graham_scan();
+
+    return result;
+}
+
+ConvexHull andrew_merge(ConvexHull a, ConvexHull b) {
+    ConvexHull result;
+
+    // we can avoid the find pivot step because of top_point_first
+    if(b.boundary[0] <= a.boundary[0])
+        swap(a, b);
+
+    result.boundary = a.boundary;
+    result.boundary.insert(result.boundary.end(),
+                           b.boundary.begin(), b.boundary.end());
+
+/** if we modified graham scan to work top to bottom as proposed in lect754.pdf we could replace the
+ angle sort with a simple merge sort type algorithm. furthermore, we could do the graham scan
+ online, avoiding a bunch of memory copies.  That would probably be linear. -- njh*/
+    result.andrew_scan();
+
+    return result;
+}
+
+//TODO: reinstate
+/*ConvexCover::ConvexCover(Path const &sp) : path(&sp) {
+    cc.reserve(sp.size());
+    for(Geom::Path::const_iterator it(sp.begin()), end(sp.end()); it != end; ++it) {
+        cc.push_back(ConvexHull((*it).begin(), (*it).end()));
+    }
+}*/
+
+double ConvexHull::centroid_and_area(Geom::Point& centroid) const {
+    const unsigned n = boundary.size();
+    if (n < 2)
+        return 0;
+    if(n < 3) {
+        centroid = (boundary[0] + boundary[1])/2;
+        return 0;
+    }
+    Geom::Point centroid_tmp(0,0);
+    double atmp = 0;
+    for (unsigned i = n-1, j = 0; j < n; i = j, j++) {
+        const double ai = cross(boundary[j], boundary[i]);
+        atmp += ai;
+        centroid_tmp += (boundary[j] + boundary[i])*ai; // first moment.
+    }
+    if (atmp != 0) {
+        centroid = centroid_tmp / (3 * atmp);
+    }
+    return atmp / 2;
+}
+
+// TODO: This can be made lg(n) using golden section/fibonacci search three starting points, say 0,
+// n/2, n-1 construct a new point, say (n/2 + n)/2 throw away the furthest boundary point iterate
+// until interval is a single value
+Point const * ConvexHull::furthest(Point direction) const {
+    Point const * p = &boundary[0];
+    double d = dot(*p, direction);
+    for(unsigned i = 1; i < boundary.size(); i++) {
+        double dd = dot(boundary[i], direction);
+        if(d < dd) {
+            p = &boundary[i];
+            d = dd;
+        }
+    }
+    return p;
+}
+
+
+// returns (a, (b,c)), three points which define the narrowest diameter of the hull as the pair of
+// lines going through b,c, and through a, parallel to b,c TODO: This can be made linear time by
+// moving point tc incrementally from the previous value (it can only move in one direction).  It
+// is currently n*O(furthest)
+double ConvexHull::narrowest_diameter(Point &a, Point &b, Point &c) {
+    Point tb = boundary.back();
+    double d = std::numeric_limits<double>::max();
+    for(unsigned i = 0; i < boundary.size(); i++) {
+        Point tc = boundary[i];
+        Point n = -rot90(tb-tc);
+        Point ta = *furthest(n);
+        double td = dot(n, ta-tb)/dot(n,n);
+        if(td < d) {
+            a = ta;
+            b = tb;
+            c = tc;
+            d = td;
+        }
+        tb = tc;
+    }
+    return d;
+}
+#endif
+
+};
+
+/*
+  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 :