Adding upstream version 1.3.upstream/1.3upstream

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

1 files changed, 1161 insertions, 0 deletions

diff --git a/src/2geom/path.cpp b/src/2geom/path.cpp
new file mode 100644
index 0000000..aeff503
--- /dev/null
+++ b/src/2geom/path.cpp
@@ -0,0 +1,1161 @@
+/** @file
+ * @brief Path - a sequence of contiguous curves (implementation file)
+ *//*
+ * Authors:
+ *   MenTaLguY <mental@rydia.net>
+ *   Marco Cecchetti <mrcekets at gmail.com>
+ *   Krzysztof Kosiński <tweenk.pl@gmail.com>
+ *
+ * Copyright 2007-2014  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/path.h>
+#include <2geom/pathvector.h>
+#include <2geom/transforms.h>
+#include <2geom/circle.h>
+#include <2geom/ellipse.h>
+#include <2geom/convex-hull.h>
+#include <2geom/svg-path-writer.h>
+#include <2geom/sweeper.h>
+#include <algorithm>
+#include <limits>
+#include <optional>
+
+using std::swap;
+using namespace Geom::PathInternal;
+
+namespace Geom {
+
+// this represents an empty interval
+PathInterval::PathInterval()
+    : _from(0, 0.0)
+    , _to(0, 0.0)
+    , _path_size(1)
+    , _cross_start(false)
+    , _reverse(false)
+{}
+
+PathInterval::PathInterval(PathTime const &from, PathTime const &to, bool cross_start, size_type path_size)
+    : _from(from)
+    , _to(to)
+    , _path_size(path_size)
+    , _cross_start(cross_start)
+    , _reverse((to < from) ^ cross_start)
+{
+    if (_reverse) {
+        _to.normalizeForward(_path_size);
+        if (cross_start && _to < to) {
+            // Normalization made us cross start (closed path),
+            // so we don't need to cross the start anymore.
+            _cross_start = false;
+        }
+        if (_from != _to) {
+            _from.normalizeBackward(_path_size);
+            if (cross_start && _from > from) {
+                // Normalization backwards made us logically cross
+                // the start – we shouldn't cross the start again.
+                _cross_start = false;
+            }
+        }
+    } else {
+        _from.normalizeForward(_path_size);
+        if (cross_start && _from < from) {
+            _cross_start = false;
+        }
+        if (_from != _to) {
+            _to.normalizeBackward(_path_size);
+            if (cross_start && _to > to) {
+                _cross_start = false;
+            }
+        }
+    }
+
+    if (_from == _to) {
+        _reverse = false;
+        _cross_start = false;
+    }
+}
+
+bool PathInterval::contains(PathTime const &pos) const {
+    if (_cross_start) {
+        if (_reverse) {
+            return pos >= _to || _from >= pos;
+        } else {
+            return pos >= _from || _to >= pos;
+        }
+    } else {
+        if (_reverse) {
+            return _to <= pos && pos <= _from;
+        } else {
+            return _from <= pos && pos <= _to;
+        }
+    }
+}
+
+PathInterval::size_type PathInterval::curveCount() const
+{
+    if (isDegenerate()) return 0;
+    if (_cross_start) {
+        if (_reverse) {
+            return _path_size - _to.curve_index + _from.curve_index + 1;
+        } else {
+            return _path_size - _from.curve_index + _to.curve_index + 1;
+        }
+    } else {
+        if (_reverse) {
+            return _from.curve_index - _to.curve_index + 1;
+        } else {
+            return _to.curve_index - _from.curve_index + 1;
+        }
+    }
+}
+
+PathTime PathInterval::inside(Coord min_dist) const
+{
+    // If there is some node further than min_dist (in time coord) from the ends,
+    // return that node. Otherwise, return the middle.
+    PathTime result(0, 0.0);
+
+    if (!_cross_start && _from.curve_index == _to.curve_index) {
+        PathTime result(_from.curve_index, lerp(0.5, _from.t, _to.t));
+        return result;
+    }
+    // If _cross_start, then we can be sure that at least one node is in the domain.
+    // If dcurve == 0, it actually means that all curves are included in the domain
+
+    if (_reverse) {
+        size_type dcurve = (_path_size + _from.curve_index - _to.curve_index) % _path_size;
+        bool from_close = _from.t < min_dist;
+        bool to_close = _to.t > 1 - min_dist;
+
+        if (dcurve == 0) {
+            dcurve = _path_size;
+        }
+
+        if (dcurve == 1) {
+            if (from_close || to_close) {
+                result.curve_index = _from.curve_index;
+                Coord tmid = _from.t - ((1 - _to.t) + _from.t) * 0.5;
+                if (tmid < 0) {
+                    result.curve_index = (_path_size + result.curve_index - 1) % _path_size;
+                    tmid += 1;
+                }
+                result.t = tmid;
+                return result;
+            }
+
+            result.curve_index = _from.curve_index;
+            return result;
+        }
+
+        result.curve_index = (_to.curve_index + 1) % _path_size;
+        if (to_close) {
+            if (dcurve == 2) {
+                result.t = 0.5;
+            } else {
+                result.curve_index = (result.curve_index + 1) % _path_size;
+            }
+        }
+        return result;
+    } else {
+        size_type dcurve = (_path_size + _to.curve_index - _from.curve_index) % _path_size;
+        bool from_close = _from.t > 1 - min_dist;
+        bool to_close = _to.t < min_dist;
+
+        if (dcurve == 0) {
+            dcurve = _path_size;
+        }
+
+        if (dcurve == 1) {
+            if (from_close || to_close) {
+                result.curve_index = _from.curve_index;
+                Coord tmid = ((1 - _from.t) + _to.t) * 0.5 + _from.t;
+                if (tmid >= 1) {
+                    result.curve_index = (result.curve_index + 1) % _path_size;
+                    tmid -= 1;
+                }
+                result.t = tmid;
+                return result;
+            }
+
+            result.curve_index = _to.curve_index;
+            return result;
+        }
+
+        result.curve_index = (_from.curve_index + 1) % _path_size;
+        if (from_close) {
+            if (dcurve == 2) {
+                result.t = 0.5;
+            } else {
+                result.curve_index = (result.curve_index + 1) % _path_size;
+            }
+        }
+        return result;
+    }
+
+    result.curve_index = _reverse ? _from.curve_index : _to.curve_index;
+    return result;
+}
+
+PathInterval PathInterval::from_direction(PathTime const &from, PathTime const &to, bool reversed, size_type path_size)
+{
+    PathInterval result;
+    result._from = from;
+    result._to = to;
+    result._path_size = path_size;
+
+    if (reversed) {
+        result._to.normalizeForward(path_size);
+        if (result._from != result._to) {
+            result._from.normalizeBackward(path_size);
+        }
+    } else {
+        result._from.normalizeForward(path_size);
+        if (result._from != result._to) {
+            result._to.normalizeBackward(path_size);
+        }
+    }
+
+    if (result._from == result._to) {
+        result._reverse = false;
+        result._cross_start = false;
+    } else {
+        result._reverse = reversed;
+        if (reversed) {
+            result._cross_start = from < to;
+        } else {
+            result._cross_start = to < from;
+        }
+    }
+    return result;
+}
+
+
+Path::Path(Rect const &r)
+    : _data(new PathData())
+    , _closing_seg(new ClosingSegment(r.corner(3), r.corner(0)))
+    , _closed(true)
+    , _exception_on_stitch(true)
+{
+    for (unsigned i = 0; i < 3; ++i) {
+        _data->curves.push_back(new LineSegment(r.corner(i), r.corner(i+1)));
+    }
+    _data->curves.push_back(_closing_seg);
+}
+
+Path::Path(ConvexHull const &ch)
+    : _data(new PathData())
+    , _closing_seg(new ClosingSegment(Point(), Point()))
+    , _closed(true)
+    , _exception_on_stitch(true)
+{
+    if (ch.empty()) {
+        _data->curves.push_back(_closing_seg);
+        return;
+    }
+
+    _closing_seg->setInitial(ch.back());
+    _closing_seg->setFinal(ch.front());
+
+    Point last = ch.front();
+
+    for (std::size_t i = 1; i < ch.size(); ++i) {
+        _data->curves.push_back(new LineSegment(last, ch[i]));
+        last = ch[i];
+    }
+
+    _data->curves.push_back(_closing_seg);
+    _closed = true;
+}
+
+Path::Path(Circle const &c)
+    : _data(new PathData())
+    , _closing_seg(NULL)
+    , _closed(true)
+    , _exception_on_stitch(true)
+{
+    Point p1 = c.pointAt(0);
+    Point p2 = c.pointAt(M_PI);
+    _data->curves.push_back(new EllipticalArc(p1, c.radius(), c.radius(), 0, false, true, p2));
+    _data->curves.push_back(new EllipticalArc(p2, c.radius(), c.radius(), 0, false, true, p1));
+    _closing_seg = new ClosingSegment(p1, p1);
+    _data->curves.push_back(_closing_seg);
+}
+
+Path::Path(Ellipse const &e)
+    : _data(new PathData())
+    , _closing_seg(NULL)
+    , _closed(true)
+    , _exception_on_stitch(true)
+{
+    Point p1 = e.pointAt(0);
+    Point p2 = e.pointAt(M_PI);
+    _data->curves.push_back(new EllipticalArc(p1, e.rays(), e.rotationAngle(), false, true, p2));
+    _data->curves.push_back(new EllipticalArc(p2, e.rays(), e.rotationAngle(), false, true, p1));
+    _closing_seg = new ClosingSegment(p1, p1);
+    _data->curves.push_back(_closing_seg);
+}
+
+void Path::close(bool c)
+{
+    if (c == _closed) return;
+    if (c && _data->curves.size() >= 2) {
+        // when closing, if last segment is linear and ends at initial point,
+        // replace it with the closing segment
+        Sequence::iterator last = _data->curves.end() - 2;
+        if (last->isLineSegment() && last->finalPoint() == initialPoint()) {
+            _closing_seg->setInitial(last->initialPoint());
+            _data->curves.erase(last);
+        }
+    }
+    _closed = c;
+}
+
+void Path::clear()
+{
+    _unshare();
+    _data->curves.pop_back().release();
+    _data->curves.clear();
+    _closing_seg->setInitial(Point(0, 0));
+    _closing_seg->setFinal(Point(0, 0));
+    _data->curves.push_back(_closing_seg);
+    _closed = false;
+}
+
+OptRect Path::boundsFast() const
+{
+    OptRect bounds;
+    if (empty()) {
+        return bounds;
+    }
+    // if the path is not empty, we look for cached bounds
+    if (_data->fast_bounds) {
+        return _data->fast_bounds;
+    }
+
+    bounds = front().boundsFast();
+    const_iterator iter = begin();
+    // the closing path segment can be ignored, because it will always
+    // lie within the bbox of the rest of the path
+    if (iter != end()) {
+        for (++iter; iter != end(); ++iter) {
+            bounds.unionWith(iter->boundsFast());
+        }
+    }
+    _data->fast_bounds = bounds;
+    return bounds;
+}
+
+OptRect Path::boundsExact() const
+{
+    OptRect bounds;
+    if (empty())
+        return bounds;
+    bounds = front().boundsExact();
+    const_iterator iter = begin();
+    // the closing path segment can be ignored, because it will always lie within the bbox of the rest of the path
+    if (iter != end()) {
+        for (++iter; iter != end(); ++iter) {
+            bounds.unionWith(iter->boundsExact());
+        }
+    }
+    return bounds;
+}
+
+Piecewise<D2<SBasis> > Path::toPwSb() const
+{
+    Piecewise<D2<SBasis> > ret;
+    ret.push_cut(0);
+    unsigned i = 1;
+    bool degenerate = true;
+    // pw<d2<>> is always open. so if path is closed, add closing segment as well to pwd2.
+    for (const_iterator it = begin(); it != end_default(); ++it) {
+        if (!it->isDegenerate()) {
+            ret.push(it->toSBasis(), i++);
+            degenerate = false;
+        }
+    }
+    if (degenerate) {
+        // if path only contains degenerate curves, no second cut is added
+        // so we need to create at least one segment manually
+        ret = Piecewise<D2<SBasis> >(initialPoint());
+    }
+    return ret;
+}
+
+template <typename iter>
+iter inc(iter const &x, unsigned n) {
+    iter ret = x;
+    for (unsigned i = 0; i < n; i++)
+        ret++;
+    return ret;
+}
+
+bool Path::operator==(Path const &other) const
+{
+    if (this == &other)
+        return true;
+    if (_closed != other._closed)
+        return false;
+    return _data->curves == other._data->curves;
+}
+
+void Path::start(Point const &p) {
+    if (_data->curves.size() > 1) {
+        clear();
+    }
+    _closing_seg->setInitial(p);
+    _closing_seg->setFinal(p);
+}
+
+Interval Path::timeRange() const
+{
+    Interval ret(0, size_default());
+    return ret;
+}
+
+Curve const &Path::curveAt(Coord t, Coord *rest) const
+{
+    PathTime pos = _factorTime(t);
+    if (rest) {
+        *rest = pos.t;
+    }
+    return at(pos.curve_index);
+}
+
+Point Path::pointAt(Coord t) const
+{
+    return pointAt(_factorTime(t));
+}
+
+Coord Path::valueAt(Coord t, Dim2 d) const
+{
+    return valueAt(_factorTime(t), d);
+}
+
+Curve const &Path::curveAt(PathTime const &pos) const
+{
+    return at(pos.curve_index);
+}
+Point Path::pointAt(PathTime const &pos) const
+{
+    return at(pos.curve_index).pointAt(pos.t);
+}
+Coord Path::valueAt(PathTime const &pos, Dim2 d) const
+{
+    return at(pos.curve_index).valueAt(pos.t, d);
+}
+
+std::vector<PathTime> Path::roots(Coord v, Dim2 d) const
+{
+    std::vector<PathTime> res;
+    for (unsigned i = 0; i < size(); i++) {
+        std::vector<Coord> temp = (*this)[i].roots(v, d);
+        for (double j : temp)
+            res.emplace_back(i, j);
+    }
+    return res;
+}
+
+
+// The class below implements sweepline optimization for curve intersection in paths.
+// Instead of O(N^2), this takes O(N + X), where X is the number of overlaps
+// between the bounding boxes of curves.
+
+struct CurveIntersectionSweepSet
+{
+public:
+    struct CurveRecord {
+        boost::intrusive::list_member_hook<> _hook;
+        Curve const *curve;
+        Rect bounds;
+        std::size_t index;
+        unsigned which;
+
+        CurveRecord(Curve const *pc, std::size_t idx, unsigned w)
+            : curve(pc)
+            , bounds(curve->boundsFast())
+            , index(idx)
+            , which(w)
+        {}
+    };
+
+    typedef std::vector<CurveRecord>::const_iterator ItemIterator;
+
+    CurveIntersectionSweepSet(std::vector<PathIntersection> &result,
+                              Path const &a, Path const &b, Coord precision)
+        : _result(result)
+        , _precision(precision)
+        , _sweep_dir(X)
+    {
+        std::size_t asz = a.size(), bsz = b.size();
+        _records.reserve(asz + bsz);
+
+        for (std::size_t i = 0; i < asz; ++i) {
+            _records.emplace_back(&a[i], i, 0);
+        }
+        for (std::size_t i = 0; i < bsz; ++i) {
+            _records.emplace_back(&b[i], i, 1);
+        }
+
+        OptRect abb = a.boundsFast() | b.boundsFast();
+        if (abb && abb->height() > abb->width()) {
+            _sweep_dir = Y;
+        }
+    }
+
+    std::vector<CurveRecord> const &items() { return _records; }
+    Interval itemBounds(ItemIterator ii) {
+        return ii->bounds[_sweep_dir];
+    }
+
+    void addActiveItem(ItemIterator ii) {
+        unsigned w = ii->which;
+        unsigned ow = (w+1) % 2;
+
+        _active[w].push_back(const_cast<CurveRecord&>(*ii));
+
+        for (auto & i : _active[ow]) {
+            if (!ii->bounds.intersects(i.bounds)) continue;
+            std::vector<CurveIntersection> cx = ii->curve->intersect(*i.curve, _precision);
+            for (auto & k : cx) {
+                PathTime tw(ii->index, k.first), tow(i.index, k.second);
+                _result.emplace_back(
+                    w == 0 ? tw : tow,
+                    w == 0 ? tow : tw,
+                    k.point());
+            }
+        }
+    }
+    void removeActiveItem(ItemIterator ii) {
+        ActiveCurveList &acl = _active[ii->which];
+        acl.erase(acl.iterator_to(*ii));
+    }
+
+private:
+    typedef boost::intrusive::list
+        < CurveRecord
+        , boost::intrusive::member_hook
+            < CurveRecord
+            , boost::intrusive::list_member_hook<>
+            , &CurveRecord::_hook
+            >
+        > ActiveCurveList;
+
+    std::vector<CurveRecord> _records;
+    std::vector<PathIntersection> &_result;
+    ActiveCurveList _active[2];
+    Coord _precision;
+    Dim2 _sweep_dir;
+};
+
+std::vector<PathIntersection> Path::intersect(Path const &other, Coord precision) const
+{
+    std::vector<PathIntersection> result;
+
+    CurveIntersectionSweepSet cisset(result, *this, other, precision);
+    Sweeper<CurveIntersectionSweepSet> sweeper(cisset);
+    sweeper.process();
+
+    // preprocessing to remove duplicate intersections at endpoints
+    std::size_t asz = size(), bsz = other.size();
+    for (auto & i : result) {
+        i.first.normalizeForward(asz);
+        i.second.normalizeForward(bsz);
+    }
+    std::sort(result.begin(), result.end());
+    result.erase(std::unique(result.begin(), result.end()), result.end());
+
+    return result;
+}
+
+int Path::winding(Point const &p) const {
+    int wind = 0;
+
+    /* To handle all the edge cases, we consider the maximum Y edge of the bounding box
+     * as not included in box. This way paths that contain linear horizontal
+     * segments will be treated correctly. */
+    for (const_iterator i = begin(); i != end_closed(); ++i) {
+        Rect bounds = i->boundsFast();
+
+        if (bounds.height() == 0) continue;
+        if (p[X] > bounds.right() || !bounds[Y].lowerContains(p[Y])) {
+            // Ray doesn't intersect bbox, so we ignore this segment
+            continue;
+        }
+
+        if (p[X] < bounds.left()) {
+            /* Ray intersects the curve's bbox, but the point is outside it.
+             * The winding contribution is exactly the same as that
+             * of a linear segment with the same initial and final points. */
+            Point ip = i->initialPoint();
+            Point fp = i->finalPoint();
+            Rect eqbox(ip, fp);
+
+            if (eqbox[Y].lowerContains(p[Y])) {
+                /* The ray intersects the equivalent linear segment.
+                 * Determine winding contribution based on its derivative. */
+                if (ip[Y] < fp[Y]) {
+                    wind += 1;
+                } else if (ip[Y] > fp[Y]) {
+                    wind -= 1;
+                } else {
+                    // should never happen, because bounds.height() was not zero
+                    assert(false);
+                }
+            }
+        } else {
+            // point is inside bbox
+            wind += i->winding(p);
+        }
+    }
+    return wind;
+}
+
+std::vector<double> Path::allNearestTimes(Point const &_point, double from, double to) const
+{
+    // TODO from and to are not used anywhere.
+    // rewrite this to simplify.
+    using std::swap;
+
+    if (from > to)
+        swap(from, to);
+    const Path &_path = *this;
+    unsigned int sz = _path.size();
+    if (_path.closed())
+        ++sz;
+    if (from < 0 || to > sz) {
+        THROW_RANGEERROR("[from,to] interval out of bounds");
+    }
+    double sif, st = modf(from, &sif);
+    double eif, et = modf(to, &eif);
+    unsigned int si = static_cast<unsigned int>(sif);
+    unsigned int ei = static_cast<unsigned int>(eif);
+    if (si == sz) {
+        --si;
+        st = 1;
+    }
+    if (ei == sz) {
+        --ei;
+        et = 1;
+    }
+    if (si == ei) {
+        std::vector<double> all_nearest = _path[si].allNearestTimes(_point, st, et);
+        for (double & i : all_nearest) {
+            i = si + i;
+        }
+        return all_nearest;
+    }
+    std::vector<double> all_t;
+    std::vector<std::vector<double> > all_np;
+    all_np.push_back(_path[si].allNearestTimes(_point, st));
+    std::vector<unsigned int> ni;
+    ni.push_back(si);
+    double dsq;
+    double mindistsq = distanceSq(_point, _path[si].pointAt(all_np.front().front()));
+    Rect bb(Geom::Point(0, 0), Geom::Point(0, 0));
+    for (unsigned int i = si + 1; i < ei; ++i) {
+        bb = (_path[i].boundsFast());
+        dsq = distanceSq(_point, bb);
+        if (mindistsq < dsq)
+            continue;
+        all_t = _path[i].allNearestTimes(_point);
+        dsq = distanceSq(_point, _path[i].pointAt(all_t.front()));
+        if (mindistsq > dsq) {
+            all_np.clear();
+            all_np.push_back(all_t);
+            ni.clear();
+            ni.push_back(i);
+            mindistsq = dsq;
+        } else if (mindistsq == dsq) {
+            all_np.push_back(all_t);
+            ni.push_back(i);
+        }
+    }
+    bb = (_path[ei].boundsFast());
+    dsq = distanceSq(_point, bb);
+    if (mindistsq >= dsq) {
+        all_t = _path[ei].allNearestTimes(_point, 0, et);
+        dsq = distanceSq(_point, _path[ei].pointAt(all_t.front()));
+        if (mindistsq > dsq) {
+            for (double & i : all_t) {
+                i = ei + i;
+            }
+            return all_t;
+        } else if (mindistsq == dsq) {
+            all_np.push_back(all_t);
+            ni.push_back(ei);
+        }
+    }
+    std::vector<double> all_nearest;
+    for (unsigned int i = 0; i < all_np.size(); ++i) {
+        for (unsigned int j = 0; j < all_np[i].size(); ++j) {
+            all_nearest.push_back(ni[i] + all_np[i][j]);
+        }
+    }
+    all_nearest.erase(std::unique(all_nearest.begin(), all_nearest.end()), all_nearest.end());
+    return all_nearest;
+}
+
+std::vector<Coord> Path::nearestTimePerCurve(Point const &p) const
+{
+    // return a single nearest time for each curve in this path
+    std::vector<Coord> np;
+    for (const_iterator it = begin(); it != end_default(); ++it) {
+        np.push_back(it->nearestTime(p));
+    }
+    return np;
+}
+
+PathTime Path::nearestTime(Point const &p, Coord *dist) const
+{
+    Coord mindist = std::numeric_limits<Coord>::max();
+    PathTime ret;
+
+    if (_data->curves.size() == 1) {
+        // naked moveto
+        ret.curve_index = 0;
+        ret.t = 0;
+        if (dist) {
+            *dist = distance(_closing_seg->initialPoint(), p);
+        }
+        return ret;
+    }
+
+    for (size_type i = 0; i < size_default(); ++i) {
+        Curve const &c = at(i);
+        if (distance(p, c.boundsFast()) >= mindist) continue;
+
+        Coord t = c.nearestTime(p);
+        Coord d = distance(c.pointAt(t), p);
+        if (d < mindist) {
+            mindist = d;
+            ret.curve_index = i;
+            ret.t = t;
+        }
+    }
+    if (dist) {
+        *dist = mindist;
+    }
+
+    return ret;
+}
+
+std::vector<Point> Path::nodes() const
+{
+    std::vector<Point> result;
+    size_type path_size = size_closed();
+    for (size_type i = 0; i < path_size; ++i) {
+        result.push_back(_data->curves[i].initialPoint());
+    }
+    return result;
+}
+
+void Path::appendPortionTo(Path &ret, double from, double to) const
+{
+    if (!(from >= 0 && to >= 0)) {
+        THROW_RANGEERROR("from and to must be >=0 in Path::appendPortionTo");
+    }
+    if (to == 0)
+        to = size() + 0.999999;
+    if (from == to) {
+        return;
+    }
+    double fi, ti;
+    double ff = modf(from, &fi), tf = modf(to, &ti);
+    if (tf == 0) {
+        ti--;
+        tf = 1;
+    }
+    const_iterator fromi = inc(begin(), (unsigned)fi);
+    if (fi == ti && from < to) {
+        ret.append(fromi->portion(ff, tf));
+        return;
+    }
+    const_iterator toi = inc(begin(), (unsigned)ti);
+    if (ff != 1.) {
+        // fromv->setInitial(ret.finalPoint());
+        ret.append(fromi->portion(ff, 1.));
+    }
+    if (from >= to) {
+        const_iterator ender = end();
+        if (ender->initialPoint() == ender->finalPoint())
+            ++ender;
+        ret.insert(ret.end(), ++fromi, ender);
+        ret.insert(ret.end(), begin(), toi);
+    } else {
+        ret.insert(ret.end(), ++fromi, toi);
+    }
+    ret.append(toi->portion(0., tf));
+}
+
+void Path::appendPortionTo(Path &target, PathInterval const &ival,
+                           std::optional<Point> const &p_from, std::optional<Point> const &p_to) const
+{
+    assert(ival.pathSize() == size_closed());
+
+    if (ival.isDegenerate()) {
+        Point stitch_to = p_from ? *p_from : pointAt(ival.from());
+        target.stitchTo(stitch_to);
+        return;
+    }
+
+    PathTime const &from = ival.from(), &to = ival.to();
+
+    bool reverse = ival.reverse();
+    int di = reverse ? -1 : 1;
+    size_type s = size_closed();
+
+    if (!ival.crossesStart() && from.curve_index == to.curve_index) {
+        Curve *c = (*this)[from.curve_index].portion(from.t, to.t);
+        if (p_from) {
+            c->setInitial(*p_from);
+        }
+        if (p_to) {
+            c->setFinal(*p_to);
+        }
+        target.append(c);
+    } else {
+        Curve *c_first = (*this)[from.curve_index].portion(from.t, reverse ? 0 : 1);
+        if (p_from) {
+            c_first->setInitial(*p_from);
+        }
+        target.append(c_first);
+
+        for (size_type i = (from.curve_index + s + di) % s; i != to.curve_index;
+             i = (i + s + di) % s)
+        {
+            if (reverse) {
+                target.append((*this)[i].reverse());
+            } else {
+                target.append((*this)[i].duplicate());
+            }
+        }
+
+        Curve *c_last = (*this)[to.curve_index].portion(reverse ? 1 : 0, to.t);
+        if (p_to) {
+            c_last->setFinal(*p_to);
+        }
+        target.append(c_last);
+    }
+}
+
+Path Path::reversed() const
+{
+    typedef std::reverse_iterator<Sequence::const_iterator> RIter;
+
+    Path ret(finalPoint());
+    if (empty()) return ret;
+
+    ret._data->curves.pop_back(); // this also deletes the closing segment from ret
+
+    RIter iter(_includesClosingSegment() ? _data->curves.end() : _data->curves.end() - 1);
+    RIter rend(_data->curves.begin());
+
+    if (_closed) {
+        // when the path is closed, there are two cases:
+        if (front().isLineSegment()) {
+            // 1. initial segment is linear: it becomes the new closing segment.
+            rend = RIter(_data->curves.begin() + 1);
+            ret._closing_seg = new ClosingSegment(front().finalPoint(), front().initialPoint());
+        } else {
+            // 2. initial segment is not linear: the closing segment becomes degenerate.
+            // However, skip it if it's already degenerate.
+            Point fp = finalPoint();
+            ret._closing_seg = new ClosingSegment(fp, fp);
+        }
+    } else {
+        // when the path is open, we reverse all real curves, and add a reversed closing segment.
+        ret._closing_seg = static_cast<ClosingSegment *>(_closing_seg->reverse());
+    }
+
+    for (; iter != rend; ++iter) {
+        ret._data->curves.push_back(iter->reverse());
+    }
+    ret._data->curves.push_back(ret._closing_seg);
+    ret._closed = _closed;
+    return ret;
+}
+
+
+void Path::insert(iterator pos, Curve const &curve)
+{
+    _unshare();
+    Sequence::iterator seq_pos(seq_iter(pos));
+    Sequence source;
+    source.push_back(curve.duplicate());
+    do_update(seq_pos, seq_pos, source);
+}
+
+void Path::erase(iterator pos)
+{
+    _unshare();
+    Sequence::iterator seq_pos(seq_iter(pos));
+
+    Sequence stitched;
+    do_update(seq_pos, seq_pos + 1, stitched);
+}
+
+void Path::erase(iterator first, iterator last)
+{
+    _unshare();
+    Sequence::iterator seq_first = seq_iter(first);
+    Sequence::iterator seq_last = seq_iter(last);
+
+    Sequence stitched;
+    do_update(seq_first, seq_last, stitched);
+}
+
+void Path::stitchTo(Point const &p)
+{
+    if (!empty() && _closing_seg->initialPoint() != p) {
+        if (_exception_on_stitch) {
+            THROW_CONTINUITYERROR();
+        }
+        _unshare();
+        do_append(new StitchSegment(_closing_seg->initialPoint(), p));
+    }
+}
+
+void Path::replace(iterator replaced, Curve const &curve)
+{
+    replace(replaced, replaced + 1, curve);
+}
+
+void Path::replace(iterator first_replaced, iterator last_replaced, Curve const &curve)
+{
+    _unshare();
+    Sequence::iterator seq_first_replaced(seq_iter(first_replaced));
+    Sequence::iterator seq_last_replaced(seq_iter(last_replaced));
+    Sequence source(1);
+    source.push_back(curve.duplicate());
+
+    do_update(seq_first_replaced, seq_last_replaced, source);
+}
+
+void Path::replace(iterator replaced, Path const &path)
+{
+    replace(replaced, path.begin(), path.end());
+}
+
+void Path::replace(iterator first, iterator last, Path const &path)
+{
+    replace(first, last, path.begin(), path.end());
+}
+
+void Path::snapEnds(Coord precision)
+{
+    if (!_closed) return;
+    if (_data->curves.size() > 1 && are_near(_closing_seg->length(precision), 0, precision)) {
+        _unshare();
+        _closing_seg->setInitial(_closing_seg->finalPoint());
+        (_data->curves.end() - 1)->setFinal(_closing_seg->finalPoint());
+    }
+}
+
+Path Path::withoutDegenerateCurves() const
+{
+    Sequence cleaned;
+    cleaned.reserve(size());
+
+    for (auto it = begin(); it != end_open(); ++it) {
+        if (!it->isDegenerate()) {
+            cleaned.push_back(it->duplicate());
+        }
+    }
+
+    Path result;
+    result._closed = _closed;
+    result.do_update(result._data->curves.begin(), result._data->curves.end(), cleaned);
+    return result;
+}
+
+// Replace curves between first and last with the contents of source.
+void Path::do_update(Sequence::iterator first, Sequence::iterator last, Sequence &source)
+{
+    // TODO: handle cases where first > last in closed paths?
+    bool last_beyond_closing_segment = (last == _data->curves.end());
+
+    // special case:
+    // if do_update replaces the closing segment, we have to regenerate it
+    if (source.empty()) {
+        if (first == last) return; // nothing to do
+
+        // only removing some segments
+        if ((!_closed && first == _data->curves.begin()) || (!_closed && last == _data->curves.end() - 1) || last_beyond_closing_segment) {
+            // just adjust the closing segment
+            // do nothing
+        } else if (first->initialPoint() != (last - 1)->finalPoint()) {
+            if (_exception_on_stitch) {
+                THROW_CONTINUITYERROR();
+            }
+            source.push_back(new StitchSegment(first->initialPoint(), (last - 1)->finalPoint()));
+        }
+    } else {
+        // replacing
+        if (first == _data->curves.begin() && last == _data->curves.end()) {
+            // special case: replacing everything should work the same in open and closed curves
+            _data->curves.erase(_data->curves.begin(), _data->curves.end() - 1);
+            _closing_seg->setFinal(source.front().initialPoint());
+            _closing_seg->setInitial(source.back().finalPoint());
+            _data->curves.transfer(_data->curves.begin(), source.begin(), source.end(), source);
+            return;
+        }
+
+        // stitch in front
+        if (!_closed && first == _data->curves.begin()) {
+            // not necessary to stitch in front
+        } else if (first->initialPoint() != source.front().initialPoint()) {
+            if (_exception_on_stitch) {
+                THROW_CONTINUITYERROR();
+            }
+            source.insert(source.begin(), new StitchSegment(first->initialPoint(), source.front().initialPoint()));
+        }
+
+        // stitch at the end
+        if ((!_closed && last == _data->curves.end() - 1) || last_beyond_closing_segment) {
+            // repurpose the closing segment as the stitch segment
+            // do nothing
+        } else if (source.back().finalPoint() != (last - 1)->finalPoint()) {
+            if (_exception_on_stitch) {
+                THROW_CONTINUITYERROR();
+            }
+            source.push_back(new StitchSegment(source.back().finalPoint(), (last - 1)->finalPoint()));
+        }
+    }
+
+    // do not erase the closing segment, adjust it instead
+    if (last_beyond_closing_segment) {
+        --last;
+    }
+    _data->curves.erase(first, last);
+    _data->curves.transfer(first, source.begin(), source.end(), source);
+
+    // adjust closing segment
+    if (size_open() == 0) {
+        _closing_seg->setFinal(_closing_seg->initialPoint());
+    } else {
+        _closing_seg->setInitial(back_open().finalPoint());
+        _closing_seg->setFinal(front().initialPoint());
+    }
+
+    checkContinuity();
+}
+
+void Path::do_append(Curve *c)
+{
+    if (&_data->curves.front() == _closing_seg) {
+        _closing_seg->setFinal(c->initialPoint());
+    } else {
+        // if we can't freely move the closing segment, we check whether
+        // the new curve connects with the last non-closing curve
+        if (c->initialPoint() != _closing_seg->initialPoint()) {
+            THROW_CONTINUITYERROR();
+        }
+        if (_closed && c->isLineSegment() &&
+            c->finalPoint() == _closing_seg->finalPoint())
+        {
+            // appending a curve that matches the closing segment has no effect
+            delete c;
+            return;
+        }
+    }
+    _data->curves.insert(_data->curves.end() - 1, c);
+    _closing_seg->setInitial(c->finalPoint());
+}
+
+void Path::checkContinuity() const
+{
+    Sequence::const_iterator i = _data->curves.begin(), j = _data->curves.begin();
+    ++j;
+    for (; j != _data->curves.end(); ++i, ++j) {
+        if (i->finalPoint() != j->initialPoint()) {
+            THROW_CONTINUITYERROR();
+        }
+    }
+    if (_data->curves.front().initialPoint() != _data->curves.back().finalPoint()) {
+        THROW_CONTINUITYERROR();
+    }
+}
+
+// breaks time value into integral and fractional part
+PathTime Path::_factorTime(Coord t) const
+{
+    size_type sz = size_default();
+    if (t < 0 || t > sz) {
+        THROW_RANGEERROR("parameter t out of bounds");
+    }
+
+    PathTime ret;
+    Coord k;
+    ret.t = modf(t, &k);
+    ret.curve_index = k;
+    if (ret.curve_index == sz) {
+        --ret.curve_index;
+        ret.t = 1;
+    }
+    return ret;
+}
+
+Piecewise<D2<SBasis> > paths_to_pw(PathVector const &paths)
+{
+    Piecewise<D2<SBasis> > ret = paths[0].toPwSb();
+    for (unsigned i = 1; i < paths.size(); i++) {
+        ret.concat(paths[i].toPwSb());
+    }
+    return ret;
+}
+
+bool are_near(Path const &a, Path const &b, Coord precision)
+{
+    if (a.size() != b.size()) return false;
+
+    for (unsigned i = 0; i < a.size(); ++i) {
+        if (!a[i].isNear(b[i], precision)) return false;
+    }
+    return true;
+}
+
+std::ostream &operator<<(std::ostream &out, Path const &path)
+{
+    SVGPathWriter pw;
+    pw.feed(path);
+    out << pw.str();
+    return out;
+}
+
+} // end namespace Geom
+
+/*
+  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 :