Adding upstream version 1.0.2.upstream/1.0.2upstream

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

1 files changed, 945 insertions, 0 deletions

diff --git a/src/2geom/piecewise.h b/src/2geom/piecewise.h
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+/** @file
+ * @brief Piecewise function class
+ *//*
+ * Copyright 2007 Michael Sloan <mgsloan@gmail.com>
+ *
+ * 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, output 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.
+ *
+ */
+
+#ifndef LIB2GEOM_SEEN_PIECEWISE_H
+#define LIB2GEOM_SEEN_PIECEWISE_H
+
+#include <vector>
+#include <map>
+#include <utility>
+#include <boost/concept_check.hpp>
+#include <2geom/concepts.h>
+#include <2geom/math-utils.h>
+#include <2geom/sbasis.h>
+
+
+namespace Geom {
+/**
+ * @brief Function defined as discrete pieces.
+ *
+ * The Piecewise class manages a sequence of elements of a type as segments and
+ * the ’cuts’ between them. These cuts are time values which separate the pieces.
+ * This function representation allows for more interesting functions, as it provides
+ * a viable output for operations such as inversion, which may require multiple
+ * SBasis to properly invert the original.
+ *
+ * As for technical details, while the actual SBasis segments begin on the first
+ * cut and end on the last, the function is defined throughout all inputs by ex-
+ * tending the first and last segments. The exact switching between segments is
+ * arbitrarily such that beginnings (t=0) have preference over endings (t=1). This
+ * only matters if it is discontinuous at the location.
+ * \f[
+ *      f(t) \rightarrow \left\{ 
+ *      \begin{array}{cc}
+ *      s_1,& t <= c_2 \\
+ *      s_2,& c_2 <= t <= c_3\\
+ *      \ldots \\
+ *      s_n,& c_n <= t
+ *      \end{array}\right.
+ * \f]
+ *
+ * @ingroup Fragments
+ */
+template <typename T>
+class Piecewise {
+  BOOST_CLASS_REQUIRE(T, Geom, FragmentConcept);
+
+  public:
+    std::vector<double> cuts;
+    std::vector<T> segs;
+    //segs[i] stretches from cuts[i] to cuts[i+1].
+
+    Piecewise() {}
+
+    explicit Piecewise(const T &s) {
+        push_cut(0.);
+        push_seg(s);
+        push_cut(1.);
+    }
+
+    unsigned input_dim(){return 1;}
+
+    typedef typename T::output_type output_type;
+
+    explicit Piecewise(const output_type & v) {
+        push_cut(0.);
+        push_seg(T(v));
+        push_cut(1.);
+    }
+
+    inline void reserve(unsigned i) { segs.reserve(i); cuts.reserve(i + 1); }
+
+    inline T const& operator[](unsigned i) const { return segs[i]; }
+    inline T&       operator[](unsigned i)       { return segs[i]; }
+    inline output_type operator()(double t) const { return valueAt(t); }
+    inline output_type valueAt(double t) const {
+        unsigned n = segN(t);
+        return segs[n](segT(t, n));
+    }
+    inline output_type firstValue() const {
+        return valueAt(cuts.front());
+    }
+    inline output_type lastValue() const {
+        return valueAt(cuts.back());
+    }
+
+    /**
+    *  The size of the returned vector equals n_derivs+1.
+    */
+    std::vector<output_type> valueAndDerivatives(double t, unsigned n_derivs) const {
+        unsigned n = segN(t);
+        std::vector<output_type> ret, val = segs[n].valueAndDerivatives(segT(t, n), n_derivs);
+        double mult = 1;
+        for(unsigned i = 0; i < val.size(); i++) {
+            ret.push_back(val[i] * mult);
+            mult /= cuts[n+1] - cuts[n];
+        }
+        return ret;
+    }
+
+    //TODO: maybe it is not a good idea to have this?
+    Piecewise<T> operator()(SBasis f);
+    Piecewise<T> operator()(Piecewise<SBasis>f);
+
+    inline unsigned size() const { return segs.size(); }
+    inline bool empty() const { return segs.empty(); }
+    inline void clear() {
+        segs.clear();
+        cuts.clear();
+    }
+
+    /**Convenience/implementation hiding function to add segment/cut pairs.
+     * Asserts that basic size and order invariants are correct
+     */
+    inline void push(const T &s, double to) {
+        assert(cuts.size() - segs.size() == 1);
+        push_seg(s);
+        push_cut(to);
+    }
+    inline void push(T &&s, double to) {
+        assert(cuts.size() - segs.size() == 1);
+        push_seg(s);
+        push_cut(to);
+    }
+    //Convenience/implementation hiding function to add cuts.
+    inline void push_cut(double c) {
+        ASSERT_INVARIANTS(cuts.empty() || c > cuts.back());
+        cuts.push_back(c);
+    }
+    //Convenience/implementation hiding function to add segments.
+    inline void push_seg(const T &s) { segs.push_back(s); }
+    inline void push_seg(T &&s) { segs.emplace_back(s); }
+
+    /**Returns the segment index which corresponds to a 'global' piecewise time.
+     * Also takes optional low/high parameters to expedite the search for the segment.
+     */
+    inline unsigned segN(double t, int low = 0, int high = -1) const {
+        high = (high == -1) ? size() : high;
+        if(t < cuts[0]) return 0;
+        if(t >= cuts[size()]) return size() - 1;
+        while(low < high) {
+            int mid = (high + low) / 2; //Lets not plan on having huge (> INT_MAX / 2) cut sequences
+            double mv = cuts[mid];
+            if(mv < t) {
+                if(t < cuts[mid + 1]) return mid; else low = mid + 1;
+            } else if(t < mv) {
+                if(cuts[mid - 1] < t) return mid - 1; else high = mid - 1;
+            } else {
+                return mid;
+            }
+        }
+        return low;
+    }
+
+    /**Returns the time within a segment, given the 'global' piecewise time.
+     * Also takes an optional index parameter which may be used for efficiency or to find the time on a
+     * segment outside its range.  If it is left to its default, -1, it will call segN to find the index.
+     */
+    inline double segT(double t, int i = -1) const {
+        if(i == -1) i = segN(t);
+        assert(i >= 0);
+        return (t - cuts[i]) / (cuts[i+1] - cuts[i]);
+    }
+
+    inline double mapToDomain(double t, unsigned i) const {
+        return (1-t)*cuts[i] + t*cuts[i+1]; //same as: t * (cuts[i+1] - cuts[i]) + cuts[i]
+    }
+
+    //Offsets the piecewise domain
+    inline void offsetDomain(double o) {
+        assert(std::isfinite(o));
+        if(o != 0)
+            for(unsigned i = 0; i <= size(); i++)
+                cuts[i] += o;
+    }
+
+    //Scales the domain of the function by a value. 0 will result in an empty Piecewise.
+    inline void scaleDomain(double s) {
+        assert(s > 0);
+        if(s == 0) {
+            cuts.clear(); segs.clear();
+            return;
+        }
+        for(unsigned i = 0; i <= size(); i++)
+            cuts[i] *= s;
+    }
+
+    //Retrieves the domain in interval form
+    inline Interval domain() const { return Interval(cuts.front(), cuts.back()); }
+
+    //Transforms the domain into another interval
+    inline void setDomain(Interval dom) {
+        if(empty()) return;
+        /* dom can not be empty
+        if(dom.empty()) {
+            cuts.clear(); segs.clear();
+            return;
+        }*/
+        double cf = cuts.front();
+        double o = dom.min() - cf, s = dom.extent() / (cuts.back() - cf);
+        for(unsigned i = 0; i <= size(); i++)
+            cuts[i] = (cuts[i] - cf) * s + o;
+        //fix floating point precision errors.
+        cuts[0] = dom.min();
+        cuts[size()] = dom.max();
+    }
+
+    //Concatenates this Piecewise function with another, offsetting time of the other to match the end.
+    inline void concat(const Piecewise<T> &other) {
+        if(other.empty()) return;
+
+        if(empty()) {
+            cuts = other.cuts; segs = other.segs;
+            return;
+        }
+
+        segs.insert(segs.end(), other.segs.begin(), other.segs.end());
+        double t = cuts.back() - other.cuts.front();
+        cuts.reserve(cuts.size() + other.size());
+        for(unsigned i = 0; i < other.size(); i++)
+            push_cut(other.cuts[i + 1] + t);
+    }
+
+    //Like concat, but ensures continuity.
+    inline void continuousConcat(const Piecewise<T> &other) {
+        boost::function_requires<AddableConcept<typename T::output_type> >();
+        if(other.empty()) return;
+
+        if(empty()) { segs = other.segs; cuts = other.cuts; return; }
+
+        typename T::output_type y = segs.back().at1() - other.segs.front().at0();
+        double t = cuts.back() - other.cuts.front();
+        reserve(size() + other.size());
+        for(unsigned i = 0; i < other.size(); i++)
+            push(other[i] + y, other.cuts[i + 1] + t);
+    }
+
+    //returns true if the Piecewise<T> meets some basic invariants.
+    inline bool invariants() const {
+        // segs between cuts
+        if(!(segs.size() + 1 == cuts.size() || (segs.empty() && cuts.empty())))
+            return false;
+        // cuts in order
+        for(unsigned i = 0; i < segs.size(); i++)
+            if(cuts[i] >= cuts[i+1])
+                return false;
+        return true;
+    }
+
+};
+
+/**
+ *  ...
+ *  \return ...
+ *  \relates Piecewise
+ */
+template<typename T>
+inline typename FragmentConcept<T>::BoundsType bounds_fast(const Piecewise<T> &f) {
+    boost::function_requires<FragmentConcept<T> >();
+
+    if(f.empty()) return typename FragmentConcept<T>::BoundsType();
+    typename FragmentConcept<T>::BoundsType ret(bounds_fast(f[0]));
+    for(unsigned i = 1; i < f.size(); i++)
+        ret.unionWith(bounds_fast(f[i]));
+    return ret;
+}
+
+/**
+ *  ...
+ *  \return ...
+ *  \relates Piecewise
+ */
+template<typename T>
+inline typename FragmentConcept<T>::BoundsType bounds_exact(const Piecewise<T> &f) {
+    boost::function_requires<FragmentConcept<T> >();
+
+    if(f.empty()) return typename FragmentConcept<T>::BoundsType();
+    typename FragmentConcept<T>::BoundsType ret(bounds_exact(f[0]));
+    for(unsigned i = 1; i < f.size(); i++)
+        ret.unionWith(bounds_exact(f[i]));
+    return ret;
+}
+
+/**
+ *  ...
+ *  \return ...
+ *  \relates Piecewise
+ */
+template<typename T>
+inline typename FragmentConcept<T>::BoundsType bounds_local(const Piecewise<T> &f, const OptInterval &_m) {
+    boost::function_requires<FragmentConcept<T> >();
+
+    if(f.empty() || !_m) return typename FragmentConcept<T>::BoundsType();
+    Interval const &m = *_m;
+    if(m.isSingular()) return typename FragmentConcept<T>::BoundsType(f(m.min()));
+
+    unsigned fi = f.segN(m.min()), ti = f.segN(m.max());
+    double ft = f.segT(m.min(), fi), tt = f.segT(m.max(), ti);
+
+    if(fi == ti) return bounds_local(f[fi], Interval(ft, tt));
+
+    typename FragmentConcept<T>::BoundsType ret(bounds_local(f[fi], Interval(ft, 1.)));
+    for(unsigned i = fi + 1; i < ti; i++)
+        ret.unionWith(bounds_exact(f[i]));
+    if(tt != 0.) ret.unionWith(bounds_local(f[ti], Interval(0., tt)));
+
+    return ret;
+}
+
+/**
+ *  Returns a portion of a piece of a Piecewise<T>, given the piece's index and a to/from time.
+ *  \relates Piecewise
+ */
+template<typename T>
+T elem_portion(const Piecewise<T> &a, unsigned i, double from, double to) {
+    assert(i < a.size());
+    double rwidth = 1 / (a.cuts[i+1] - a.cuts[i]);
+    return portion( a[i], (from - a.cuts[i]) * rwidth, (to - a.cuts[i]) * rwidth );
+}
+
+/**Piecewise<T> partition(const Piecewise<T> &pw, std::vector<double> const &c);
+ * Further subdivides the Piecewise<T> such that there is a cut at every value in c.
+ * Precondition: c sorted lower to higher.
+ *
+ * //Given Piecewise<T> a and b:
+ * Piecewise<T> ac = a.partition(b.cuts);
+ * Piecewise<T> bc = b.partition(a.cuts);
+ * //ac.cuts should be equivalent to bc.cuts
+ *
+ * \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> partition(const Piecewise<T> &pw, std::vector<double> const &c) {
+    assert(pw.invariants());
+    if(c.empty()) return Piecewise<T>(pw);
+
+    Piecewise<T> ret = Piecewise<T>();
+    ret.reserve(c.size() + pw.cuts.size() - 1);
+
+    if(pw.empty()) {
+        ret.cuts = c;
+        for(unsigned i = 0; i < c.size() - 1; i++)
+            ret.push_seg(T());
+        return ret;
+    }
+
+    unsigned si = 0, ci = 0;     //Segment index, Cut index
+
+    //if the cuts have something earlier than the Piecewise<T>, add portions of the first segment
+    while(ci < c.size() && c[ci] < pw.cuts.front()) {
+        bool isLast = (ci == c.size()-1 || c[ci + 1] >= pw.cuts.front());
+        ret.push_cut(c[ci]);
+        ret.push_seg( elem_portion(pw, 0, c[ci], isLast ? pw.cuts.front() : c[ci + 1]) );
+        ci++;
+    }
+
+    ret.push_cut(pw.cuts[0]);
+    double prev = pw.cuts[0];    //previous cut
+    //Loop which handles cuts within the Piecewise<T> domain
+    //Should have the cuts = segs + 1 invariant
+    while(si < pw.size() && ci <= c.size()) {
+        if(ci == c.size() && prev <= pw.cuts[si]) { //cuts exhausted, straight copy the rest
+            ret.segs.insert(ret.segs.end(), pw.segs.begin() + si, pw.segs.end());
+            ret.cuts.insert(ret.cuts.end(), pw.cuts.begin() + si + 1, pw.cuts.end());
+            return ret;
+        }else if(ci == c.size() || c[ci] >= pw.cuts[si + 1]) {  //no more cuts within this segment, finalize
+            if(prev > pw.cuts[si]) {      //segment already has cuts, so portion is required
+                ret.push_seg(portion(pw[si], pw.segT(prev, si), 1.0));
+            } else {                     //plain copy is fine
+                ret.push_seg(pw[si]);
+            }
+            ret.push_cut(pw.cuts[si + 1]);
+            prev = pw.cuts[si + 1];
+            si++;
+        } else if(c[ci] == pw.cuts[si]){                  //coincident
+            //Already finalized the seg with the code immediately above
+            ci++;
+        } else {                                         //plain old subdivision
+            ret.push(elem_portion(pw, si, prev, c[ci]), c[ci]);
+            prev = c[ci];
+            ci++;
+        }
+    }
+
+    //input cuts extend further than this Piecewise<T>, extend the last segment.
+    while(ci < c.size()) {
+        if(c[ci] > prev) {
+            ret.push(elem_portion(pw, pw.size() - 1, prev, c[ci]), c[ci]);
+            prev = c[ci];
+        }
+        ci++;
+    }
+    return ret;
+}
+
+/**
+ *  Returns a Piecewise<T> with a defined domain of [min(from, to), max(from, to)].
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> portion(const Piecewise<T> &pw, double from, double to) {
+    if(pw.empty() || from == to) return Piecewise<T>();
+
+    Piecewise<T> ret;
+
+    double temp = from;
+    from = std::min(from, to);
+    to = std::max(temp, to);
+
+    unsigned i = pw.segN(from);
+    ret.push_cut(from);
+    if(i == pw.size() - 1 || to <= pw.cuts[i + 1]) {    //to/from inhabit the same segment
+        ret.push(elem_portion(pw, i, from, to), to);
+        return ret;
+    }
+    ret.push_seg(portion( pw[i], pw.segT(from, i), 1.0 ));
+    i++;
+    unsigned fi = pw.segN(to, i);
+    ret.reserve(fi - i + 1);
+    if (to == pw.cuts[fi]) fi-=1;
+
+    ret.segs.insert(ret.segs.end(), pw.segs.begin() + i, pw.segs.begin() + fi);  //copy segs
+    ret.cuts.insert(ret.cuts.end(), pw.cuts.begin() + i, pw.cuts.begin() + fi + 1);  //and their cuts
+
+    ret.push_seg( portion(pw[fi], 0.0, pw.segT(to, fi)));
+    if(to != ret.cuts.back()) ret.push_cut(to);
+    ret.invariants();
+    return ret;
+}
+
+//TODO: seems like these should be mutating
+/**
+ *  ...
+ *  \return ...
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> remove_short_cuts(Piecewise<T> const &f, double tol) {
+    if(f.empty()) return f;
+    Piecewise<T> ret;
+    ret.reserve(f.size());
+    ret.push_cut(f.cuts[0]);
+    for(unsigned i=0; i<f.size(); i++){
+        if (f.cuts[i+1]-f.cuts[i] >= tol || i==f.size()-1) {
+            ret.push(f[i], f.cuts[i+1]);
+        }
+    }
+    return ret;
+}
+
+//TODO: seems like these should be mutating
+/**
+ *  ...
+ *  \return ...
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> remove_short_cuts_extending(Piecewise<T> const &f, double tol) {
+    if(f.empty()) return f;
+    Piecewise<T> ret;
+    ret.reserve(f.size());
+    ret.push_cut(f.cuts[0]);
+    double last = f.cuts[0]; // last cut included
+    for(unsigned i=0; i<f.size(); i++){
+        if (f.cuts[i+1]-f.cuts[i] >= tol) {
+            ret.push(elem_portion(f, i, last, f.cuts[i+1]), f.cuts[i+1]);
+            last = f.cuts[i+1];
+        }
+    }
+    return ret;
+}
+
+/**
+ *  ...
+ *  \return ...
+ *  \relates Piecewise
+ */
+template<typename T>
+std::vector<double> roots(const Piecewise<T> &pw) {
+    std::vector<double> ret;
+    for(unsigned i = 0; i < pw.size(); i++) {
+        std::vector<double> sr = roots(pw[i]);
+        for (unsigned j = 0; j < sr.size(); j++) ret.push_back(sr[j] * (pw.cuts[i + 1] - pw.cuts[i]) + pw.cuts[i]);
+
+    }
+    return ret;
+}
+
+//IMPL: OffsetableConcept
+/**
+ *  ...
+ *  \return \f$ a + b = \f$
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> operator+(Piecewise<T> const &a, typename T::output_type b) {
+    boost::function_requires<OffsetableConcept<T> >();
+//TODO:empty
+    Piecewise<T> ret;
+    ret.segs.reserve(a.size());
+    ret.cuts = a.cuts;
+    for(unsigned i = 0; i < a.size();i++)
+        ret.push_seg(a[i] + b);
+    return ret;
+}
+template<typename T>
+Piecewise<T> operator-(Piecewise<T> const &a, typename T::output_type b) {
+    boost::function_requires<OffsetableConcept<T> >();
+//TODO: empty
+    Piecewise<T> ret;
+    ret.segs.reserve(a.size());
+    ret.cuts = a.cuts;
+    for(unsigned i = 0; i < a.size();i++)
+        ret.push_seg(a[i] - b);
+    return ret;
+}
+template<typename T>
+Piecewise<T>& operator+=(Piecewise<T>& a, typename T::output_type b) {
+    boost::function_requires<OffsetableConcept<T> >();
+
+    if(a.empty()) { a.push_cut(0.); a.push(T(b), 1.); return a; }
+
+    for(unsigned i = 0; i < a.size();i++)
+        a[i] += b;
+    return a;
+}
+template<typename T>
+Piecewise<T>& operator-=(Piecewise<T>& a, typename T::output_type b) {
+    boost::function_requires<OffsetableConcept<T> >();
+
+    if(a.empty()) { a.push_cut(0.); a.push(T(-b), 1.); return a; }
+
+    for(unsigned i = 0;i < a.size();i++)
+        a[i] -= b;
+    return a;
+}
+
+//IMPL: ScalableConcept
+/**
+ *  ...
+ *  \return \f$ -a = \f$
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> operator-(Piecewise<T> const &a) {
+    boost::function_requires<ScalableConcept<T> >();
+
+    Piecewise<T> ret;
+    ret.segs.reserve(a.size());
+    ret.cuts = a.cuts;
+    for(unsigned i = 0; i < a.size();i++)
+        ret.push_seg(- a[i]);
+    return ret;
+}
+/**
+ *  ...
+ *  \return \f$ a * b = \f$
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> operator*(Piecewise<T> const &a, double b) {
+    boost::function_requires<ScalableConcept<T> >();
+
+    if(a.empty()) return Piecewise<T>();
+
+    Piecewise<T> ret;
+    ret.segs.reserve(a.size());
+    ret.cuts = a.cuts;
+    for(unsigned i = 0; i < a.size();i++)
+        ret.push_seg(a[i] * b);
+    return ret;
+}
+/**
+ *  ...
+ *  \return \f$ a * b = \f$
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> operator*(Piecewise<T> const &a, T b) {
+    boost::function_requires<ScalableConcept<T> >();
+
+    if(a.empty()) return Piecewise<T>();
+
+    Piecewise<T> ret;
+    ret.segs.reserve(a.size());
+    ret.cuts = a.cuts;
+    for(unsigned i = 0; i < a.size();i++)
+        ret.push_seg(a[i] * b);
+    return ret;
+}
+/**
+ *  ...
+ *  \return \f$ a / b = \f$
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> operator/(Piecewise<T> const &a, double b) {
+    boost::function_requires<ScalableConcept<T> >();
+
+    //FIXME: b == 0?
+    if(a.empty()) return Piecewise<T>();
+
+    Piecewise<T> ret;
+    ret.segs.reserve(a.size());
+    ret.cuts = a.cuts;
+    for(unsigned i = 0; i < a.size();i++)
+        ret.push_seg(a[i] / b);
+    return ret;
+}
+template<typename T>
+Piecewise<T>& operator*=(Piecewise<T>& a, double b) {
+    boost::function_requires<ScalableConcept<T> >();
+
+    for(unsigned i = 0; i < a.size();i++)
+        a[i] *= b;
+    return a;
+}
+template<typename T>
+Piecewise<T>& operator/=(Piecewise<T>& a, double b) {
+    boost::function_requires<ScalableConcept<T> >();
+
+    //FIXME: b == 0?
+
+    for(unsigned i = 0; i < a.size();i++)
+        a[i] /= b;
+    return a;
+}
+
+//IMPL: AddableConcept
+/**
+ *  ...
+ *  \return \f$ a + b = \f$
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> operator+(Piecewise<T> const &a, Piecewise<T> const &b) {
+    boost::function_requires<AddableConcept<T> >();
+
+    Piecewise<T> pa = partition(a, b.cuts), pb = partition(b, a.cuts);
+    Piecewise<T> ret;
+    assert(pa.size() == pb.size());
+    ret.segs.reserve(pa.size());
+    ret.cuts = pa.cuts;
+    for (unsigned i = 0; i < pa.size(); i++)
+        ret.push_seg(pa[i] + pb[i]);
+    return ret;
+}
+/**
+ *  ...
+ *  \return \f$ a - b = \f$
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> operator-(Piecewise<T> const &a, Piecewise<T> const &b) {
+    boost::function_requires<AddableConcept<T> >();
+
+    Piecewise<T> pa = partition(a, b.cuts), pb = partition(b, a.cuts);
+    Piecewise<T> ret = Piecewise<T>();
+    assert(pa.size() == pb.size());
+    ret.segs.reserve(pa.size());
+    ret.cuts = pa.cuts;
+    for (unsigned i = 0; i < pa.size(); i++)
+        ret.push_seg(pa[i] - pb[i]);
+    return ret;
+}
+template<typename T>
+inline Piecewise<T>& operator+=(Piecewise<T> &a, Piecewise<T> const &b) {
+    a = a+b;
+    return a;
+}
+template<typename T>
+inline Piecewise<T>& operator-=(Piecewise<T> &a, Piecewise<T> const &b) {
+    a = a-b;
+    return a;
+}
+
+/**
+ *  ...
+ *  \return \f$ a \cdot b = \f$
+ *  \relates Piecewise
+ */
+template<typename T1,typename T2>
+Piecewise<T2> operator*(Piecewise<T1> const &a, Piecewise<T2> const &b) {
+    //function_requires<MultiplicableConcept<T1> >();
+    //function_requires<MultiplicableConcept<T2> >();
+
+    Piecewise<T1> pa = partition(a, b.cuts);
+    Piecewise<T2> pb = partition(b, a.cuts);
+    Piecewise<T2> ret = Piecewise<T2>();
+    assert(pa.size() == pb.size());
+    ret.segs.reserve(pa.size());
+    ret.cuts = pa.cuts;
+    for (unsigned i = 0; i < pa.size(); i++)
+        ret.push_seg(pa[i] * pb[i]);
+    return ret;
+}
+
+/**
+ *  ...
+ *  \return \f$ a \cdot b \f$
+ *  \relates Piecewise
+ */
+template<typename T>
+inline Piecewise<T>& operator*=(Piecewise<T> &a, Piecewise<T> const &b) {
+    a = a * b;
+    return a;
+}
+
+Piecewise<SBasis> divide(Piecewise<SBasis> const &a, Piecewise<SBasis> const &b, unsigned k);
+//TODO: replace divide(a,b,k) by divide(a,b,tol,k)?
+//TODO: atm, relative error is ~(tol/a)%. Find a way to make it independent of a.
+//Nota: the result is 'truncated' where b is smaller than 'zero': ~ a/max(b,zero).
+Piecewise<SBasis>
+divide(Piecewise<SBasis> const &a, Piecewise<SBasis> const &b, double tol, unsigned k, double zero=1.e-3);
+Piecewise<SBasis>
+divide(SBasis const &a, Piecewise<SBasis> const &b, double tol, unsigned k, double zero=1.e-3);
+Piecewise<SBasis>
+divide(Piecewise<SBasis> const &a, SBasis const &b, double tol, unsigned k, double zero=1.e-3);
+Piecewise<SBasis>
+divide(SBasis const &a, SBasis const &b, double tol, unsigned k, double zero=1.e-3);
+
+//Composition: functions called compose_* are pieces of compose that are factored out in pw.cpp.
+std::map<double,unsigned> compose_pullback(std::vector<double> const &cuts, SBasis const &g);
+int compose_findSegIdx(std::map<double,unsigned>::iterator  const &cut,
+                       std::map<double,unsigned>::iterator  const &next,
+                       std::vector<double>  const &levels,
+                       SBasis const &g);
+
+/**
+ *  ...
+ *  \return ...
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> compose(Piecewise<T> const &f, SBasis const &g){
+    /// \todo add concept check
+    Piecewise<T> result;
+    if (f.empty()) return result;
+    if (g.isZero()) return Piecewise<T>(f(0));
+    if (f.size()==1){
+        double t0 = f.cuts[0], width = f.cuts[1] - t0;
+        return (Piecewise<T>) compose(f.segs[0],compose(Linear(-t0 / width, (1-t0) / width), g));
+    }
+
+    //first check bounds...
+    Interval bs = *bounds_fast(g);
+    if (f.cuts.front() > bs.max()  || bs.min() > f.cuts.back()){
+        int idx = (bs.max() < f.cuts[1]) ? 0 : f.cuts.size()-2;
+        double t0 = f.cuts[idx], width = f.cuts[idx+1] - t0;
+        return (Piecewise<T>) compose(f.segs[idx],compose(Linear(-t0 / width, (1-t0) / width), g));
+    }
+
+    std::vector<double> levels;//we can forget first and last cuts...
+    levels.insert(levels.begin(),f.cuts.begin()+1,f.cuts.end()-1);
+    //TODO: use a std::vector<pairs<double,unsigned> > instead of a map<double,unsigned>.
+    std::map<double,unsigned> cuts_pb = compose_pullback(levels,g);
+
+    //-- Compose each piece of g with the relevant seg of f.
+    result.cuts.push_back(0.);
+    std::map<double,unsigned>::iterator cut=cuts_pb.begin();
+    std::map<double,unsigned>::iterator next=cut; next++;
+    while(next!=cuts_pb.end()){
+        //assert(std::abs(int((*cut).second-(*next).second))<1);
+        //TODO: find a way to recover from this error? the root finder missed some root;
+        //  the levels/variations of f might be too close/fast...
+        int idx = compose_findSegIdx(cut,next,levels,g);
+        double t0=(*cut).first;
+        double t1=(*next).first;
+
+        if (!are_near(t0,t1,EPSILON*EPSILON)) { // prevent adding cuts that are extremely close together and that may cause trouble with rounding e.g. when reversing the path
+            SBasis sub_g=compose(g, Linear(t0,t1));
+            sub_g=compose(Linear(-f.cuts[idx]/(f.cuts[idx+1]-f.cuts[idx]),
+                                 (1-f.cuts[idx])/(f.cuts[idx+1]-f.cuts[idx])),sub_g);
+            result.push(compose(f[idx],sub_g),t1);
+        }
+
+        cut++;
+        next++;
+    }
+    return(result);
+}
+
+/**
+ *  ...
+ *  \return ...
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> compose(Piecewise<T> const &f, Piecewise<SBasis> const &g){
+/// \todo add concept check
+  Piecewise<T> result;
+  for(unsigned i = 0; i < g.segs.size(); i++){
+      Piecewise<T> fgi=compose(f, g.segs[i]);
+      fgi.setDomain(Interval(g.cuts[i], g.cuts[i+1]));
+      result.concat(fgi);
+  }
+  return result;
+}
+
+/*
+Piecewise<D2<SBasis> > compose(D2<SBasis2d> const &sb2d, Piecewise<D2<SBasis> > const &pwd2sb){
+/// \todo add concept check
+  Piecewise<D2<SBasis> > result;
+  result.push_cut(0.);
+  for(unsigned i = 0; i < pwd2sb.size(); i++){
+     result.push(compose_each(sb2d,pwd2sb[i]),i+1);
+  }
+  return result;
+}*/
+
+/** Compose an SBasis with the inverse of another.
+ * WARNING: It's up to the user to check that the second SBasis is indeed
+ * invertible (i.e. strictly increasing or decreasing).
+ *  \return \f$ f \cdot g^{-1} \f$
+ *  \relates Piecewise
+ */
+Piecewise<SBasis> pw_compose_inverse(SBasis const &f, SBasis const &g, unsigned order, double zero);
+
+
+
+template <typename T>
+Piecewise<T> Piecewise<T>::operator()(SBasis f){return compose((*this),f);}
+template <typename T>
+Piecewise<T> Piecewise<T>::operator()(Piecewise<SBasis>f){return compose((*this),f);}
+
+/**
+ *  ...
+ *  \return ...
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> integral(Piecewise<T> const &a) {
+    Piecewise<T> result;
+    result.segs.resize(a.segs.size());
+    result.cuts = a.cuts;
+    typename T::output_type c = a.segs[0].at0();
+    for(unsigned i = 0; i < a.segs.size(); i++){
+        result.segs[i] = integral(a.segs[i])*(a.cuts[i+1]-a.cuts[i]);
+        result.segs[i]+= c-result.segs[i].at0();
+        c = result.segs[i].at1();
+    }
+    return result;
+}
+
+/**
+ *  ...
+ *  \return ...
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> derivative(Piecewise<T> const &a) {
+    Piecewise<T> result;
+    result.segs.resize(a.segs.size());
+    result.cuts = a.cuts;
+    for(unsigned i = 0; i < a.segs.size(); i++){
+        result.segs[i] = derivative(a.segs[i])/(a.cuts[i+1]-a.cuts[i]);
+    }
+    return result;
+}
+
+std::vector<double> roots(Piecewise<SBasis> const &f);
+
+std::vector<std::vector<double> >multi_roots(Piecewise<SBasis> const &f, std::vector<double> const &values);
+
+//TODO: implement level_sets directly for pwsb instead of sb (and derive it fo sb).
+//It should be faster than the reverse as the algorithm may jump over full cut intervals.
+std::vector<Interval> level_set(Piecewise<SBasis> const &f, Interval const &level, double tol=1e-5);
+std::vector<Interval> level_set(Piecewise<SBasis> const &f, double v, double vtol, double tol=1e-5);
+//std::vector<Interval> level_sets(Piecewise<SBasis> const &f, std::vector<Interval> const &levels, double tol=1e-5);
+//std::vector<Interval> level_sets(Piecewise<SBasis> const &f, std::vector<double> &v, double vtol, double tol=1e-5);
+
+
+/**
+ *  ...
+ *  \return ...
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> reverse(Piecewise<T> const &f) {
+    Piecewise<T> ret = Piecewise<T>();
+    ret.reserve(f.size());
+    double start = f.cuts[0];
+    double end = f.cuts.back();
+    for (unsigned i = 0; i < f.cuts.size(); i++) {
+        double x = f.cuts[f.cuts.size() - 1 - i];
+        ret.push_cut(end - (x - start));
+    }
+    for (unsigned i = 0; i < f.segs.size(); i++)
+        ret.push_seg(reverse(f[f.segs.size() - i - 1]));
+    return ret;
+}
+
+/**
+ *  Interpolates between a and b.
+ *  \return a if t = 0, b if t = 1, or an interpolation between a and b for t in [0,1]
+ *  \relates Piecewise
+ */
+template<typename T>
+Piecewise<T> lerp(double t, Piecewise<T> const &a, Piecewise<T> b) {
+    // Make sure both paths have the same number of segments and cuts at the same locations
+    b.setDomain(a.domain());
+    Piecewise<T> pA = partition(a, b.cuts);
+    Piecewise<T> pB = partition(b, a.cuts);
+
+    return (pA*(1-t)  +  pB*t);
+}
+
+}
+#endif //LIB2GEOM_SEEN_PIECEWISE_H
+/*
+  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 :