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diff --git a/src/3rdparty/2geom/include/2geom/piecewise.h b/src/3rdparty/2geom/include/2geom/piecewise.h new file mode 100644 index 0000000..e34df15 --- /dev/null +++ b/src/3rdparty/2geom/include/2geom/piecewise.h @@ -0,0 +1,945 @@ +/** @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 (double & j : sr) ret.push_back(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 : |