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+/** @file
+ * @brief Polynomial in symmetric power basis (S-basis)
+ *//*
+ * Authors:
+ * Nathan Hurst <njh@mail.csse.monash.edu.au>
+ * Michael Sloan <mgsloan@gmail.com>
+ *
+ * Copyright (C) 2006-2007 authors
+ *
+ * This library is free software; you can redistribute it and/or
+ * modify it either under the terms of the GNU Lesser General Public
+ * License version 2.1 as published by the Free Software Foundation
+ * (the "LGPL") or, at your option, under the terms of the Mozilla
+ * Public License Version 1.1 (the "MPL"). If you do not alter this
+ * notice, a recipient may use your version of this file under either
+ * the MPL or the LGPL.
+ *
+ * You should have received a copy of the LGPL along with this library
+ * in the file COPYING-LGPL-2.1; if not, write to the Free Software
+ * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * You should have received a copy of the MPL along with this library
+ * in the file COPYING-MPL-1.1
+ *
+ * The contents of this file are subject to the Mozilla Public License
+ * Version 1.1 (the "License"); you may not use this file except in
+ * compliance with the License. You may obtain a copy of the License at
+ * http://www.mozilla.org/MPL/
+ *
+ * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
+ * OF ANY KIND, either express or implied. See the LGPL or the MPL for
+ * the specific language governing rights and limitations.
+ */
+
+#ifndef LIB2GEOM_SEEN_SBASIS_H
+#define LIB2GEOM_SEEN_SBASIS_H
+#include <vector>
+#include <cassert>
+#include <iostream>
+
+#include <2geom/linear.h>
+#include <2geom/interval.h>
+#include <2geom/utils.h>
+#include <2geom/exception.h>
+
+//#define USE_SBASISN 1
+
+
+#if defined(USE_SBASIS_OF)
+
+#include "sbasis-of.h"
+
+#elif defined(USE_SBASISN)
+
+#include "sbasisN.h"
+namespace Geom{
+
+/*** An empty SBasis is identically 0. */
+class SBasis : public SBasisN<1>;
+
+};
+#else
+
+namespace Geom {
+
+/**
+ * @brief Polynomial in symmetric power basis
+ * @ingroup Fragments
+ */
+class SBasis {
+ std::vector<Linear> d;
+ void push_back(Linear const&l) { d.push_back(l); }
+
+public:
+ // As part of our migration away from SBasis isa vector we provide this minimal set of vector interface methods.
+ size_t size() const {return d.size();}
+ typedef std::vector<Linear>::iterator iterator;
+ typedef std::vector<Linear>::const_iterator const_iterator;
+ Linear operator[](unsigned i) const {
+ return d[i];
+ }
+ Linear& operator[](unsigned i) { return d.at(i); }
+ const_iterator begin() const { return d.begin();}
+ const_iterator end() const { return d.end();}
+ iterator begin() { return d.begin();}
+ iterator end() { return d.end();}
+ bool empty() const { return d.size() == 1 && d[0][0] == 0 && d[0][1] == 0; }
+ Linear &back() {return d.back();}
+ Linear const &back() const {return d.back();}
+ void pop_back() {
+ if (d.size() > 1) {
+ d.pop_back();
+ } else {
+ d[0][0] = 0;
+ d[0][1] = 0;
+ }
+ }
+ void resize(unsigned n) { d.resize(std::max<unsigned>(n, 1));}
+ void resize(unsigned n, Linear const& l) { d.resize(std::max<unsigned>(n, 1), l);}
+ void reserve(unsigned n) { d.reserve(n);}
+ void clear() {
+ d.resize(1);
+ d[0][0] = 0;
+ d[0][1] = 0;
+ }
+ void insert(iterator before, const_iterator src_begin, const_iterator src_end) { d.insert(before, src_begin, src_end);}
+ Linear& at(unsigned i) { return d.at(i);}
+ //void insert(Linear* before, int& n, Linear const &l) { d.insert(std::vector<Linear>::iterator(before), n, l);}
+ bool operator==(SBasis const&B) const { return d == B.d;}
+ bool operator!=(SBasis const&B) const { return d != B.d;}
+
+ SBasis()
+ : d(1, Linear(0, 0))
+ {}
+ explicit SBasis(double a)
+ : d(1, Linear(a, a))
+ {}
+ explicit SBasis(double a, double b)
+ : d(1, Linear(a, b))
+ {}
+ SBasis(SBasis const &a)
+ : d(a.d)
+ {}
+ SBasis(std::vector<Linear> const &ls)
+ : d(ls)
+ {}
+ SBasis(Linear const &bo)
+ : d(1, bo)
+ {}
+ SBasis(Linear* bo)
+ : d(1, bo ? *bo : Linear(0, 0))
+ {}
+ explicit SBasis(size_t n, Linear const&l) : d(n, l) {}
+
+ SBasis(Coord c0, Coord c1, Coord c2, Coord c3)
+ : d(2)
+ {
+ d[0][0] = c0;
+ d[1][0] = c1;
+ d[1][1] = c2;
+ d[0][1] = c3;
+ }
+ SBasis(Coord c0, Coord c1, Coord c2, Coord c3, Coord c4, Coord c5)
+ : d(3)
+ {
+ d[0][0] = c0;
+ d[1][0] = c1;
+ d[2][0] = c2;
+ d[2][1] = c3;
+ d[1][1] = c4;
+ d[0][1] = c5;
+ }
+ SBasis(Coord c0, Coord c1, Coord c2, Coord c3, Coord c4, Coord c5,
+ Coord c6, Coord c7)
+ : d(4)
+ {
+ d[0][0] = c0;
+ d[1][0] = c1;
+ d[2][0] = c2;
+ d[3][0] = c3;
+ d[3][1] = c4;
+ d[2][1] = c5;
+ d[1][1] = c6;
+ d[0][1] = c7;
+ }
+ SBasis(Coord c0, Coord c1, Coord c2, Coord c3, Coord c4, Coord c5,
+ Coord c6, Coord c7, Coord c8, Coord c9)
+ : d(5)
+ {
+ d[0][0] = c0;
+ d[1][0] = c1;
+ d[2][0] = c2;
+ d[3][0] = c3;
+ d[4][0] = c4;
+ d[4][1] = c5;
+ d[3][1] = c6;
+ d[2][1] = c7;
+ d[1][1] = c8;
+ d[0][1] = c9;
+ }
+
+ // construct from a sequence of coefficients
+ template <typename Iter>
+ SBasis(Iter first, Iter last) {
+ assert(std::distance(first, last) % 2 == 0);
+ assert(std::distance(first, last) >= 2);
+ for (; first != last; ++first) {
+ --last;
+ push_back(Linear(*first, *last));
+ }
+ }
+
+ //IMPL: FragmentConcept
+ typedef double output_type;
+ inline bool isZero(double eps=EPSILON) const {
+ assert(size() > 0);
+ for(unsigned i = 0; i < size(); i++) {
+ if(!(*this)[i].isZero(eps)) return false;
+ }
+ return true;
+ }
+ inline bool isConstant(double eps=EPSILON) const {
+ assert(size() > 0);
+ if(!(*this)[0].isConstant(eps)) return false;
+ for (unsigned i = 1; i < size(); i++) {
+ if(!(*this)[i].isZero(eps)) return false;
+ }
+ return true;
+ }
+
+ bool isFinite() const;
+ inline Coord at0() const { return (*this)[0][0]; }
+ inline Coord &at0() { return (*this)[0][0]; }
+ inline Coord at1() const { return (*this)[0][1]; }
+ inline Coord &at1() { return (*this)[0][1]; }
+
+ int degreesOfFreedom() const { return size()*2;}
+
+ double valueAt(double t) const {
+ assert(size() > 0);
+ double s = t*(1-t);
+ double p0 = 0, p1 = 0;
+ for(unsigned k = size(); k > 0; k--) {
+ const Linear &lin = (*this)[k-1];
+ p0 = p0*s + lin[0];
+ p1 = p1*s + lin[1];
+ }
+ return (1-t)*p0 + t*p1;
+ }
+ //double valueAndDerivative(double t, double &der) const {
+ //}
+ double operator()(double t) const {
+ return valueAt(t);
+ }
+
+ std::vector<double> valueAndDerivatives(double t, unsigned n) const;
+
+ SBasis toSBasis() const { return SBasis(*this); }
+
+ double tailError(unsigned tail) const;
+
+// compute f(g)
+ SBasis operator()(SBasis const & g) const;
+
+//MUTATOR PRISON
+ //remove extra zeros
+ void normalize() {
+ while(size() > 1 && back().isZero(0))
+ pop_back();
+ }
+
+ void truncate(unsigned k) { if(k < size()) resize(std::max<size_t>(k, 1)); }
+private:
+ void derive(); // in place version
+};
+
+//TODO: figure out how to stick this in linear, while not adding an sbasis dep
+inline SBasis Linear::toSBasis() const { return SBasis(*this); }
+
+//implemented in sbasis-roots.cpp
+OptInterval bounds_exact(SBasis const &a);
+OptInterval bounds_fast(SBasis const &a, int order = 0);
+OptInterval bounds_local(SBasis const &a, const OptInterval &t, int order = 0);
+
+/** Returns a function which reverses the domain of a.
+ \param a sbasis function
+ \relates SBasis
+
+useful for reversing a parameteric curve.
+*/
+inline SBasis reverse(SBasis const &a) {
+ SBasis result(a.size(), Linear());
+
+ for(unsigned k = 0; k < a.size(); k++)
+ result[k] = reverse(a[k]);
+ return result;
+}
+
+//IMPL: ScalableConcept
+inline SBasis operator-(const SBasis& p) {
+ if(p.isZero()) return SBasis();
+ SBasis result(p.size(), Linear());
+
+ for(unsigned i = 0; i < p.size(); i++) {
+ result[i] = -p[i];
+ }
+ return result;
+}
+SBasis operator*(SBasis const &a, double k);
+inline SBasis operator*(double k, SBasis const &a) { return a*k; }
+inline SBasis operator/(SBasis const &a, double k) { return a*(1./k); }
+SBasis& operator*=(SBasis& a, double b);
+inline SBasis& operator/=(SBasis& a, double b) { return (a*=(1./b)); }
+
+//IMPL: AddableConcept
+SBasis operator+(const SBasis& a, const SBasis& b);
+SBasis operator-(const SBasis& a, const SBasis& b);
+SBasis& operator+=(SBasis& a, const SBasis& b);
+SBasis& operator-=(SBasis& a, const SBasis& b);
+
+//TODO: remove?
+/*inline SBasis operator+(const SBasis & a, Linear const & b) {
+ if(b.isZero()) return a;
+ if(a.isZero()) return b;
+ SBasis result(a);
+ result[0] += b;
+ return result;
+}
+inline SBasis operator-(const SBasis & a, Linear const & b) {
+ if(b.isZero()) return a;
+ SBasis result(a);
+ result[0] -= b;
+ return result;
+}
+inline SBasis& operator+=(SBasis& a, const Linear& b) {
+ if(a.isZero())
+ a.push_back(b);
+ else
+ a[0] += b;
+ return a;
+}
+inline SBasis& operator-=(SBasis& a, const Linear& b) {
+ if(a.isZero())
+ a.push_back(-b);
+ else
+ a[0] -= b;
+ return a;
+ }*/
+
+//IMPL: OffsetableConcept
+inline SBasis operator+(const SBasis & a, double b) {
+ if(a.isZero()) return Linear(b, b);
+ SBasis result(a);
+ result[0] += b;
+ return result;
+}
+inline SBasis operator-(const SBasis & a, double b) {
+ if(a.isZero()) return Linear(-b, -b);
+ SBasis result(a);
+ result[0] -= b;
+ return result;
+}
+inline SBasis& operator+=(SBasis& a, double b) {
+ if(a.isZero())
+ a = SBasis(Linear(b,b));
+ else
+ a[0] += b;
+ return a;
+}
+inline SBasis& operator-=(SBasis& a, double b) {
+ if(a.isZero())
+ a = SBasis(Linear(-b,-b));
+ else
+ a[0] -= b;
+ return a;
+}
+
+SBasis shift(SBasis const &a, int sh);
+SBasis shift(Linear const &a, int sh);
+
+inline SBasis truncate(SBasis const &a, unsigned terms) {
+ SBasis c;
+ c.insert(c.begin(), a.begin(), a.begin() + std::min(terms, (unsigned)a.size()));
+ return c;
+}
+
+SBasis multiply(SBasis const &a, SBasis const &b);
+// This performs a multiply and accumulate operation in about the same time as multiply. return a*b + c
+SBasis multiply_add(SBasis const &a, SBasis const &b, SBasis c);
+
+SBasis integral(SBasis const &c);
+SBasis derivative(SBasis const &a);
+
+SBasis sqrt(SBasis const &a, int k);
+
+// return a kth order approx to 1/a)
+SBasis reciprocal(Linear const &a, int k);
+SBasis divide(SBasis const &a, SBasis const &b, int k);
+
+inline SBasis operator*(SBasis const & a, SBasis const & b) {
+ return multiply(a, b);
+}
+
+inline SBasis& operator*=(SBasis& a, SBasis const & b) {
+ a = multiply(a, b);
+ return a;
+}
+
+/** Returns the degree of the first non zero coefficient.
+ \param a sbasis function
+ \param tol largest abs val considered 0
+ \return first non zero coefficient
+ \relates SBasis
+*/
+inline unsigned
+valuation(SBasis const &a, double tol=0){
+ unsigned val=0;
+ while( val<a.size() &&
+ fabs(a[val][0])<tol &&
+ fabs(a[val][1])<tol )
+ val++;
+ return val;
+}
+
+// a(b(t))
+SBasis compose(SBasis const &a, SBasis const &b);
+SBasis compose(SBasis const &a, SBasis const &b, unsigned k);
+SBasis inverse(SBasis a, int k);
+//compose_inverse(f,g)=compose(f,inverse(g)), but is numerically more stable in some good cases...
+//TODO: requires g(0)=0 & g(1)=1 atm. generalization should be obvious.
+SBasis compose_inverse(SBasis const &f, SBasis const &g, unsigned order=2, double tol=1e-3);
+
+/** Returns the sbasis on domain [0,1] that was t on [from, to]
+ \param t sbasis function
+ \param from,to interval
+ \return sbasis
+ \relates SBasis
+*/
+SBasis portion(const SBasis &t, double from, double to);
+inline SBasis portion(const SBasis &t, Interval const &ivl) { return portion(t, ivl.min(), ivl.max()); }
+
+// compute f(g)
+inline SBasis
+SBasis::operator()(SBasis const & g) const {
+ return compose(*this, g);
+}
+
+inline std::ostream &operator<< (std::ostream &out_file, const Linear &bo) {
+ out_file << "{" << bo[0] << ", " << bo[1] << "}";
+ return out_file;
+}
+
+inline std::ostream &operator<< (std::ostream &out_file, const SBasis & p) {
+ for(unsigned i = 0; i < p.size(); i++) {
+ if (i != 0) {
+ out_file << " + ";
+ }
+ out_file << p[i] << "s^" << i;
+ }
+ return out_file;
+}
+
+// These are deprecated, use sbasis-math.h versions if possible
+SBasis sin(Linear bo, int k);
+SBasis cos(Linear bo, int k);
+
+std::vector<double> roots(SBasis const & s);
+std::vector<double> roots(SBasis const & s, Interval const inside);
+std::vector<std::vector<double> > multi_roots(SBasis const &f,
+ std::vector<double> const &levels,
+ double htol=1e-7,
+ double vtol=1e-7,
+ double a=0,
+ double b=1);
+
+//--------- Levelset like functions -----------------------------------------------------
+
+/** Solve f(t) = v +/- tolerance. The collection of intervals where
+ * v - vtol <= f(t) <= v+vtol
+ * is returned (with a precision tol on the boundaries).
+ \param f sbasis function
+ \param level the value of v.
+ \param vtol: error tolerance on v.
+ \param a, b limit search on domain [a,b]
+ \param tol: tolerance on the result bounds.
+ \returns a vector of intervals.
+*/
+std::vector<Interval> level_set (SBasis const &f,
+ double level,
+ double vtol = 1e-5,
+ double a=0.,
+ double b=1.,
+ double tol = 1e-5);
+
+/** Solve f(t)\in I=[u,v], which defines a collection of intervals (J_k). More precisely,
+ * a collection (J'_k) is returned with J'_k = J_k up to a given tolerance.
+ \param f sbasis function
+ \param level: the given interval of deisred values for f.
+ \param a, b limit search on domain [a,b]
+ \param tol: tolerance on the bounds of the result.
+ \returns a vector of intervals.
+*/
+std::vector<Interval> level_set (SBasis const &f,
+ Interval const &level,
+ double a=0.,
+ double b=1.,
+ double tol = 1e-5);
+
+/** 'Solve' f(t) = v +/- tolerance for several values of v at once.
+ \param f sbasis function
+ \param levels vector of values, that should be sorted.
+ \param vtol: error tolerance on v.
+ \param a, b limit search on domain [a,b]
+ \param tol: the bounds of the returned intervals are exact up to that tolerance.
+ \returns a vector of vectors of intervals.
+*/
+std::vector<std::vector<Interval> > level_sets (SBasis const &f,
+ std::vector<double> const &levels,
+ double a=0.,
+ double b=1.,
+ double vtol = 1e-5,
+ double tol = 1e-5);
+
+/** 'Solve' f(t)\in I=[u,v] for several intervals I at once.
+ \param f sbasis function
+ \param levels vector of 'y' intervals, that should be disjoints and sorted.
+ \param a, b limit search on domain [a,b]
+ \param tol: the bounds of the returned intervals are exact up to that tolerance.
+ \returns a vector of vectors of intervals.
+*/
+std::vector<std::vector<Interval> > level_sets (SBasis const &f,
+ std::vector<Interval> const &levels,
+ double a=0.,
+ double b=1.,
+ double tol = 1e-5);
+
+}
+#endif
+
+/*
+ Local Variables:
+ mode:c++
+ c-file-style:"stroustrup"
+ c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
+ indent-tabs-mode:nil
+ fill-column:99
+ End:
+*/
+// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:fileencoding=utf-8:textwidth=99 :
+#endif
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