/*
* sbasis.cpp - S-power basis function class + supporting classes
*
* 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.
*/
#include <cmath>
#include <2geom/sbasis.h>
#include <2geom/math-utils.h>
namespace Geom{
/** bound the error from term truncation
\param tail first term to chop
\returns the largest possible error this truncation could give
*/
double SBasis::tailError(unsigned tail) const {
Interval bs = *bounds_fast(*this, tail);
return std::max(fabs(bs.min()),fabs(bs.max()));
}
/** test all coefficients are finite
*/
bool SBasis::isFinite() const {
for(unsigned i = 0; i < size(); i++) {
if(!(*this)[i].isFinite())
return false;
}
return true;
}
/** Compute the value and the first n derivatives
\param t position to evaluate
\param n number of derivatives (not counting value)
\returns a vector with the value and the n derivative evaluations
There is an elegant way to compute the value and n derivatives for a polynomial using a variant of horner's rule. Someone will someday work out how for sbasis.
*/
std::vector<double> SBasis::valueAndDerivatives(double t, unsigned n) const {
std::vector<double> ret(n+1);
ret[0] = valueAt(t);
SBasis tmp = *this;
for(unsigned i = 1; i < n+1; i++) {
tmp.derive();
ret[i] = tmp.valueAt(t);
}
return ret;
}
/** Compute the pointwise sum of a and b (Exact)
\param a,b sbasis functions
\returns sbasis equal to a+b
*/
SBasis operator+(const SBasis& a, const SBasis& b) {
const unsigned out_size = std::max(a.size(), b.size());
const unsigned min_size = std::min(a.size(), b.size());
SBasis result(out_size, Linear());
for(unsigned i = 0; i < min_size; i++) {
result[i] = a[i] + b[i];
}
for(unsigned i = min_size; i < a.size(); i++)
result[i] = a[i];
for(unsigned i = min_size; i < b.size(); i++)
result[i] = b[i];
assert(result.size() == out_size);
return result;
}
/** Compute the pointwise difference of a and b (Exact)
\param a,b sbasis functions
\returns sbasis equal to a-b
*/
SBasis operator-(const SBasis& a, const SBasis& b) {
const unsigned out_size = std::max(a.size(), b.size());
const unsigned min_size = std::min(a.size(), b.size());
SBasis result(out_size, Linear());
for(unsigned i = 0; i < min_size; i++) {
result[i] = a[i] - b[i];
}
for(unsigned i = min_size; i < a.size(); i++)
result[i] = a[i];
for(unsigned i = min_size; i < b.size(); i++)
result[i] = -b[i];
assert(result.size() == out_size);
return result;
}
/** Compute the pointwise sum of a and b and store in a (Exact)
\param a,b sbasis functions
\returns sbasis equal to a+b
*/
SBasis& operator+=(SBasis& a, const SBasis& b) {
const unsigned out_size = std::max(a.size(), b.size());
const unsigned min_size = std::min(a.size(), b.size());
a.resize(out_size);
for(unsigned i = 0; i < min_size; i++)
a[i] += b[i];
for(unsigned i = min_size; i < b.size(); i++)
a[i] = b[i];
assert(a.size() == out_size);
return a;
}
/** Compute the pointwise difference of a and b and store in a (Exact)
\param a,b sbasis functions
\returns sbasis equal to a-b
*/
SBasis& operator-=(SBasis& a, const SBasis& b) {
const unsigned out_size = std::max(a.size(), b.size());
const unsigned min_size = std::min(a.size(), b.size());
a.resize(out_size);
for(unsigned i = 0; i < min_size; i++)
a[i] -= b[i];
for(unsigned i = min_size; i < b.size(); i++)
a[i] = -b[i];
assert(a.size() == out_size);
return a;
}
/** Compute the pointwise product of a and b (Exact)
\param a,b sbasis functions
\returns sbasis equal to a*b
*/
SBasis operator*(SBasis const &a, double k) {
SBasis c(a.size(), Linear());
for(unsigned i = 0; i < a.size(); i++)
c[i] = a[i] * k;
return c;
}
/** Compute the pointwise product of a and b and store the value in a (Exact)
\param a,b sbasis functions
\returns sbasis equal to a*b
*/
SBasis& operator*=(SBasis& a, double b) {
if (a.isZero()) return a;
if (b == 0)
a.clear();
else
for(unsigned i = 0; i < a.size(); i++)
a[i] *= b;
return a;
}
/** multiply a by x^sh in place (Exact)
\param a sbasis function
\param sh power
\returns a
*/
SBasis shift(SBasis const &a, int sh) {
size_t n = a.size()+sh;
SBasis c(n, Linear());
size_t m = std::max(0, sh);
for(int i = 0; i < sh; i++)
c[i] = Linear(0,0);
for(size_t i = m, j = std::max(0,-sh); i < n; i++, j++)
c[i] = a[j];
return c;
}
/** multiply a by x^sh (Exact)
\param a linear function
\param sh power
\returns a* x^sh
*/
SBasis shift(Linear const &a, int sh) {
size_t n = 1+sh;
SBasis c(n, Linear());
for(int i = 0; i < sh; i++)
c[i] = Linear(0,0);
if(sh >= 0)
c[sh] = a;
return c;
}
#if 0
SBasis multiply(SBasis const &a, SBasis const &b) {
// c = {a0*b0 - shift(1, a.Tri*b.Tri), a1*b1 - shift(1, a.Tri*b.Tri)}
// shift(1, a.Tri*b.Tri)
SBasis c(a.size() + b.size(), Linear(0,0));
if(a.isZero() || b.isZero())
return c;
for(unsigned j = 0; j < b.size(); j++) {
for(unsigned i = j; i < a.size()+j; i++) {
double tri = b[j].tri()*a[i-j].tri();
c[i+1/*shift*/] += Linear(-tri);
}
}
for(unsigned j = 0; j < b.size(); j++) {
for(unsigned i = j; i < a.size()+j; i++) {
for(unsigned dim = 0; dim < 2; dim++)
c[i][dim] += b[j][dim]*a[i-j][dim];
}
}
c.normalize();
//assert(!(0 == c.back()[0] && 0 == c.back()[1]));
return c;
}
#else
/** Compute the pointwise product of a and b adding c (Exact)
\param a,b,c sbasis functions
\returns sbasis equal to a*b+c
The added term is almost free
*/
SBasis multiply_add(SBasis const &a, SBasis const &b, SBasis c) {
if(a.isZero() || b.isZero())
return c;
c.resize(a.size() + b.size(), Linear(0,0));
for(unsigned j = 0; j < b.size(); j++) {
for(unsigned i = j; i < a.size()+j; i++) {
double tri = b[j].tri()*a[i-j].tri();
c[i+1/*shift*/] += Linear(-tri);
}
}
for(unsigned j = 0; j < b.size(); j++) {
for(unsigned i = j; i < a.size()+j; i++) {
for(unsigned dim = 0; dim < 2; dim++)
c[i][dim] += b[j][dim]*a[i-j][dim];
}
}
c.normalize();
//assert(!(0 == c.back()[0] && 0 == c.back()[1]));
return c;
}
/** Compute the pointwise product of a and b (Exact)
\param a,b sbasis functions
\returns sbasis equal to a*b
*/
SBasis multiply(SBasis const &a, SBasis const &b) {
if(a.isZero() || b.isZero()) {
SBasis c(1, Linear(0,0));
return c;
}
SBasis c(a.size() + b.size(), Linear(0,0));
return multiply_add(a, b, c);
}
#endif
/** Compute the integral of a (Exact)
\param a sbasis functions
\returns sbasis integral(a)
*/
SBasis integral(SBasis const &c) {
SBasis a;
a.resize(c.size() + 1, Linear(0,0));
a[0] = Linear(0,0);
for(unsigned k = 1; k < c.size() + 1; k++) {
double ahat = -c[k-1].tri()/(2*k);
a[k][0] = a[k][1] = ahat;
}
double aTri = 0;
for(int k = c.size()-1; k >= 0; k--) {
aTri = (c[k].hat() + (k+1)*aTri/2)/(2*k+1);
a[k][0] -= aTri/2;
a[k][1] += aTri/2;
}
a.normalize();
return a;
}
/** Compute the derivative of a (Exact)
\param a sbasis functions
\returns sbasis da/dt
*/
SBasis derivative(SBasis const &a) {
SBasis c;
c.resize(a.size(), Linear(0,0));
if(a.isZero())
return c;
for(unsigned k = 0; k < a.size()-1; k++) {
double d = (2*k+1)*(a[k][1] - a[k][0]);
c[k][0] = d + (k+1)*a[k+1][0];
c[k][1] = d - (k+1)*a[k+1][1];
}
int k = a.size()-1;
double d = (2*k+1)*(a[k][1] - a[k][0]);
if (d == 0 && k > 0) {
c.pop_back();
} else {
c[k][0] = d;
c[k][1] = d;
}
return c;
}
/** Compute the derivative of this inplace (Exact)
*/
void SBasis::derive() { // in place version
if(isZero()) return;
for(unsigned k = 0; k < size()-1; k++) {
double d = (2*k+1)*((*this)[k][1] - (*this)[k][0]);
(*this)[k][0] = d + (k+1)*(*this)[k+1][0];
(*this)[k][1] = d - (k+1)*(*this)[k+1][1];
}
int k = size()-1;
double d = (2*k+1)*((*this)[k][1] - (*this)[k][0]);
if (d == 0 && k > 0) {
pop_back();
} else {
(*this)[k][0] = d;
(*this)[k][1] = d;
}
}
/** Compute the sqrt of a
\param a sbasis functions
\returns sbasis \f[ \sqrt{a} \f]
It is recommended to use the piecewise version unless you have good reason.
TODO: convert int k to unsigned k, and remove cast
*/
SBasis sqrt(SBasis const &a, int k) {
SBasis c;
if(a.isZero() || k == 0)
return c;
c.resize(k, Linear(0,0));
c[0] = Linear(std::sqrt(a[0][0]), std::sqrt(a[0][1]));
SBasis r = a - multiply(c, c); // remainder
for(unsigned i = 1; i <= (unsigned)k && i<r.size(); i++) {
Linear ci(r[i][0]/(2*c[0][0]), r[i][1]/(2*c[0][1]));
SBasis cisi = shift(ci, i);
r -= multiply(shift((c*2 + cisi), i), SBasis(ci));
r.truncate(k+1);
c += cisi;
if(r.tailError(i) == 0) // if exact
break;
}
return c;
}
/** Compute the recpirocal of a
\param a sbasis functions
\returns sbasis 1/a
It is recommended to use the piecewise version unless you have good reason.
*/
SBasis reciprocal(Linear const &a, int k) {
SBasis c;
assert(!a.isZero());
c.resize(k, Linear(0,0));
double r_s0 = (a.tri()*a.tri())/(-a[0]*a[1]);
double r_s0k = 1;
for(unsigned i = 0; i < (unsigned)k; i++) {
c[i] = Linear(r_s0k/a[0], r_s0k/a[1]);
r_s0k *= r_s0;
}
return c;
}
/** Compute a / b to k terms
\param a,b sbasis functions
\returns sbasis a/b
It is recommended to use the piecewise version unless you have good reason.
*/
SBasis divide(SBasis const &a, SBasis const &b, int k) {
SBasis c;
assert(!a.isZero());
SBasis r = a; // remainder
k++;
r.resize(k, Linear(0,0));
c.resize(k, Linear(0,0));
for(unsigned i = 0; i < (unsigned)k; i++) {
Linear ci(r[i][0]/b[0][0], r[i][1]/b[0][1]); //H0
c[i] += ci;
r -= shift(multiply(ci,b), i);
r.truncate(k+1);
if(r.tailError(i) == 0) // if exact
break;
}
return c;
}
/** Compute a composed with b
\param a,b sbasis functions
\returns sbasis a(b(t))
return a0 + s(a1 + s(a2 +... where s = (1-u)u; ak =(1 - u)a^0_k + ua^1_k
*/
SBasis compose(SBasis const &a, SBasis const &b) {
SBasis s = multiply((SBasis(Linear(1,1))-b), b);
SBasis r;
for(int i = a.size()-1; i >= 0; i--) {
r = multiply_add(r, s, SBasis(Linear(a[i][0])) - b*a[i][0] + b*a[i][1]);
}
return r;
}
/** Compute a composed with b to k terms
\param a,b sbasis functions
\returns sbasis a(b(t))
return a0 + s(a1 + s(a2 +... where s = (1-u)u; ak =(1 - u)a^0_k + ua^1_k
*/
SBasis compose(SBasis const &a, SBasis const &b, unsigned k) {
SBasis s = multiply((SBasis(Linear(1,1))-b), b);
SBasis r;
for(int i = a.size()-1; i >= 0; i--) {
r = multiply_add(r, s, SBasis(Linear(a[i][0])) - b*a[i][0] + b*a[i][1]);
}
r.truncate(k);
return r;
}
SBasis portion(const SBasis &t, double from, double to) {
double fv = t.valueAt(from);
double tv = t.valueAt(to);
SBasis ret = compose(t, Linear(from, to));
ret.at0() = fv;
ret.at1() = tv;
return ret;
}
/*
Inversion algorithm. The notation is certainly very misleading. The
pseudocode should say:
c(v) := 0
r(u) := r_0(u) := u
for i:=0 to k do
c_i(v) := H_0(r_i(u)/(t_1)^i; u)
c(v) := c(v) + c_i(v)*t^i
r(u) := r(u) ? c_i(u)*(t(u))^i
endfor
*/
//#define DEBUG_INVERSION 1
/** find the function a^-1 such that a^-1 composed with a to k terms is the identity function
\param a sbasis function
\returns sbasis a^-1 s.t. a^-1(a(t)) = 1
The function must have 'unit range'("a00 = 0 and a01 = 1") and be monotonic.
*/
SBasis inverse(SBasis a, int k) {
assert(a.size() > 0);
double a0 = a[0][0];
if(a0 != 0) {
a -= a0;
}
double a1 = a[0][1];
assert(a1 != 0); // not invertable.
if(a1 != 1) {
a /= a1;
}
SBasis c(k, Linear()); // c(v) := 0
if(a.size() >= 2 && k == 2) {
c[0] = Linear(0,1);
Linear t1(1+a[1][0], 1-a[1][1]); // t_1
c[1] = Linear(-a[1][0]/t1[0], -a[1][1]/t1[1]);
} else if(a.size() >= 2) { // non linear
SBasis r = Linear(0,1); // r(u) := r_0(u) := u
Linear t1(1./(1+a[1][0]), 1./(1-a[1][1])); // 1./t_1
Linear one(1,1);
Linear t1i = one; // t_1^0
SBasis one_minus_a = SBasis(one) - a;
SBasis t = multiply(one_minus_a, a); // t(u)
SBasis ti(one); // t(u)^0
#ifdef DEBUG_INVERSION
std::cout << "a=" << a << std::endl;
std::cout << "1-a=" << one_minus_a << std::endl;
std::cout << "t1=" << t1 << std::endl;
//assert(t1 == t[1]);
#endif
//c.resize(k+1, Linear(0,0));
for(unsigned i = 0; i < (unsigned)k; i++) { // for i:=0 to k do
#ifdef DEBUG_INVERSION
std::cout << "-------" << i << ": ---------" <<std::endl;
std::cout << "r=" << r << std::endl
<< "c=" << c << std::endl
<< "ti=" << ti << std::endl
<< std::endl;
#endif
if(r.size() <= i) // ensure enough space in the remainder, probably not needed
r.resize(i+1, Linear(0,0));
Linear ci(r[i][0]*t1i[0], r[i][1]*t1i[1]); // c_i(v) := H_0(r_i(u)/(t_1)^i; u)
#ifdef DEBUG_INVERSION
std::cout << "t1i=" << t1i << std::endl;
std::cout << "ci=" << ci << std::endl;
#endif
for(int dim = 0; dim < 2; dim++) // t1^-i *= 1./t1
t1i[dim] *= t1[dim];
c[i] = ci; // c(v) := c(v) + c_i(v)*t^i
// change from v to u parameterisation
SBasis civ = one_minus_a*ci[0] + a*ci[1];
// r(u) := r(u) - c_i(u)*(t(u))^i
// We can truncate this to the number of final terms, as no following terms can
// contribute to the result.
r -= multiply(civ,ti);
r.truncate(k);
if(r.tailError(i) == 0)
break; // yay!
ti = multiply(ti,t);
}
#ifdef DEBUG_INVERSION
std::cout << "##########################" << std::endl;
#endif
} else
c = Linear(0,1); // linear
c -= a0; // invert the offset
c /= a1; // invert the slope
return c;
}
/** Compute the sine of a to k terms
\param b linear function
\returns sbasis sin(a)
It is recommended to use the piecewise version unless you have good reason.
*/
SBasis sin(Linear b, int k) {
SBasis s(k+2, Linear());
s[0] = Linear(std::sin(b[0]), std::sin(b[1]));
double tr = s[0].tri();
double t2 = b.tri();
s[1] = Linear(std::cos(b[0])*t2 - tr, -std::cos(b[1])*t2 + tr);
t2 *= t2;
for(int i = 0; i < k; i++) {
Linear bo(4*(i+1)*s[i+1][0] - 2*s[i+1][1],
-2*s[i+1][0] + 4*(i+1)*s[i+1][1]);
bo -= s[i]*(t2/(i+1));
s[i+2] = bo/double(i+2);
}
return s;
}
/** Compute the cosine of a
\param b linear function
\returns sbasis cos(a)
It is recommended to use the piecewise version unless you have good reason.
*/
SBasis cos(Linear bo, int k) {
return sin(Linear(bo[0] + M_PI/2,
bo[1] + M_PI/2),
k);
}
/** compute fog^-1.
\param f,g sbasis functions
\returns sbasis f(g^-1(t)).
("zero" = double comparison threshold. *!*we might divide by "zero"*!*)
TODO: compute order according to tol?
TODO: requires g(0)=0 & g(1)=1 atm... adaptation to other cases should be obvious!
*/
SBasis compose_inverse(SBasis const &f, SBasis const &g, unsigned order, double zero){
SBasis result(order, Linear(0.)); //result
SBasis r=f; //remainder
SBasis Pk=Linear(1)-g,Qk=g,sg=Pk*Qk;
Pk.truncate(order);
Qk.truncate(order);
Pk.resize(order,Linear(0.));
Qk.resize(order,Linear(0.));
r.resize(order,Linear(0.));
int vs = valuation(sg,zero);
if (vs == 0) { // to prevent infinite loop
return result;
}
for (unsigned k=0; k<order; k+=vs){
double p10 = Pk.at(k)[0];// we have to solve the linear system:
double p01 = Pk.at(k)[1];//
double q10 = Qk.at(k)[0];// p10*a + q10*b = r10
double q01 = Qk.at(k)[1];// &
double r10 = r.at(k)[0];// p01*a + q01*b = r01
double r01 = r.at(k)[1];//
double a,b;
double det = p10*q01-p01*q10;
//TODO: handle det~0!!
if (fabs(det)<zero){
a=b=0;
}else{
a=( q01*r10-q10*r01)/det;
b=(-p01*r10+p10*r01)/det;
}
result[k] = Linear(a,b);
r=r-Pk*a-Qk*b;
Pk=Pk*sg;
Qk=Qk*sg;
Pk.resize(order,Linear(0.)); // truncates if too high order, expands with zeros if too low
Qk.resize(order,Linear(0.));
r.resize(order,Linear(0.));
}
result.normalize();
return result;
}
}
/*
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 :