From 55d43e4e27e0ba58a47fad70957dfa989aa173ad Mon Sep 17 00:00:00 2001
From: "Johan B. C. Engelen" <jbc.engelen@swissonline.ch>
Date: Tue, 14 Aug 2007 20:54:48 +0000
Subject: Commit LivePathEffect branch to trunk! (disabled
 extension/internal/bitmap/*.* in build.xml to fix compilation)

(bzr r3472)
---
 src/2geom/sbasis.cpp | 490 +++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 490 insertions(+)
 create mode 100644 src/2geom/sbasis.cpp

(limited to 'src/2geom/sbasis.cpp')

diff --git a/src/2geom/sbasis.cpp b/src/2geom/sbasis.cpp
new file mode 100644
index 000000000..5bf0d2876
--- /dev/null
+++ b/src/2geom/sbasis.cpp
@@ -0,0 +1,490 @@
+/*
+ *  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 "sbasis.h"
+#include "isnan.h"
+
+namespace Geom{
+
+/*** At some point we should work on tighter bounds for the error.  It is clear that the error is
+ * bounded by the L1 norm over the tail of the series, but this is very loose, leading to far too
+ * many cubic beziers.  I've changed this to be \sum _i=tail ^\infty |hat a_i| 2^-i but I have no
+ * evidence that this is correct.
+ */
+
+/*
+double SBasis::tail_error(unsigned tail) const {
+    double err = 0, s = 1./(1<<(2*tail)); // rough
+    for(unsigned i = tail; i < size(); i++) {
+        err += (fabs((*this)[i][0]) + fabs((*this)[i][1]))*s;
+        s /= 4;
+    }
+    return err;
+}
+*/
+
+double SBasis::tailError(unsigned tail) const {
+  Interval bs = bounds_fast(*this, tail);
+  return std::max(fabs(bs.min()),fabs(bs.max()));
+}
+
+bool SBasis::isFinite() const {
+    for(unsigned i = 0; i < size(); i++) {
+        if(!(*this)[i].isFinite())
+            return false;
+    }
+    return true;
+}
+
+SBasis operator+(const SBasis& a, const SBasis& b) {
+    SBasis result;
+    const unsigned out_size = std::max(a.size(), b.size());
+    const unsigned min_size = std::min(a.size(), b.size());
+    result.reserve(out_size);
+    
+    for(unsigned i = 0; i < min_size; i++) {
+        result.push_back(a[i] + b[i]);
+    }
+    for(unsigned i = min_size; i < a.size(); i++)
+        result.push_back(a[i]);
+    for(unsigned i = min_size; i < b.size(); i++)
+        result.push_back(b[i]);
+
+    assert(result.size() == out_size);
+    return result;
+}
+
+SBasis operator-(const SBasis& a, const SBasis& b) {
+    SBasis result;
+    const unsigned out_size = std::max(a.size(), b.size());
+    const unsigned min_size = std::min(a.size(), b.size());
+    result.reserve(out_size);
+    
+    for(unsigned i = 0; i < min_size; i++) {
+        result.push_back(a[i] - b[i]);
+    }
+    for(unsigned i = min_size; i < a.size(); i++)
+        result.push_back(a[i]);
+    for(unsigned i = min_size; i < b.size(); i++)
+        result.push_back(-b[i]);
+
+    assert(result.size() == out_size);
+    return result;
+}
+
+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.reserve(out_size);
+        
+    for(unsigned i = 0; i < min_size; i++)
+        a[i] += b[i];
+    for(unsigned i = min_size; i < b.size(); i++)
+        a.push_back(b[i]);
+    
+    assert(a.size() == out_size);
+    return a;
+}
+
+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.reserve(out_size);
+        
+    for(unsigned i = 0; i < min_size; i++)
+        a[i] -= b[i];
+    for(unsigned i = min_size; i < b.size(); i++)
+        a.push_back(-b[i]);
+    
+    assert(a.size() == out_size);
+    return a;
+}
+
+SBasis operator*(SBasis const &a, double k) {
+    SBasis c;
+    c.reserve(a.size());
+    for(unsigned i = 0; i < a.size(); i++)
+        c.push_back(a[i] * k);
+    return c;
+}
+
+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;
+}
+
+SBasis shift(SBasis const &a, int sh) {
+    SBasis c = a;
+    if(sh > 0) {
+        c.insert(c.begin(), sh, Linear(0,0));
+    } else {
+        //TODO: truncate
+    }
+    return c;
+}
+
+SBasis shift(Linear const &a, int sh) {
+    SBasis c;
+    if(sh > 0) {
+        c.insert(c.begin(), sh, Linear(0,0));
+        c.push_back(a);
+    }
+    return c;
+}
+
+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;
+    if(a.isZero() || b.isZero())
+        return c;
+    c.resize(a.size() + b.size(), Linear(0,0));
+    c[0] = Linear(0,0);
+    for(unsigned j = 0; j < b.size(); j++) {
+        for(unsigned i = j; i < a.size()+j; i++) {
+            double tri = Tri(b[j])*Tri(a[i-j]);
+            c[i+1/*shift*/] += Linear(Hat(-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;
+}
+
+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 = -Tri(c[k-1])/(2*k);
+        a[k] = Hat(ahat);
+    }
+    double aTri = 0;
+    for(int k = c.size()-1; k >= 0; k--) {
+        aTri = (Hat(c[k]).d + (k+1)*aTri/2)/(2*k+1);
+        a[k][0] -= aTri/2;
+        a[k][1] += aTri/2;
+    }
+    a.normalize();
+    return a;
+}
+
+SBasis derivative(SBasis const &a) {
+    SBasis c;
+    c.resize(a.size(), Linear(0,0));
+    
+    for(unsigned k = 0; k < a.size(); k++) {
+        double d = (2*k+1)*Tri(a[k]);
+        
+        for(unsigned dim = 0; dim < 2; dim++) {
+            c[k][dim] = d;
+            if(k+1 < a.size()) {
+                if(dim)
+                    c[k][dim] = d - (k+1)*a[k+1][dim];
+                else
+                    c[k][dim] = d + (k+1)*a[k+1][dim];
+            }
+        }
+    }
+    
+    return c;
+}
+
+//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 and 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;
+}
+
+// return a kth order approx to 1/a)
+SBasis reciprocal(Linear const &a, int k) {
+    SBasis c;
+    assert(!a.isZero());
+    c.resize(k, Linear(0,0));
+    double r_s0 = (Tri(a)*Tri(a))/(-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;
+}
+
+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;
+}
+
+// a(b)
+// 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 = SBasis(Linear(Hat(a[i][0]))) - b*a[i][0] + b*a[i][1] + multiply(r,s);
+    }
+    return r;
+}
+
+// a(b)
+// 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 = SBasis(Linear(Hat(a[i][0]))) - b*a[i][0] + b*a[i][1] + multiply(r,s);
+    }
+    r.truncate(k);
+    return r;
+}
+
+/*
+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 
+
+SBasis inverse(SBasis a, int k) {
+    assert(a.size() > 0);
+// the function should have 'unit range'("a00 = 0 and a01 = 1") and be monotonic.
+    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;                           // c(v) := 0
+    if(a.size() >= 2 && k == 2) {
+        c.push_back(Linear(0,1));
+        Linear t1(1+a[1][0], 1-a[1][1]);    // t_1
+        c.push_back(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;
+}
+
+SBasis sin(Linear b, int k) {
+    SBasis s = Linear(std::sin(b[0]), std::sin(b[1]));
+    Tri tr(s[0]);
+    double t2 = Tri(b);
+    s.push_back(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.push_back(bo/double(i+2));
+    }
+    
+    return s;
+}
+
+SBasis cos(Linear bo, int k) {
+    return sin(Linear(bo[0] + M_PI/2,
+                      bo[1] + M_PI/2),
+               k);
+}
+
+//compute fog^-1. ("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; //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);
+    
+    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){
+            det = zero;
+            a=b=0;
+        }else{
+            a=( q01*r10-q10*r01)/det;
+            b=(-p01*r10+p10*r01)/det;
+        }
+        result.push_back(Linear(a,b));
+        r=r-Pk*a-Qk*b;
+        
+        Pk=Pk*sg;
+        Qk=Qk*sg;
+        Pk.truncate(order);
+        Qk.truncate(order);
+        r.truncate(order);
+    }
+    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:encoding=utf-8:textwidth=99 :
-- 
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