--- /dev/null
+/*! ========================================================================
+** Extended Template Library
+** Bezier Template Class Implementation
+** $Id$
+**
+** Copyright (c) 2002 Robert B. Quattlebaum Jr.
+** Copyright (c) 2007 Chris Moore
+**
+** This package is free software; you can redistribute it and/or
+** modify it under the terms of the GNU General Public License as
+** published by the Free Software Foundation; either version 2 of
+** the License, or (at your option) any later version.
+**
+** This package is distributed in the hope that it will be useful,
+** but WITHOUT ANY WARRANTY; without even the implied warranty of
+** MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+** General Public License for more details.
+**
+** === N O T E S ===========================================================
+**
+** This is an internal header file, included by other ETL headers.
+** You should not attempt to use it directly.
+**
+** ========================================================================= */
+
+/* === S T A R T =========================================================== */
+
+#ifndef __ETL__BEZIER_H
+#define __ETL__BEZIER_H
+
+/* === H E A D E R S ======================================================= */
+
+#include "_curve_func.h"
+#include <cmath> // for ldexp
+// #include <ETL/fixed> // not used
+
+/* === M A C R O S ========================================================= */
+
+#define MAXDEPTH 64 /* Maximum depth for recursion */
+
+/* take binary sign of a, either -1, or 1 if >= 0 */
+#define SGN(a) (((a)<0) ? -1 : 1)
+
+/* find minimum of a and b */
+#ifndef MIN
+#define MIN(a,b) (((a)<(b))?(a):(b))
+#endif
+
+/* find maximum of a and b */
+#ifndef MAX
+#define MAX(a,b) (((a)>(b))?(a):(b))
+#endif
+
+#define BEZIER_EPSILON (ldexp(1.0,-MAXDEPTH-1)) /*Flatness control value */
+//#define BEZIER_EPSILON 0.00005 /*Flatness control value */
+#define DEGREE 3 /* Cubic Bezier curve */
+#define W_DEGREE 5 /* Degree of eqn to find roots of */
+
+/* === T Y P E D E F S ===================================================== */
+
+/* === C L A S S E S & S T R U C T S ======================================= */
+
+_ETL_BEGIN_NAMESPACE
+
+template<typename V,typename T> class bezier;
+
+//! Cubic Bezier Curve Base Class
+// This generic implementation uses the DeCasteljau algorithm.
+// Works for just about anything that has an affine combination function
+template <typename V,typename T=float>
+class bezier_base : public std::unary_function<T,V>
+{
+public:
+ typedef V value_type;
+ typedef T time_type;
+
+private:
+ value_type a,b,c,d;
+ time_type r,s;
+
+protected:
+ affine_combo<value_type,time_type> affine_func;
+
+public:
+ bezier_base():r(0.0),s(1.0) { }
+ bezier_base(
+ const value_type &a, const value_type &b, const value_type &c, const value_type &d,
+ const time_type &r=0.0, const time_type &s=1.0):
+ a(a),b(b),c(c),d(d),r(r),s(s) { sync(); }
+
+ void sync()
+ {
+ }
+
+ value_type
+ operator()(time_type t)const
+ {
+ t=(t-r)/(s-r);
+ return
+ affine_func(
+ affine_func(
+ affine_func(a,b,t),
+ affine_func(b,c,t)
+ ,t),
+ affine_func(
+ affine_func(b,c,t),
+ affine_func(c,d,t)
+ ,t)
+ ,t);
+ }
+
+ /*
+ void evaluate(time_type t, value_type &f, value_type &df) const
+ {
+ t=(t-r)/(s-r);
+
+ value_type p1 = affine_func(
+ affine_func(a,b,t),
+ affine_func(b,c,t)
+ ,t);
+ value_type p2 = affine_func(
+ affine_func(b,c,t),
+ affine_func(c,d,t)
+ ,t);
+
+ f = affine_func(p1,p2,t);
+ df = (p2-p1)*3;
+ }
+ */
+
+ void set_rs(time_type new_r, time_type new_s) { r=new_r; s=new_s; }
+ void set_r(time_type new_r) { r=new_r; }
+ void set_s(time_type new_s) { s=new_s; }
+ const time_type &get_r()const { return r; }
+ const time_type &get_s()const { return s; }
+ time_type get_dt()const { return s-r; }
+
+ bool intersect_hull(const bezier_base<value_type,time_type> &x)const
+ {
+ return 0;
+ }
+
+ //! Bezier curve intersection function
+ /*! Calculates the time of intersection
+ ** for the calling curve.
+ **
+ ** I still have not figured out a good generic
+ ** method of doing this for a bi-infinite
+ ** cubic bezier curve calculated with the DeCasteljau
+ ** algorithm.
+ **
+ ** One method, although it does not work for the
+ ** entire bi-infinite curve, is to iteratively
+ ** intersect the hulls. However, we would only detect
+ ** intersections that occur between R and S.
+ **
+ ** It is entirely possible that a new construct similar
+ ** to the affine combination function will be necessary
+ ** for this to work properly.
+ **
+ ** For now, this function is BROKEN. (although it works
+ ** for the floating-point specializations, using newton's method)
+ */
+ time_type intersect(const bezier_base<value_type,time_type> &x, time_type near=0.0)const
+ {
+ return 0;
+ }
+
+ /* subdivide at some time t into 2 separate curves left and right
+
+ b0 l1
+ * 0+1 l2
+ b1 * 1+2*1+2 l3
+ * 1+2 * 0+3*1+3*2+3 l4,r1
+ b2 * 1+2*2+2 r2 *
+ * 2+3 r3 *
+ b3 r4 *
+ *
+
+ 0.1 2.3 -> 0.1 2 3 4 5.6
+ */
+/* void subdivide(bezier_base *left, bezier_base *right, const time_type &time = (time_type)0.5) const
+ {
+ time_type t = (time-r)/(s-r);
+ bezier_base lt,rt;
+
+ value_type temp;
+
+ //1st stage points to keep
+ lt.a = a;
+ rt.d = d;
+
+ //2nd stage calc
+ lt.b = affine_func(a,b,t);
+ temp = affine_func(b,c,t);
+ rt.c = affine_func(c,d,t);
+
+ //3rd stage calc
+ lt.c = affine_func(lt.b,temp,t);
+ rt.b = affine_func(temp,rt.c,t);
+
+ //last stage calc
+ lt.d = rt.a = affine_func(lt.c,rt.b,t);
+
+ //set the time range for l,r (the inside values should be 1, 0 respectively)
+ lt.r = r;
+ rt.s = s;
+
+ //give back the curves
+ if(left) *left = lt;
+ if(right) *right = rt;
+ }
+ */
+ value_type &
+ operator[](int i)
+ { return (&a)[i]; }
+
+ const value_type &
+ operator[](int i) const
+ { return (&a)[i]; }
+};
+
+
+#if 1
+// Fast float implementation of a cubic bezier curve
+template <>
+class bezier_base<float,float> : public std::unary_function<float,float>
+{
+public:
+ typedef float value_type;
+ typedef float time_type;
+private:
+ affine_combo<value_type,time_type> affine_func;
+ value_type a,b,c,d;
+ time_type r,s;
+
+ value_type _coeff[4];
+ time_type drs; // reciprocal of (s-r)
+public:
+ bezier_base():r(0.0),s(1.0),drs(1.0) { }
+ bezier_base(
+ const value_type &a, const value_type &b, const value_type &c, const value_type &d,
+ const time_type &r=0.0, const time_type &s=1.0):
+ a(a),b(b),c(c),d(d),r(r),s(s),drs(1.0/(s-r)) { sync(); }
+
+ void sync()
+ {
+// drs=1.0/(s-r);
+ _coeff[0]= a;
+ _coeff[1]= b*3 - a*3;
+ _coeff[2]= c*3 - b*6 + a*3;
+ _coeff[3]= d - c*3 + b*3 - a;
+ }
+
+ // Cost Summary: 4 products, 3 sums, and 1 difference.
+ inline value_type
+ operator()(time_type t)const
+ { t-=r; t*=drs; return _coeff[0]+(_coeff[1]+(_coeff[2]+(_coeff[3])*t)*t)*t; }
+
+ void set_rs(time_type new_r, time_type new_s) { r=new_r; s=new_s; drs=1.0/(s-r); }
+ void set_r(time_type new_r) { r=new_r; drs=1.0/(s-r); }
+ void set_s(time_type new_s) { s=new_s; drs=1.0/(s-r); }
+ const time_type &get_r()const { return r; }
+ const time_type &get_s()const { return s; }
+ time_type get_dt()const { return s-r; }
+
+ //! Bezier curve intersection function
+ /*! Calculates the time of intersection
+ ** for the calling curve.
+ */
+ time_type intersect(const bezier_base<value_type,time_type> &x, time_type t=0.0,int i=15)const
+ {
+ //BROKEN - the time values of the 2 curves should be independent
+ value_type system[4];
+ system[0]=_coeff[0]-x._coeff[0];
+ system[1]=_coeff[1]-x._coeff[1];
+ system[2]=_coeff[2]-x._coeff[2];
+ system[3]=_coeff[3]-x._coeff[3];
+
+ t-=r;
+ t*=drs;
+
+ // Newton's method
+ // Inner loop cost summary: 7 products, 5 sums, 1 difference
+ for(;i;i--)
+ t-= (system[0]+(system[1]+(system[2]+(system[3])*t)*t)*t)/
+ (system[1]+(system[2]*2+(system[3]*3)*t)*t);
+
+ t*=(s-r);
+ t+=r;
+
+ return t;
+ }
+
+ value_type &
+ operator[](int i)
+ { return (&a)[i]; }
+
+ const value_type &
+ operator[](int i) const
+ { return (&a)[i]; }
+};
+
+
+// Fast double implementation of a cubic bezier curve
+template <>
+class bezier_base<double,float> : public std::unary_function<float,double>
+{
+public:
+ typedef double value_type;
+ typedef float time_type;
+private:
+ affine_combo<value_type,time_type> affine_func;
+ value_type a,b,c,d;
+ time_type r,s;
+
+ value_type _coeff[4];
+ time_type drs; // reciprocal of (s-r)
+public:
+ bezier_base():r(0.0),s(1.0),drs(1.0) { }
+ bezier_base(
+ const value_type &a, const value_type &b, const value_type &c, const value_type &d,
+ const time_type &r=0.0, const time_type &s=1.0):
+ a(a),b(b),c(c),d(d),r(r),s(s),drs(1.0/(s-r)) { sync(); }
+
+ void sync()
+ {
+// drs=1.0/(s-r);
+ _coeff[0]= a;
+ _coeff[1]= b*3 - a*3;
+ _coeff[2]= c*3 - b*6 + a*3;
+ _coeff[3]= d - c*3 + b*3 - a;
+ }
+
+ // 4 products, 3 sums, and 1 difference.
+ inline value_type
+ operator()(time_type t)const
+ { t-=r; t*=drs; return _coeff[0]+(_coeff[1]+(_coeff[2]+(_coeff[3])*t)*t)*t; }
+
+ void set_rs(time_type new_r, time_type new_s) { r=new_r; s=new_s; drs=1.0/(s-r); }
+ void set_r(time_type new_r) { r=new_r; drs=1.0/(s-r); }
+ void set_s(time_type new_s) { s=new_s; drs=1.0/(s-r); }
+ const time_type &get_r()const { return r; }
+ const time_type &get_s()const { return s; }
+ time_type get_dt()const { return s-r; }
+
+ //! Bezier curve intersection function
+ /*! Calculates the time of intersection
+ ** for the calling curve.
+ */
+ time_type intersect(const bezier_base<value_type,time_type> &x, time_type t=0.0,int i=15)const
+ {
+ //BROKEN - the time values of the 2 curves should be independent
+ value_type system[4];
+ system[0]=_coeff[0]-x._coeff[0];
+ system[1]=_coeff[1]-x._coeff[1];
+ system[2]=_coeff[2]-x._coeff[2];
+ system[3]=_coeff[3]-x._coeff[3];
+
+ t-=r;
+ t*=drs;
+
+ // Newton's method
+ // Inner loop: 7 products, 5 sums, 1 difference
+ for(;i;i--)
+ t-= (system[0]+(system[1]+(system[2]+(system[3])*t)*t)*t)/
+ (system[1]+(system[2]*2+(system[3]*3)*t)*t);
+
+ t*=(s-r);
+ t+=r;
+
+ return t;
+ }
+
+ value_type &
+ operator[](int i)
+ { return (&a)[i]; }
+
+ const value_type &
+ operator[](int i) const
+ { return (&a)[i]; }
+};
+
+//#ifdef __FIXED__
+
+// Fast double implementation of a cubic bezier curve
+/*
+template <>
+template <class T,unsigned int FIXED_BITS>
+class bezier_base<fixed_base<T,FIXED_BITS> > : std::unary_function<fixed_base<T,FIXED_BITS>,fixed_base<T,FIXED_BITS> >
+{
+public:
+ typedef fixed_base<T,FIXED_BITS> value_type;
+ typedef fixed_base<T,FIXED_BITS> time_type;
+
+private:
+ affine_combo<value_type,time_type> affine_func;
+ value_type a,b,c,d;
+ time_type r,s;
+
+ value_type _coeff[4];
+ time_type drs; // reciprocal of (s-r)
+public:
+ bezier_base():r(0.0),s(1.0),drs(1.0) { }
+ bezier_base(
+ const value_type &a, const value_type &b, const value_type &c, const value_type &d,
+ const time_type &r=0, const time_type &s=1):
+ a(a),b(b),c(c),d(d),r(r),s(s),drs(1.0/(s-r)) { sync(); }
+
+ void sync()
+ {
+ drs=time_type(1)/(s-r);
+ _coeff[0]= a;
+ _coeff[1]= b*3 - a*3;
+ _coeff[2]= c*3 - b*6 + a*3;
+ _coeff[3]= d - c*3 + b*3 - a;
+ }
+
+ // 4 products, 3 sums, and 1 difference.
+ inline value_type
+ operator()(time_type t)const
+ { t-=r; t*=drs; return _coeff[0]+(_coeff[1]+(_coeff[2]+(_coeff[3])*t)*t)*t; }
+
+ void set_rs(time_type new_r, time_type new_s) { r=new_r; s=new_s; drs=time_type(1)/(s-r); }
+ void set_r(time_type new_r) { r=new_r; drs=time_type(1)/(s-r); }
+ void set_s(time_type new_s) { s=new_s; drs=time_type(1)/(s-r); }
+ const time_type &get_r()const { return r; }
+ const time_type &get_s()const { return s; }
+ time_type get_dt()const { return s-r; }
+
+ //! Bezier curve intersection function
+ //! Calculates the time of intersection
+ // for the calling curve.
+ //
+ time_type intersect(const bezier_base<value_type,time_type> &x, time_type t=0,int i=15)const
+ {
+ value_type system[4];
+ system[0]=_coeff[0]-x._coeff[0];
+ system[1]=_coeff[1]-x._coeff[1];
+ system[2]=_coeff[2]-x._coeff[2];
+ system[3]=_coeff[3]-x._coeff[3];
+
+ t-=r;
+ t*=drs;
+
+ // Newton's method
+ // Inner loop: 7 products, 5 sums, 1 difference
+ for(;i;i--)
+ t-=(time_type) ( (system[0]+(system[1]+(system[2]+(system[3])*t)*t)*t)/
+ (system[1]+(system[2]*2+(system[3]*3)*t)*t) );
+
+ t*=(s-r);
+ t+=r;
+
+ return t;
+ }
+
+ value_type &
+ operator[](int i)
+ { return (&a)[i]; }
+
+ const value_type &
+ operator[](int i) const
+ { return (&a)[i]; }
+};
+*/
+//#endif
+
+#endif
+
+
+
+template <typename V, typename T>
+class bezier_iterator
+{
+public:
+
+ struct iterator_category {};
+ typedef V value_type;
+ typedef T difference_type;
+ typedef V reference;
+
+private:
+ difference_type t;
+ difference_type dt;
+ bezier_base<V,T> curve;
+
+public:
+
+/*
+ reference
+ operator*(void)const { return curve(t); }
+ const surface_iterator&
+
+ operator++(void)
+ { t+=dt; return &this; }
+
+ const surface_iterator&
+ operator++(int)
+ { hermite_iterator _tmp=*this; t+=dt; return _tmp; }
+
+ const surface_iterator&
+ operator--(void)
+ { t-=dt; return &this; }
+
+ const surface_iterator&
+ operator--(int)
+ { hermite_iterator _tmp=*this; t-=dt; return _tmp; }
+
+
+ surface_iterator
+ operator+(difference_type __n) const
+ { return surface_iterator(data+__n[0]+__n[1]*pitch,pitch); }
+
+ surface_iterator
+ operator-(difference_type __n) const
+ { return surface_iterator(data-__n[0]-__n[1]*pitch,pitch); }
+*/
+
+};
+
+template <typename V,typename T=float>
+class bezier : public bezier_base<V,T>
+{
+public:
+ typedef V value_type;
+ typedef T time_type;
+ typedef float distance_type;
+ typedef bezier_iterator<V,T> iterator;
+ typedef bezier_iterator<V,T> const_iterator;
+
+ distance_func<value_type> dist;
+
+ using bezier_base<V,T>::get_r;
+ using bezier_base<V,T>::get_s;
+ using bezier_base<V,T>::get_dt;
+
+public:
+ bezier() { }
+ bezier(const value_type &a, const value_type &b, const value_type &c, const value_type &d):
+ bezier_base<V,T>(a,b,c,d) { }
+
+
+ const_iterator begin()const;
+ const_iterator end()const;
+
+ time_type find_closest(bool fast, const value_type& x, int i=7)const
+ {
+ if (!fast)
+ {
+ value_type array[4] = {
+ bezier<V,T>::operator[](0),
+ bezier<V,T>::operator[](1),
+ bezier<V,T>::operator[](2),
+ bezier<V,T>::operator[](3)};
+ return NearestPointOnCurve(x, array);
+ }
+ else
+ {
+ time_type r(0), s(1);
+ float t((r+s)*0.5); /* half way between r and s */
+
+ for(;i;i--)
+ {
+ // compare 33% of the way between r and s with 67% of the way between r and s
+ if(dist(operator()((s-r)*(1.0/3.0)+r), x) <
+ dist(operator()((s-r)*(2.0/3.0)+r), x))
+ s=t;
+ else
+ r=t;
+ t=((r+s)*0.5);
+ }
+ return t;
+ }
+ }
+
+ distance_type find_distance(time_type r, time_type s, int steps=7)const
+ {
+ const time_type inc((s-r)/steps);
+ if (!inc) return 0;
+ distance_type ret(0);
+ value_type last(operator()(r));
+
+ for(r+=inc;r<s;r+=inc)
+ {
+ const value_type n(operator()(r));
+ ret+=dist.uncook(dist(last,n));
+ last=n;
+ }
+ ret+=dist.uncook(dist(last,operator()(r)))*(s-(r-inc))/inc;
+
+ return ret;
+ }
+
+ distance_type length()const { return find_distance(get_r(),get_s()); }
+
+ /* subdivide at some time t into 2 separate curves left and right
+
+ b0 l1
+ * 0+1 l2
+ b1 * 1+2*1+2 l3
+ * 1+2 * 0+3*1+3*2+3 l4,r1
+ b2 * 1+2*2+2 r2 *
+ * 2+3 r3 *
+ b3 r4 *
+ *
+
+ 0.1 2.3 -> 0.1 2 3 4 5.6
+ */
+ void subdivide(bezier *left, bezier *right, const time_type &time = (time_type)0.5) const
+ {
+ time_type t=(time-get_r())/get_dt();
+ bezier lt,rt;
+
+ value_type temp;
+ const value_type& a((*this)[0]);
+ const value_type& b((*this)[1]);
+ const value_type& c((*this)[2]);
+ const value_type& d((*this)[3]);
+
+ //1st stage points to keep
+ lt[0] = a;
+ rt[3] = d;
+
+ //2nd stage calc
+ lt[1] = affine_func(a,b,t);
+ temp = affine_func(b,c,t);
+ rt[2] = affine_func(c,d,t);
+
+ //3rd stage calc
+ lt[2] = affine_func(lt[1],temp,t);
+ rt[1] = affine_func(temp,rt[2],t);
+
+ //last stage calc
+ lt[3] = rt[0] = affine_func(lt[2],rt[1],t);
+
+ //set the time range for l,r (the inside values should be 1, 0 respectively)
+ lt.set_r(get_r());
+ rt.set_s(get_s());
+
+ lt.sync();
+ rt.sync();
+
+ //give back the curves
+ if(left) *left = lt;
+ if(right) *right = rt;
+ }
+
+
+ void evaluate(time_type t, value_type &f, value_type &df) const
+ {
+ t=(t-get_r())/get_dt();
+
+ const value_type& a((*this)[0]);
+ const value_type& b((*this)[1]);
+ const value_type& c((*this)[2]);
+ const value_type& d((*this)[3]);
+
+ const value_type p1 = affine_func(
+ affine_func(a,b,t),
+ affine_func(b,c,t)
+ ,t);
+ const value_type p2 = affine_func(
+ affine_func(b,c,t),
+ affine_func(c,d,t)
+ ,t);
+
+ f = affine_func(p1,p2,t);
+ df = (p2-p1)*3;
+ }
+
+private:
+ /*
+ * Bezier :
+ * Evaluate a Bezier curve at a particular parameter value
+ * Fill in control points for resulting sub-curves if "Left" and
+ * "Right" are non-null.
+ *
+ * int degree; Degree of bezier curve
+ * value_type *VT; Control pts
+ * time_type t; Parameter value
+ * value_type *Left; RETURN left half ctl pts
+ * value_type *Right; RETURN right half ctl pts
+ */
+ static value_type Bezier(value_type *VT, int degree, time_type t, value_type *Left, value_type *Right)
+ {
+ int i, j; /* Index variables */
+ value_type Vtemp[W_DEGREE+1][W_DEGREE+1];
+
+ /* Copy control points */
+ for (j = 0; j <= degree; j++)
+ Vtemp[0][j] = VT[j];
+
+ /* Triangle computation */
+ for (i = 1; i <= degree; i++)
+ for (j =0 ; j <= degree - i; j++)
+ {
+ Vtemp[i][j][0] = (1.0 - t) * Vtemp[i-1][j][0] + t * Vtemp[i-1][j+1][0];
+ Vtemp[i][j][1] = (1.0 - t) * Vtemp[i-1][j][1] + t * Vtemp[i-1][j+1][1];
+ }
+
+ if (Left != NULL)
+ for (j = 0; j <= degree; j++)
+ Left[j] = Vtemp[j][0];
+
+ if (Right != NULL)
+ for (j = 0; j <= degree; j++)
+ Right[j] = Vtemp[degree-j][j];
+
+ return (Vtemp[degree][0]);
+ }
+
+ /*
+ * CrossingCount :
+ * Count the number of times a Bezier control polygon
+ * crosses the 0-axis. This number is >= the number of roots.
+ *
+ * value_type *VT; Control pts of Bezier curve
+ */
+ static int CrossingCount(value_type *VT)
+ {
+ int i;
+ int n_crossings = 0; /* Number of zero-crossings */
+ int sign, old_sign; /* Sign of coefficients */
+
+ sign = old_sign = SGN(VT[0][1]);
+ for (i = 1; i <= W_DEGREE; i++)
+ {
+ sign = SGN(VT[i][1]);
+ if (sign != old_sign) n_crossings++;
+ old_sign = sign;
+ }
+
+ return n_crossings;
+ }
+
+ /*
+ * ControlPolygonFlatEnough :
+ * Check if the control polygon of a Bezier curve is flat enough
+ * for recursive subdivision to bottom out.
+ *
+ * value_type *VT; Control points
+ */
+ static int ControlPolygonFlatEnough(value_type *VT)
+ {
+ int i; /* Index variable */
+ distance_type distance[W_DEGREE]; /* Distances from pts to line */
+ distance_type max_distance_above; /* maximum of these */
+ distance_type max_distance_below;
+ time_type intercept_1, intercept_2, left_intercept, right_intercept;
+ distance_type a, b, c; /* Coefficients of implicit */
+ /* eqn for line from VT[0]-VT[deg] */
+ /* Find the perpendicular distance */
+ /* from each interior control point to */
+ /* line connecting VT[0] and VT[W_DEGREE] */
+ {
+ distance_type abSquared;
+
+ /* Derive the implicit equation for line connecting first *
+ * and last control points */
+ a = VT[0][1] - VT[W_DEGREE][1];
+ b = VT[W_DEGREE][0] - VT[0][0];
+ c = VT[0][0] * VT[W_DEGREE][1] - VT[W_DEGREE][0] * VT[0][1];
+
+ abSquared = (a * a) + (b * b);
+
+ for (i = 1; i < W_DEGREE; i++)
+ {
+ /* Compute distance from each of the points to that line */
+ distance[i] = a * VT[i][0] + b * VT[i][1] + c;
+ if (distance[i] > 0.0) distance[i] = (distance[i] * distance[i]) / abSquared;
+ if (distance[i] < 0.0) distance[i] = -(distance[i] * distance[i]) / abSquared;
+ }
+ }
+
+ /* Find the largest distance */
+ max_distance_above = max_distance_below = 0.0;
+
+ for (i = 1; i < W_DEGREE; i++)
+ {
+ if (distance[i] < 0.0) max_distance_below = MIN(max_distance_below, distance[i]);
+ if (distance[i] > 0.0) max_distance_above = MAX(max_distance_above, distance[i]);
+ }
+
+ /* Implicit equation for "above" line */
+ intercept_1 = -(c + max_distance_above)/a;
+
+ /* Implicit equation for "below" line */
+ intercept_2 = -(c + max_distance_below)/a;
+
+ /* Compute intercepts of bounding box */
+ left_intercept = MIN(intercept_1, intercept_2);
+ right_intercept = MAX(intercept_1, intercept_2);
+
+ return 0.5 * (right_intercept-left_intercept) < BEZIER_EPSILON ? 1 : 0;
+ }
+
+ /*
+ * ComputeXIntercept :
+ * Compute intersection of chord from first control point to last
+ * with 0-axis.
+ *
+ * value_type *VT; Control points
+ */
+ static time_type ComputeXIntercept(value_type *VT)
+ {
+ distance_type YNM = VT[W_DEGREE][1] - VT[0][1];
+ return (YNM*VT[0][0] - (VT[W_DEGREE][0] - VT[0][0])*VT[0][1]) / YNM;
+ }
+
+ /*
+ * FindRoots :
+ * Given a 5th-degree equation in Bernstein-Bezier form, find
+ * all of the roots in the interval [0, 1]. Return the number
+ * of roots found.
+ *
+ * value_type *w; The control points
+ * time_type *t; RETURN candidate t-values
+ * int depth; The depth of the recursion
+ */
+ static int FindRoots(value_type *w, time_type *t, int depth)
+ {
+ int i;
+ value_type Left[W_DEGREE+1]; /* New left and right */
+ value_type Right[W_DEGREE+1]; /* control polygons */
+ int left_count; /* Solution count from */
+ int right_count; /* children */
+ time_type left_t[W_DEGREE+1]; /* Solutions from kids */
+ time_type right_t[W_DEGREE+1];
+
+ switch (CrossingCount(w))
+ {
+ case 0 :
+ { /* No solutions here */
+ return 0;
+ }
+ case 1 :
+ { /* Unique solution */
+ /* Stop recursion when the tree is deep enough */
+ /* if deep enough, return 1 solution at midpoint */
+ if (depth >= MAXDEPTH)
+ {
+ t[0] = (w[0][0] + w[W_DEGREE][0]) / 2.0;
+ return 1;
+ }
+ if (ControlPolygonFlatEnough(w))
+ {
+ t[0] = ComputeXIntercept(w);
+ return 1;
+ }
+ break;
+ }
+ }
+
+ /* Otherwise, solve recursively after */
+ /* subdividing control polygon */
+ Bezier(w, W_DEGREE, 0.5, Left, Right);
+ left_count = FindRoots(Left, left_t, depth+1);
+ right_count = FindRoots(Right, right_t, depth+1);
+
+ /* Gather solutions together */
+ for (i = 0; i < left_count; i++) t[i] = left_t[i];
+ for (i = 0; i < right_count; i++) t[i+left_count] = right_t[i];
+
+ /* Send back total number of solutions */
+ return (left_count+right_count);
+ }
+
+ /*
+ * ConvertToBezierForm :
+ * Given a point and a Bezier curve, generate a 5th-degree
+ * Bezier-format equation whose solution finds the point on the
+ * curve nearest the user-defined point.
+ *
+ * value_type& P; The point to find t for
+ * value_type *VT; The control points
+ */
+ static void ConvertToBezierForm(const value_type& P, value_type *VT, value_type w[W_DEGREE+1])
+ {
+ int i, j, k, m, n, ub, lb;
+ int row, column; /* Table indices */
+ value_type c[DEGREE+1]; /* VT(i)'s - P */
+ value_type d[DEGREE]; /* VT(i+1) - VT(i) */
+ distance_type cdTable[3][4]; /* Dot product of c, d */
+ static distance_type z[3][4] = { /* Precomputed "z" for cubics */
+ {1.0, 0.6, 0.3, 0.1},
+ {0.4, 0.6, 0.6, 0.4},
+ {0.1, 0.3, 0.6, 1.0}};
+
+ /* Determine the c's -- these are vectors created by subtracting */
+ /* point P from each of the control points */
+ for (i = 0; i <= DEGREE; i++)
+ c[i] = VT[i] - P;
+
+ /* Determine the d's -- these are vectors created by subtracting */
+ /* each control point from the next */
+ for (i = 0; i <= DEGREE - 1; i++)
+ d[i] = (VT[i+1] - VT[i]) * 3.0;
+
+ /* Create the c,d table -- this is a table of dot products of the */
+ /* c's and d's */
+ for (row = 0; row <= DEGREE - 1; row++)
+ for (column = 0; column <= DEGREE; column++)
+ cdTable[row][column] = d[row] * c[column];
+
+ /* Now, apply the z's to the dot products, on the skew diagonal */
+ /* Also, set up the x-values, making these "points" */
+ for (i = 0; i <= W_DEGREE; i++)
+ {
+ w[i][0] = (distance_type)(i) / W_DEGREE;
+ w[i][1] = 0.0;
+ }
+
+ n = DEGREE;
+ m = DEGREE-1;
+ for (k = 0; k <= n + m; k++)
+ {
+ lb = MAX(0, k - m);
+ ub = MIN(k, n);
+ for (i = lb; i <= ub; i++)
+ {
+ j = k - i;
+ w[i+j][1] += cdTable[j][i] * z[j][i];
+ }
+ }
+ }
+
+ /*
+ * NearestPointOnCurve :
+ * Compute the parameter value of the point on a Bezier
+ * curve segment closest to some arbitrary, user-input point.
+ * Return the point on the curve at that parameter value.
+ *
+ * value_type& P; The user-supplied point
+ * value_type *VT; Control points of cubic Bezier
+ */
+ static time_type NearestPointOnCurve(const value_type& P, value_type VT[4])
+ {
+ value_type w[W_DEGREE+1]; /* Ctl pts of 5th-degree curve */
+ time_type t_candidate[W_DEGREE]; /* Possible roots */
+ int n_solutions; /* Number of roots found */
+ time_type t; /* Parameter value of closest pt */
+
+ /* Convert problem to 5th-degree Bezier form */
+ ConvertToBezierForm(P, VT, w);
+
+ /* Find all possible roots of 5th-degree equation */
+ n_solutions = FindRoots(w, t_candidate, 0);
+
+ /* Compare distances of P to all candidates, and to t=0, and t=1 */
+ {
+ distance_type dist, new_dist;
+ value_type p, v;
+ int i;
+
+ /* Check distance to beginning of curve, where t = 0 */
+ dist = (P - VT[0]).mag_squared();
+ t = 0.0;
+
+ /* Find distances for candidate points */
+ for (i = 0; i < n_solutions; i++)
+ {
+ p = Bezier(VT, DEGREE, t_candidate[i], (value_type *)NULL, (value_type *)NULL);
+ new_dist = (P - p).mag_squared();
+ if (new_dist < dist)
+ {
+ dist = new_dist;
+ t = t_candidate[i];
+ }
+ }
+
+ /* Finally, look at distance to end point, where t = 1.0 */
+ new_dist = (P - VT[DEGREE]).mag_squared();
+ if (new_dist < dist)
+ {
+ dist = new_dist;
+ t = 1.0;
+ }
+ }
+
+ /* Return the point on the curve at parameter value t */
+ return t;
+ }
+};
+
+_ETL_END_NAMESPACE
+
+/* === E X T E R N S ======================================================= */
+
+/* === E N D =============================================================== */
+
+#endif
projects / synfig.git / blobdiff