1 /*! ========================================================================
2 ** Extended Template Library
3 ** Bezier Template Class Implementation
6 ** Copyright (c) 2002 Robert B. Quattlebaum Jr.
8 ** This package is free software; you can redistribute it and/or
9 ** modify it under the terms of the GNU General Public License as
10 ** published by the Free Software Foundation; either version 2 of
11 ** the License, or (at your option) any later version.
13 ** This package is distributed in the hope that it will be useful,
14 ** but WITHOUT ANY WARRANTY; without even the implied warranty of
15 ** MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
16 ** General Public License for more details.
18 ** === N O T E S ===========================================================
20 ** This is an internal header file, included by other ETL headers.
21 ** You should not attempt to use it directly.
23 ** ========================================================================= */
25 /* === S T A R T =========================================================== */
27 #ifndef __ETL_BEZIER_H
28 #define __ETL_BEZIER_H
30 /* === H E A D E R S ======================================================= */
32 #include "_curve_func.h"
35 /* === M A C R O S ========================================================= */
37 #define MAXDEPTH 64 /* Maximum depth for recursion */
39 /* take binary sign of a, either -1, or 1 if >= 0 */
40 #define SGN(a) (((a)<0) ? -1 : 1)
42 /* find minimum of a and b */
44 #define MIN(a,b) (((a)<(b))?(a):(b))
47 /* find maximum of a and b */
49 #define MAX(a,b) (((a)>(b))?(a):(b))
52 #define BEZIER_EPSILON (ldexp(1.0,-MAXDEPTH-1)) /*Flatness control value */
53 //#define BEZIER_EPSILON 0.00005 /*Flatness control value */
54 #define DEGREE 3 /* Cubic Bezier curve */
55 #define W_DEGREE 5 /* Degree of eqn to find roots of */
57 /* === T Y P E D E F S ===================================================== */
59 /* === C L A S S E S & S T R U C T S ======================================= */
63 template<typename V,typename T> class bezier;
65 //! Cubic Bezier Curve Base Class
66 // This generic implementation uses the DeCasteljau algorithm.
67 // Works for just about anything that has an affine combination function
68 template <typename V,typename T=float>
69 class bezier_base : public std::unary_function<T,V>
80 affine_combo<value_type,time_type> affine_func;
83 bezier_base():r(0.0),s(1.0) { }
85 const value_type &a, const value_type &b, const value_type &c, const value_type &d,
86 const time_type &r=0.0, const time_type &s=1.0):
87 a(a),b(b),c(c),d(d),r(r),s(s) { sync(); }
94 operator()(time_type t)const
111 void evaluate(time_type t, value_type &f, value_type &df) const
115 value_type p1 = affine_func(
119 value_type p2 = affine_func(
124 f = affine_func(p1,p2,t);
129 void set_rs(time_type new_r, time_type new_s) { r=new_r; s=new_s; }
130 void set_r(time_type new_r) { r=new_r; }
131 void set_s(time_type new_s) { s=new_s; }
132 const time_type &get_r()const { return r; }
133 const time_type &get_s()const { return s; }
134 time_type get_dt()const { return s-r; }
136 bool intersect_hull(const bezier_base<value_type,time_type> &x)const
141 //! Bezier curve intersection function
142 /*! Calculates the time of intersection
143 ** for the calling curve.
145 ** I still have not figured out a good generic
146 ** method of doing this for a bi-infinite
147 ** cubic bezier curve calculated with the DeCasteljau
150 ** One method, although it does not work for the
151 ** entire bi-infinite curve, is to iteratively
152 ** intersect the hulls. However, we would only detect
153 ** intersections that occur between R and S.
155 ** It is entirely possible that a new construct similar
156 ** to the affine combination function will be necessary
157 ** for this to work properly.
159 ** For now, this function is BROKEN. (although it works
160 ** for the floating-point specializations, using newton's method)
162 time_type intersect(const bezier_base<value_type,time_type> &x, time_type near=0.0)const
167 /* subdivide at some time t into 2 separate curves left and right
172 * 1+2 * 0+3*1+3*2+3 l4,r1
178 0.1 2.3 -> 0.1 2 3 4 5.6
180 /* void subdivide(bezier_base *left, bezier_base *right, const time_type &time = (time_type)0.5) const
182 time_type t = (time-r)/(s-r);
187 //1st stage points to keep
192 lt.b = affine_func(a,b,t);
193 temp = affine_func(b,c,t);
194 rt.c = affine_func(c,d,t);
197 lt.c = affine_func(lt.b,temp,t);
198 rt.b = affine_func(temp,rt.c,t);
201 lt.d = rt.a = affine_func(lt.c,rt.b,t);
203 //set the time range for l,r (the inside values should be 1, 0 respectively)
207 //give back the curves
209 if(right) *right = rt;
217 operator[](int i) const
223 // Fast float implementation of a cubic bezier curve
225 class bezier_base<float,float> : public std::unary_function<float,float>
228 typedef float value_type;
229 typedef float time_type;
231 affine_combo<value_type,time_type> affine_func;
235 value_type _coeff[4];
236 time_type drs; // reciprocal of (s-r)
238 bezier_base():r(0.0),s(1.0),drs(1.0) { }
240 const value_type &a, const value_type &b, const value_type &c, const value_type &d,
241 const time_type &r=0.0, const time_type &s=1.0):
242 a(a),b(b),c(c),d(d),r(r),s(s),drs(1.0/(s-r)) { sync(); }
248 _coeff[1]= b*3 - a*3;
249 _coeff[2]= c*3 - b*6 + a*3;
250 _coeff[3]= d - c*3 + b*3 - a;
253 // Cost Summary: 4 products, 3 sums, and 1 difference.
255 operator()(time_type t)const
256 { t-=r; t*=drs; return _coeff[0]+(_coeff[1]+(_coeff[2]+(_coeff[3])*t)*t)*t; }
258 void set_rs(time_type new_r, time_type new_s) { r=new_r; s=new_s; drs=1.0/(s-r); }
259 void set_r(time_type new_r) { r=new_r; drs=1.0/(s-r); }
260 void set_s(time_type new_s) { s=new_s; drs=1.0/(s-r); }
261 const time_type &get_r()const { return r; }
262 const time_type &get_s()const { return s; }
263 time_type get_dt()const { return s-r; }
265 //! Bezier curve intersection function
266 /*! Calculates the time of intersection
267 ** for the calling curve.
269 time_type intersect(const bezier_base<value_type,time_type> &x, time_type t=0.0,int i=15)const
271 //BROKEN - the time values of the 2 curves should be independent
272 value_type system[4];
273 system[0]=_coeff[0]-x._coeff[0];
274 system[1]=_coeff[1]-x._coeff[1];
275 system[2]=_coeff[2]-x._coeff[2];
276 system[3]=_coeff[3]-x._coeff[3];
282 // Inner loop cost summary: 7 products, 5 sums, 1 difference
284 t-= (system[0]+(system[1]+(system[2]+(system[3])*t)*t)*t)/
285 (system[1]+(system[2]*2+(system[3]*3)*t)*t);
298 operator[](int i) const
303 // Fast double implementation of a cubic bezier curve
305 class bezier_base<double,float> : public std::unary_function<float,double>
308 typedef double value_type;
309 typedef float time_type;
311 affine_combo<value_type,time_type> affine_func;
315 value_type _coeff[4];
316 time_type drs; // reciprocal of (s-r)
318 bezier_base():r(0.0),s(1.0),drs(1.0) { }
320 const value_type &a, const value_type &b, const value_type &c, const value_type &d,
321 const time_type &r=0.0, const time_type &s=1.0):
322 a(a),b(b),c(c),d(d),r(r),s(s),drs(1.0/(s-r)) { sync(); }
328 _coeff[1]= b*3 - a*3;
329 _coeff[2]= c*3 - b*6 + a*3;
330 _coeff[3]= d - c*3 + b*3 - a;
333 // 4 products, 3 sums, and 1 difference.
335 operator()(time_type t)const
336 { t-=r; t*=drs; return _coeff[0]+(_coeff[1]+(_coeff[2]+(_coeff[3])*t)*t)*t; }
338 void set_rs(time_type new_r, time_type new_s) { r=new_r; s=new_s; drs=1.0/(s-r); }
339 void set_r(time_type new_r) { r=new_r; drs=1.0/(s-r); }
340 void set_s(time_type new_s) { s=new_s; drs=1.0/(s-r); }
341 const time_type &get_r()const { return r; }
342 const time_type &get_s()const { return s; }
343 time_type get_dt()const { return s-r; }
345 //! Bezier curve intersection function
346 /*! Calculates the time of intersection
347 ** for the calling curve.
349 time_type intersect(const bezier_base<value_type,time_type> &x, time_type t=0.0,int i=15)const
351 //BROKEN - the time values of the 2 curves should be independent
352 value_type system[4];
353 system[0]=_coeff[0]-x._coeff[0];
354 system[1]=_coeff[1]-x._coeff[1];
355 system[2]=_coeff[2]-x._coeff[2];
356 system[3]=_coeff[3]-x._coeff[3];
362 // Inner loop: 7 products, 5 sums, 1 difference
364 t-= (system[0]+(system[1]+(system[2]+(system[3])*t)*t)*t)/
365 (system[1]+(system[2]*2+(system[3]*3)*t)*t);
378 operator[](int i) const
384 // Fast double implementation of a cubic bezier curve
387 template <class T,unsigned int FIXED_BITS>
388 class bezier_base<fixed_base<T,FIXED_BITS> > : std::unary_function<fixed_base<T,FIXED_BITS>,fixed_base<T,FIXED_BITS> >
391 typedef fixed_base<T,FIXED_BITS> value_type;
392 typedef fixed_base<T,FIXED_BITS> time_type;
395 affine_combo<value_type,time_type> affine_func;
399 value_type _coeff[4];
400 time_type drs; // reciprocal of (s-r)
402 bezier_base():r(0.0),s(1.0),drs(1.0) { }
404 const value_type &a, const value_type &b, const value_type &c, const value_type &d,
405 const time_type &r=0, const time_type &s=1):
406 a(a),b(b),c(c),d(d),r(r),s(s),drs(1.0/(s-r)) { sync(); }
410 drs=time_type(1)/(s-r);
412 _coeff[1]= b*3 - a*3;
413 _coeff[2]= c*3 - b*6 + a*3;
414 _coeff[3]= d - c*3 + b*3 - a;
417 // 4 products, 3 sums, and 1 difference.
419 operator()(time_type t)const
420 { t-=r; t*=drs; return _coeff[0]+(_coeff[1]+(_coeff[2]+(_coeff[3])*t)*t)*t; }
422 void set_rs(time_type new_r, time_type new_s) { r=new_r; s=new_s; drs=time_type(1)/(s-r); }
423 void set_r(time_type new_r) { r=new_r; drs=time_type(1)/(s-r); }
424 void set_s(time_type new_s) { s=new_s; drs=time_type(1)/(s-r); }
425 const time_type &get_r()const { return r; }
426 const time_type &get_s()const { return s; }
427 time_type get_dt()const { return s-r; }
429 //! Bezier curve intersection function
430 //! Calculates the time of intersection
431 // for the calling curve.
433 time_type intersect(const bezier_base<value_type,time_type> &x, time_type t=0,int i=15)const
435 value_type system[4];
436 system[0]=_coeff[0]-x._coeff[0];
437 system[1]=_coeff[1]-x._coeff[1];
438 system[2]=_coeff[2]-x._coeff[2];
439 system[3]=_coeff[3]-x._coeff[3];
445 // Inner loop: 7 products, 5 sums, 1 difference
447 t-=(time_type) ( (system[0]+(system[1]+(system[2]+(system[3])*t)*t)*t)/
448 (system[1]+(system[2]*2+(system[3]*3)*t)*t) );
461 operator[](int i) const
471 template <typename V, typename T>
472 class bezier_iterator
476 struct iterator_category {};
477 typedef V value_type;
478 typedef T difference_type;
484 bezier_base<V,T> curve;
490 operator*(void)const { return curve(t); }
491 const surface_iterator&
494 { t+=dt; return &this; }
496 const surface_iterator&
498 { hermite_iterator _tmp=*this; t+=dt; return _tmp; }
500 const surface_iterator&
502 { t-=dt; return &this; }
504 const surface_iterator&
506 { hermite_iterator _tmp=*this; t-=dt; return _tmp; }
510 operator+(difference_type __n) const
511 { return surface_iterator(data+__n[0]+__n[1]*pitch,pitch); }
514 operator-(difference_type __n) const
515 { return surface_iterator(data-__n[0]-__n[1]*pitch,pitch); }
520 template <typename V,typename T=float>
521 class bezier : public bezier_base<V,T>
524 typedef V value_type;
526 typedef float distance_type;
527 typedef bezier_iterator<V,T> iterator;
528 typedef bezier_iterator<V,T> const_iterator;
530 distance_func<value_type> dist;
532 using bezier_base<V,T>::get_r;
533 using bezier_base<V,T>::get_s;
534 using bezier_base<V,T>::get_dt;
538 bezier(const value_type &a, const value_type &b, const value_type &c, const value_type &d):
539 bezier_base<V,T>(a,b,c,d) { }
542 const_iterator begin()const;
543 const_iterator end()const;
545 time_type find_closest(bool fast, const value_type& x, int i=7)const
549 value_type array[4] = {
550 bezier<V,T>::operator[](0),
551 bezier<V,T>::operator[](1),
552 bezier<V,T>::operator[](2),
553 bezier<V,T>::operator[](3)};
554 return NearestPointOnCurve(x, array);
558 time_type r(0), s(1);
559 float t((r+s)*0.5); /* half way between r and s */
563 // compare 33% of the way between r and s with 67% of the way between r and s
564 if(dist(operator()((s-r)*(1.0/3.0)+r), x) <
565 dist(operator()((s-r)*(2.0/3.0)+r), x))
575 distance_type find_distance(time_type r, time_type s, int steps=7)const
577 const time_type inc((s-r)/steps);
578 distance_type ret(0);
579 value_type last(operator()(r));
581 for(r+=inc;r<s;r+=inc)
583 const value_type n(operator()(r));
584 ret+=dist.uncook(dist(last,n));
587 ret+=dist.uncook(dist(last,operator()(r)))*(s-(r-inc))/inc;
592 distance_type length()const { return find_distance(get_r(),get_s()); }
594 /* subdivide at some time t into 2 separate curves left and right
599 * 1+2 * 0+3*1+3*2+3 l4,r1
605 0.1 2.3 -> 0.1 2 3 4 5.6
607 void subdivide(bezier *left, bezier *right, const time_type &time = (time_type)0.5) const
609 time_type t=(t-get_r())/get_dt();
613 const value_type& a((*this)[0]);
614 const value_type& b((*this)[1]);
615 const value_type& c((*this)[2]);
616 const value_type& d((*this)[3]);
618 //1st stage points to keep
623 lt[1] = affine_func(a,b,t);
624 temp = affine_func(b,c,t);
625 rt[2] = affine_func(c,d,t);
628 lt[2] = affine_func(lt[1],temp,t);
629 rt[1] = affine_func(temp,rt[2],t);
632 lt[3] = rt[0] = affine_func(lt[2],rt[1],t);
634 //set the time range for l,r (the inside values should be 1, 0 respectively)
641 //give back the curves
643 if(right) *right = rt;
647 void evaluate(time_type t, value_type &f, value_type &df) const
649 t=(t-get_r())/get_dt();
651 const value_type& a((*this)[0]);
652 const value_type& b((*this)[1]);
653 const value_type& c((*this)[2]);
654 const value_type& d((*this)[3]);
656 const value_type p1 = affine_func(
660 const value_type p2 = affine_func(
665 f = affine_func(p1,p2,t);
672 * Evaluate a Bezier curve at a particular parameter value
673 * Fill in control points for resulting sub-curves if "Left" and
674 * "Right" are non-null.
676 * int degree; Degree of bezier curve
677 * value_type *VT; Control pts
678 * time_type t; Parameter value
679 * value_type *Left; RETURN left half ctl pts
680 * value_type *Right; RETURN right half ctl pts
682 static value_type Bezier(value_type *VT, int degree, time_type t, value_type *Left, value_type *Right)
684 int i, j; /* Index variables */
685 value_type Vtemp[W_DEGREE+1][W_DEGREE+1];
687 /* Copy control points */
688 for (j = 0; j <= degree; j++)
691 /* Triangle computation */
692 for (i = 1; i <= degree; i++)
693 for (j =0 ; j <= degree - i; j++)
695 Vtemp[i][j][0] = (1.0 - t) * Vtemp[i-1][j][0] + t * Vtemp[i-1][j+1][0];
696 Vtemp[i][j][1] = (1.0 - t) * Vtemp[i-1][j][1] + t * Vtemp[i-1][j+1][1];
700 for (j = 0; j <= degree; j++)
701 Left[j] = Vtemp[j][0];
704 for (j = 0; j <= degree; j++)
705 Right[j] = Vtemp[degree-j][j];
707 return (Vtemp[degree][0]);
712 * Count the number of times a Bezier control polygon
713 * crosses the 0-axis. This number is >= the number of roots.
715 * value_type *VT; Control pts of Bezier curve
717 static int CrossingCount(value_type *VT)
720 int n_crossings = 0; /* Number of zero-crossings */
721 int sign, old_sign; /* Sign of coefficients */
723 sign = old_sign = SGN(VT[0][1]);
724 for (i = 1; i <= W_DEGREE; i++)
726 sign = SGN(VT[i][1]);
727 if (sign != old_sign) n_crossings++;
735 * ControlPolygonFlatEnough :
736 * Check if the control polygon of a Bezier curve is flat enough
737 * for recursive subdivision to bottom out.
739 * value_type *VT; Control points
741 static int ControlPolygonFlatEnough(value_type *VT)
743 int i; /* Index variable */
744 distance_type distance[W_DEGREE]; /* Distances from pts to line */
745 distance_type max_distance_above; /* maximum of these */
746 distance_type max_distance_below;
747 time_type intercept_1, intercept_2, left_intercept, right_intercept;
748 distance_type a, b, c; /* Coefficients of implicit */
749 /* eqn for line from VT[0]-VT[deg] */
750 /* Find the perpendicular distance */
751 /* from each interior control point to */
752 /* line connecting VT[0] and VT[W_DEGREE] */
754 distance_type abSquared;
756 /* Derive the implicit equation for line connecting first *
757 * and last control points */
758 a = VT[0][1] - VT[W_DEGREE][1];
759 b = VT[W_DEGREE][0] - VT[0][0];
760 c = VT[0][0] * VT[W_DEGREE][1] - VT[W_DEGREE][0] * VT[0][1];
762 abSquared = (a * a) + (b * b);
764 for (i = 1; i < W_DEGREE; i++)
766 /* Compute distance from each of the points to that line */
767 distance[i] = a * VT[i][0] + b * VT[i][1] + c;
768 if (distance[i] > 0.0) distance[i] = (distance[i] * distance[i]) / abSquared;
769 if (distance[i] < 0.0) distance[i] = -(distance[i] * distance[i]) / abSquared;
773 /* Find the largest distance */
774 max_distance_above = max_distance_below = 0.0;
776 for (i = 1; i < W_DEGREE; i++)
778 if (distance[i] < 0.0) max_distance_below = MIN(max_distance_below, distance[i]);
779 if (distance[i] > 0.0) max_distance_above = MAX(max_distance_above, distance[i]);
782 /* Implicit equation for "above" line */
783 intercept_1 = -(c + max_distance_above)/a;
785 /* Implicit equation for "below" line */
786 intercept_2 = -(c + max_distance_below)/a;
788 /* Compute intercepts of bounding box */
789 left_intercept = MIN(intercept_1, intercept_2);
790 right_intercept = MAX(intercept_1, intercept_2);
792 return 0.5 * (right_intercept-left_intercept) < BEZIER_EPSILON ? 1 : 0;
796 * ComputeXIntercept :
797 * Compute intersection of chord from first control point to last
800 * value_type *VT; Control points
802 static time_type ComputeXIntercept(value_type *VT)
804 distance_type YNM = VT[W_DEGREE][1] - VT[0][1];
805 return (YNM*VT[0][0] - (VT[W_DEGREE][0] - VT[0][0])*VT[0][1]) / YNM;
810 * Given a 5th-degree equation in Bernstein-Bezier form, find
811 * all of the roots in the interval [0, 1]. Return the number
814 * value_type *w; The control points
815 * time_type *t; RETURN candidate t-values
816 * int depth; The depth of the recursion
818 static int FindRoots(value_type *w, time_type *t, int depth)
821 value_type Left[W_DEGREE+1]; /* New left and right */
822 value_type Right[W_DEGREE+1]; /* control polygons */
823 int left_count; /* Solution count from */
824 int right_count; /* children */
825 time_type left_t[W_DEGREE+1]; /* Solutions from kids */
826 time_type right_t[W_DEGREE+1];
828 switch (CrossingCount(w))
831 { /* No solutions here */
835 { /* Unique solution */
836 /* Stop recursion when the tree is deep enough */
837 /* if deep enough, return 1 solution at midpoint */
838 if (depth >= MAXDEPTH)
840 t[0] = (w[0][0] + w[W_DEGREE][0]) / 2.0;
843 if (ControlPolygonFlatEnough(w))
845 t[0] = ComputeXIntercept(w);
852 /* Otherwise, solve recursively after */
853 /* subdividing control polygon */
854 Bezier(w, W_DEGREE, 0.5, Left, Right);
855 left_count = FindRoots(Left, left_t, depth+1);
856 right_count = FindRoots(Right, right_t, depth+1);
858 /* Gather solutions together */
859 for (i = 0; i < left_count; i++) t[i] = left_t[i];
860 for (i = 0; i < right_count; i++) t[i+left_count] = right_t[i];
862 /* Send back total number of solutions */
863 return (left_count+right_count);
867 * ConvertToBezierForm :
868 * Given a point and a Bezier curve, generate a 5th-degree
869 * Bezier-format equation whose solution finds the point on the
870 * curve nearest the user-defined point.
872 * value_type& P; The point to find t for
873 * value_type *VT; The control points
875 static void ConvertToBezierForm(const value_type& P, value_type *VT, value_type w[W_DEGREE+1])
877 int i, j, k, m, n, ub, lb;
878 int row, column; /* Table indices */
879 value_type c[DEGREE+1]; /* VT(i)'s - P */
880 value_type d[DEGREE]; /* VT(i+1) - VT(i) */
881 distance_type cdTable[3][4]; /* Dot product of c, d */
882 static distance_type z[3][4] = { /* Precomputed "z" for cubics */
883 {1.0, 0.6, 0.3, 0.1},
884 {0.4, 0.6, 0.6, 0.4},
885 {0.1, 0.3, 0.6, 1.0}};
887 /* Determine the c's -- these are vectors created by subtracting */
888 /* point P from each of the control points */
889 for (i = 0; i <= DEGREE; i++)
892 /* Determine the d's -- these are vectors created by subtracting */
893 /* each control point from the next */
894 for (i = 0; i <= DEGREE - 1; i++)
895 d[i] = (VT[i+1] - VT[i]) * 3.0;
897 /* Create the c,d table -- this is a table of dot products of the */
899 for (row = 0; row <= DEGREE - 1; row++)
900 for (column = 0; column <= DEGREE; column++)
901 cdTable[row][column] = d[row] * c[column];
903 /* Now, apply the z's to the dot products, on the skew diagonal */
904 /* Also, set up the x-values, making these "points" */
905 for (i = 0; i <= W_DEGREE; i++)
907 w[i][0] = (distance_type)(i) / W_DEGREE;
913 for (k = 0; k <= n + m; k++)
917 for (i = lb; i <= ub; i++)
920 w[i+j][1] += cdTable[j][i] * z[j][i];
926 * NearestPointOnCurve :
927 * Compute the parameter value of the point on a Bezier
928 * curve segment closest to some arbtitrary, user-input point.
929 * Return the point on the curve at that parameter value.
931 * value_type& P; The user-supplied point
932 * value_type *VT; Control points of cubic Bezier
934 static time_type NearestPointOnCurve(const value_type& P, value_type VT[4])
936 value_type w[W_DEGREE+1]; /* Ctl pts of 5th-degree curve */
937 time_type t_candidate[W_DEGREE]; /* Possible roots */
938 int n_solutions; /* Number of roots found */
939 time_type t; /* Parameter value of closest pt */
941 /* Convert problem to 5th-degree Bezier form */
942 ConvertToBezierForm(P, VT, w);
944 /* Find all possible roots of 5th-degree equation */
945 n_solutions = FindRoots(w, t_candidate, 0);
947 /* Compare distances of P to all candidates, and to t=0, and t=1 */
949 distance_type dist, new_dist;
953 /* Check distance to beginning of curve, where t = 0 */
954 dist = (P - VT[0]).mag_squared();
957 /* Find distances for candidate points */
958 for (i = 0; i < n_solutions; i++)
960 p = Bezier(VT, DEGREE, t_candidate[i], (value_type *)NULL, (value_type *)NULL);
961 new_dist = (P - p).mag_squared();
969 /* Finally, look at distance to end point, where t = 1.0 */
970 new_dist = (P - VT[DEGREE]).mag_squared();
978 /* Return the point on the curve at parameter value t */
985 /* === E X T E R N S ======================================================= */
987 /* === E N D =============================================================== */