/*! ========================================================================
** Extended Template Library
** Bezier Template Class Implementation
-** $Id: _bezier.h,v 1.1.1.1 2005/01/04 01:31:46 darco Exp $
+** $Id$
**
** 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
/* === H E A D E R S ======================================================= */
#include "_curve_func.h"
-#include <ETL/fixed>
+#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 ======================================= */
time_type r,s;
protected:
- affine_combo<value_type,time_type> affine_func;
+ affine_combo<value_type,time_type> affine_func;
public:
bezier_base():r(0.0),s(1.0) { }
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()
{
}
,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)
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; }
** 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.
{
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
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
+ //1st stage points to keep
lt.a = a;
rt.d = d;
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;
+ if(right) *right = rt;
}
- */
+ */
value_type &
operator[](int i)
{ return (&a)[i]; }
typedef float value_type;
typedef float time_type;
private:
- affine_combo<value_type,time_type> affine_func;
+ affine_combo<value_type,time_type> affine_func;
value_type a,b,c,d;
time_type r,s;
_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
typedef V value_type;
typedef T difference_type;
typedef V reference;
-
+
private:
difference_type t;
difference_type dt;
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); }
operator-(difference_type __n) const
{ return surface_iterator(data-__n[0]-__n[1]*pitch,pitch); }
*/
-
+
};
template <typename V,typename T=float>
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:
+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(const value_type& x, int i=7, time_type r=(0), time_type s=(1))const
+
+ time_type find_closest(bool fast, const value_type& x, int i=7)const
{
- float t((r+s)*0.5);
- for(;i;i--)
- {
- 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);
+ 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;
}
- return t;
}
-
distance_type find_distance(time_type r, time_type s, int steps=7)const
{
const time_type inc((s-r)/steps);
distance_type ret(0);
value_type last(operator()(r));
-
+
for(r+=inc;r<s;r+=inc)
{
const value_type n(operator()(r));
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
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=(t-get_r())/get_dt();
+ 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
+ //1st stage points to keep
lt[0] = a;
rt[3] = d;
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;
+ 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& 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)
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