X-Git-Url: https://git.pterodactylus.net/?a=blobdiff_plain;f=ETL%2Ftrunk%2FETL%2F_bezier.h;h=fe6b17017f1e89b13a62f40efbc9f3ed67322d58;hb=9459638ad6797b8139f1e9f0715c96076dbf0890;hp=78ae6327568e8f8c378887af08f5df33c4519e7b;hpb=36d01e1527fda602a9aed07d209eb34e6770bca8;p=synfig.git diff --git a/ETL/trunk/ETL/_bezier.h b/ETL/trunk/ETL/_bezier.h index 78ae632..fe6b170 100644 --- a/ETL/trunk/ETL/_bezier.h +++ b/ETL/trunk/ETL/_bezier.h @@ -4,6 +4,7 @@ ** $Id$ ** ** 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 @@ -24,16 +25,37 @@ /* === S T A R T =========================================================== */ -#ifndef __ETL_BEZIER_H -#define __ETL_BEZIER_H +#ifndef __ETL__BEZIER_H +#define __ETL__BEZIER_H /* === H E A D E R S ======================================================= */ #include "_curve_func.h" -#include +#include // for ldexp +// #include // 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 ======================================= */ @@ -522,24 +544,40 @@ public: 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::operator[](0), + bezier::operator[](1), + bezier::operator[](2), + bezier::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); + if (!inc) return 0; distance_type ret(0); value_type last(operator()(r)); @@ -571,7 +609,7 @@ public: */ 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; @@ -630,6 +668,319 @@ public: 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