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+/* Copyright (C) 2001-2007 Peter Selinger.
+ * This file is part of Potrace. It is free software and it is covered
+ * by the GNU General Public License. See the file COPYING for details. */
+
+/* $Id: trace.c 147 2007-04-09 00:44:09Z selinger $ */
+/* transform jaggy paths into smooth curves */
+
+#include <stdio.h>
+#include <cmath>
+#include <stdlib.h>
+#include <string.h>
+
+#include <potracelib.h>
+#include <curve.h>
+#include <lists.h>
+#include <auxiliary.h>
+#include <trace.h>
+#include <progress.h>
+
+#define INFTY 10000000 /* it suffices that this is longer than any
+ * path; it need not be really infinite */
+#define COS179 -0.999847695156 /* the cosine of 179 degrees */
+
+/* ---------------------------------------------------------------------- */
+#define SAFE_MALLOC( var, n, typ ) \
+ if( ( var = (typ*) malloc( (n) * sizeof(typ) ) ) == NULL ) \
+ goto malloc_error
+
+/* ---------------------------------------------------------------------- */
+/* auxiliary functions */
+
+/* return a direction that is 90 degrees counterclockwise from p2-p0,
+ * but then restricted to one of the major wind directions (n, nw, w, etc) */
+static inline point_t dorth_infty( dpoint_t p0, dpoint_t p2 )
+{
+ point_t r;
+
+ r.y = sign( p2.x - p0.x );
+ r.x = -sign( p2.y - p0.y );
+
+ return r;
+}
+
+
+/* return (p1-p0)x(p2-p0), the area of the parallelogram */
+static inline double dpara( dpoint_t p0, dpoint_t p1, dpoint_t p2 )
+{
+ double x1, y1, x2, y2;
+
+ x1 = p1.x - p0.x;
+ y1 = p1.y - p0.y;
+ x2 = p2.x - p0.x;
+ y2 = p2.y - p0.y;
+
+ return x1 * y2 - x2 * y1;
+}
+
+
+/* ddenom/dpara have the property that the square of radius 1 centered
+ * at p1 intersects the line p0p2 iff |dpara(p0,p1,p2)| <= ddenom(p0,p2) */
+static inline double ddenom( dpoint_t p0, dpoint_t p2 )
+{
+ point_t r = dorth_infty( p0, p2 );
+
+ return r.y * (p2.x - p0.x) - r.x * (p2.y - p0.y);
+}
+
+
+/* return 1 if a <= b < c < a, in a cyclic sense (mod n) */
+static inline int cyclic( int a, int b, int c )
+{
+ if( a<=c )
+ {
+ return a<=b && b<c;
+ }
+ else
+ {
+ return a<=b || b<c;
+ }
+}
+
+
+/* determine the center and slope of the line i..j. Assume i<j. Needs
+ * "sum" components of p to be set. */
+static void pointslope( privpath_t* pp, int i, int j, dpoint_t* ctr, dpoint_t* dir )
+{
+ /* assume i<j */
+
+ int n = pp->len;
+ sums_t* sums = pp->sums;
+
+ double x, y, x2, xy, y2;
+ double k;
+ double a, b, c, lambda2, l;
+ int r = 0; /* rotations from i to j */
+
+ while( j>=n )
+ {
+ j -= n;
+ r += 1;
+ }
+
+ while( i>=n )
+ {
+ i -= n;
+ r -= 1;
+ }
+
+ while( j<0 )
+ {
+ j += n;
+ r -= 1;
+ }
+
+ while( i<0 )
+ {
+ i += n;
+ r += 1;
+ }
+
+ x = sums[j + 1].x - sums[i].x + r * sums[n].x;
+ y = sums[j + 1].y - sums[i].y + r * sums[n].y;
+ x2 = sums[j + 1].x2 - sums[i].x2 + r * sums[n].x2;
+ xy = sums[j + 1].xy - sums[i].xy + r * sums[n].xy;
+ y2 = sums[j + 1].y2 - sums[i].y2 + r * sums[n].y2;
+ k = j + 1 - i + r * n;
+
+ ctr->x = x / k;
+ ctr->y = y / k;
+
+ a = (x2 - (double) x * x / k) / k;
+ b = (xy - (double) x * y / k) / k;
+ c = (y2 - (double) y * y / k) / k;
+
+ lambda2 = ( a + c + sqrt( (a - c) * (a - c) + 4 * b * b ) ) / 2; /* larger e.value */
+
+ /* now find e.vector for lambda2 */
+ a -= lambda2;
+ c -= lambda2;
+
+ if( fabs( a ) >= fabs( c ) )
+ {
+ l = sqrt( a * a + b * b );
+ if( l!=0 )
+ {
+ dir->x = -b / l;
+ dir->y = a / l;
+ }
+ }
+ else
+ {
+ l = sqrt( c * c + b * b );
+ if( l!=0 )
+ {
+ dir->x = -c / l;
+ dir->y = b / l;
+ }
+ }
+ if( l==0 )
+ {
+ dir->x = dir->y = 0; /* sometimes this can happen when k=4:
+ * the two eigenvalues coincide */
+ }
+}
+
+
+/* the type of (affine) quadratic forms, represented as symmetric 3x3
+ * matrices. The value of the quadratic form at a vector (x,y) is v^t
+ * Q v, where v = (x,y,1)^t. */
+typedef double quadform_t[3][3];
+
+/* Apply quadratic form Q to vector w = (w.x,w.y) */
+static inline double quadform( quadform_t Q, dpoint_t w )
+{
+ double v[3];
+ int i, j;
+ double sum;
+
+ v[0] = w.x;
+ v[1] = w.y;
+ v[2] = 1;
+ sum = 0.0;
+
+ for( i = 0; i<3; i++ )
+ {
+ for( j = 0; j<3; j++ )
+ {
+ sum += v[i] *Q[i][j] *v[j];
+ }
+ }
+
+ return sum;
+}
+
+
+/* calculate p1 x p2 */
+static inline int xprod( point_t p1, point_t p2 )
+{
+ return p1.x * p2.y - p1.y * p2.x;
+}
+
+
+/* calculate (p1-p0)x(p3-p2) */
+static inline double cprod( dpoint_t p0, dpoint_t p1, dpoint_t p2, dpoint_t p3 )
+{
+ double x1, y1, x2, y2;
+
+ x1 = p1.x - p0.x;
+ y1 = p1.y - p0.y;
+ x2 = p3.x - p2.x;
+ y2 = p3.y - p2.y;
+
+ return x1 * y2 - x2 * y1;
+}
+
+
+/* calculate (p1-p0)*(p2-p0) */
+static inline double iprod( dpoint_t p0, dpoint_t p1, dpoint_t p2 )
+{
+ double x1, y1, x2, y2;
+
+ x1 = p1.x - p0.x;
+ y1 = p1.y - p0.y;
+ x2 = p2.x - p0.x;
+ y2 = p2.y - p0.y;
+
+ return x1 * x2 + y1 * y2;
+}
+
+
+/* calculate (p1-p0)*(p3-p2) */
+static inline double iprod1( dpoint_t p0, dpoint_t p1, dpoint_t p2, dpoint_t p3 )
+{
+ double x1, y1, x2, y2;
+
+ x1 = p1.x - p0.x;
+ y1 = p1.y - p0.y;
+ x2 = p3.x - p2.x;
+ y2 = p3.y - p2.y;
+
+ return x1 * x2 + y1 * y2;
+}
+
+
+/* calculate distance between two points */
+static inline double ddist( dpoint_t p, dpoint_t q )
+{
+ return sqrt( sq( p.x - q.x ) + sq( p.y - q.y ) );
+}
+
+
+/* calculate point of a bezier curve */
+static inline dpoint_t bezier( double t, dpoint_t p0, dpoint_t p1, dpoint_t p2, dpoint_t p3 )
+{
+ double s = 1 - t;
+ dpoint_t res;
+
+ /* Note: a good optimizing compiler (such as gcc-3) reduces the
+ * following to 16 multiplications, using common subexpression
+ * elimination. */
+
+ res.x = s * s * s * p0.x + 3 * (s * s * t) * p1.x + 3 * (t * t * s) * p2.x + t * t * t * p3.x;
+ res.y = s * s * s * p0.y + 3 * (s * s * t) * p1.y + 3 * (t * t * s) * p2.y + t * t * t * p3.y;
+
+ return res;
+}
+
+
+/* calculate the point t in [0..1] on the (convex) bezier curve
+ * (p0,p1,p2,p3) which is tangent to q1-q0. Return -1.0 if there is no
+ * solution in [0..1]. */
+static double tangent( dpoint_t p0,
+ dpoint_t p1,
+ dpoint_t p2,
+ dpoint_t p3,
+ dpoint_t q0,
+ dpoint_t q1 )
+{
+ double A, B, C; /* (1-t)^2 A + 2(1-t)t B + t^2 C = 0 */
+ double a, b, c; /* a t^2 + b t + c = 0 */
+ double d, s, r1, r2;
+
+ A = cprod( p0, p1, q0, q1 );
+ B = cprod( p1, p2, q0, q1 );
+ C = cprod( p2, p3, q0, q1 );
+
+ a = A - 2 * B + C;
+ b = -2 * A + 2 * B;
+ c = A;
+
+ d = b * b - 4 * a * c;
+
+ if( a==0 || d<0 )
+ {
+ return -1.0;
+ }
+
+ s = sqrt( d );
+
+ r1 = (-b + s) / (2 * a);
+ r2 = (-b - s) / (2 * a);
+
+ if( r1 >= 0 && r1 <= 1 )
+ {
+ return r1;
+ }
+ else if( r2 >= 0 && r2 <= 1 )
+ {
+ return r2;
+ }
+ else
+ {
+ return -1.0;
+ }
+}
+
+
+/* ---------------------------------------------------------------------- */
+
+/* Preparation: fill in the sum* fields of a path (used for later
+ * rapid summing). Return 0 on success, 1 with errno set on
+ * failure. */
+static int calc_sums( privpath_t* pp )
+{
+ int i, x, y;
+ int n = pp->len;
+
+ SAFE_MALLOC( pp->sums, pp->len + 1, sums_t );
+
+ /* origin */
+ pp->x0 = pp->pt[0].x;
+ pp->y0 = pp->pt[0].y;
+
+ /* preparatory computation for later fast summing */
+ pp->sums[0].x2 = pp->sums[0].xy = pp->sums[0].y2 = pp->sums[0].x = pp->sums[0].y = 0;
+ for( i = 0; i<n; i++ )
+ {
+ x = pp->pt[i].x - pp->x0;
+ y = pp->pt[i].y - pp->y0;
+ pp->sums[i + 1].x = pp->sums[i].x + x;
+ pp->sums[i + 1].y = pp->sums[i].y + y;
+ pp->sums[i + 1].x2 = pp->sums[i].x2 + x * x;
+ pp->sums[i + 1].xy = pp->sums[i].xy + x * y;
+ pp->sums[i + 1].y2 = pp->sums[i].y2 + y * y;
+ }
+
+ return 0;
+
+malloc_error:
+ return 1;
+}
+
+
+/* ---------------------------------------------------------------------- */
+
+/* Stage 1: determine the straight subpaths (Sec. 2.2.1). Fill in the
+ * "lon" component of a path object (based on pt/len). For each i,
+ * lon[i] is the furthest index such that a straight line can be drawn
+ * from i to lon[i]. Return 1 on error with errno set, else 0. */
+
+/* this algorithm depends on the fact that the existence of straight
+ * subpaths is a triplewise property. I.e., there exists a straight
+ * line through squares i0,...,in iff there exists a straight line
+ * through i,j,k, for all i0<=i<j<k<=in. (Proof?) */
+
+/* this implementation of calc_lon is O(n^2). It replaces an older
+ * O(n^3) version. A "constraint" means that future points must
+ * satisfy xprod(constraint[0], cur) >= 0 and xprod(constraint[1],
+ * cur) <= 0. */
+
+/* Remark for Potrace 1.1: the current implementation of calc_lon is
+ * more complex than the implementation found in Potrace 1.0, but it
+ * is considerably faster. The introduction of the "nc" data structure
+ * means that we only have to test the constraints for "corner"
+ * points. On a typical input file, this speeds up the calc_lon
+ * function by a factor of 31.2, thereby decreasing its time share
+ * within the overall Potrace algorithm from 72.6% to 7.82%, and
+ * speeding up the overall algorithm by a factor of 3.36. On another
+ * input file, calc_lon was sped up by a factor of 6.7, decreasing its
+ * time share from 51.4% to 13.61%, and speeding up the overall
+ * algorithm by a factor of 1.78. In any case, the savings are
+ * substantial. */
+
+/* returns 0 on success, 1 on error with errno set */
+static int calc_lon( privpath_t* pp )
+{
+ point_t* pt = pp->pt;
+ int n = pp->len;
+ int i, j, k, k1;
+ int ct[4], dir;
+ point_t constraint[2];
+ point_t cur;
+ point_t off;
+ int* pivk = NULL; /* pivk[n] */
+ int* nc = NULL; /* nc[n]: next corner */
+ point_t dk; /* direction of k-k1 */
+ int a, b, c, d;
+
+ SAFE_MALLOC( pivk, n, int );
+ SAFE_MALLOC( nc, n, int );
+
+ /* initialize the nc data structure. Point from each point to the
+ * furthest future point to which it is connected by a vertical or
+ * horizontal segment. We take advantage of the fact that there is
+ * always a direction change at 0 (due to the path decomposition
+ * algorithm). But even if this were not so, there is no harm, as
+ * in practice, correctness does not depend on the word "furthest"
+ * above. */
+ k = 0;
+ for( i = n - 1; i>=0; i-- )
+ {
+ if( pt[i].x != pt[k].x && pt[i].y != pt[k].y )
+ {
+ k = i + 1; /* necessarily i<n-1 in this case */
+ }
+ nc[i] = k;
+ }
+
+ SAFE_MALLOC( pp->lon, n, int );
+
+ /* determine pivot points: for each i, let pivk[i] be the furthest k
+ * such that all j with i<j<k lie on a line connecting i,k. */
+
+ for( i = n - 1; i>=0; i-- )
+ {
+ ct[0] = ct[1] = ct[2] = ct[3] = 0;
+
+ /* keep track of "directions" that have occurred */
+ dir =
+ ( 3 + 3 * (pt[mod( i + 1, n )].x - pt[i].x) + (pt[mod( i + 1, n )].y - pt[i].y) ) / 2;
+ ct[dir]++;
+
+ constraint[0].x = 0;
+ constraint[0].y = 0;
+ constraint[1].x = 0;
+ constraint[1].y = 0;
+
+ /* find the next k such that no straight line from i to k */
+ k = nc[i];
+ k1 = i;
+ while( 1 )
+ {
+ dir = ( 3 + 3 * sign( pt[k].x - pt[k1].x ) + sign( pt[k].y - pt[k1].y ) ) / 2;
+ ct[dir]++;
+
+ /* if all four "directions" have occurred, cut this path */
+ if( ct[0] && ct[1] && ct[2] && ct[3] )
+ {
+ pivk[i] = k1;
+ goto foundk;
+ }
+
+ cur.x = pt[k].x - pt[i].x;
+ cur.y = pt[k].y - pt[i].y;
+
+ /* see if current constraint is violated */
+ if( xprod( constraint[0], cur ) < 0 || xprod( constraint[1], cur ) > 0 )
+ {
+ goto constraint_viol;
+ }
+
+ /* else, update constraint */
+ if( abs( cur.x ) <= 1 && abs( cur.y ) <= 1 )
+ {
+ /* no constraint */
+ }
+ else
+ {
+ off.x = cur.x + ( ( cur.y>=0 && (cur.y>0 || cur.x<0) ) ? 1 : -1 );
+ off.y = cur.y + ( ( cur.x<=0 && (cur.x<0 || cur.y<0) ) ? 1 : -1 );
+ if( xprod( constraint[0], off ) >= 0 )
+ {
+ constraint[0] = off;
+ }
+ off.x = cur.x + ( ( cur.y<=0 && (cur.y<0 || cur.x<0) ) ? 1 : -1 );
+ off.y = cur.y + ( ( cur.x>=0 && (cur.x>0 || cur.y<0) ) ? 1 : -1 );
+ if( xprod( constraint[1], off ) <= 0 )
+ {
+ constraint[1] = off;
+ }
+ }
+ k1 = k;
+ k = nc[k1];
+ if( !cyclic( k, i, k1 ) )
+ {
+ break;
+ }
+ }
+
+constraint_viol:
+
+ /* k1 was the last "corner" satisfying the current constraint, and
+ * k is the first one violating it. We now need to find the last
+ * point along k1..k which satisfied the constraint. */
+ dk.x = sign( pt[k].x - pt[k1].x );
+ dk.y = sign( pt[k].y - pt[k1].y );
+ cur.x = pt[k1].x - pt[i].x;
+ cur.y = pt[k1].y - pt[i].y;
+
+ /* find largest integer j such that xprod(constraint[0], cur+j*dk)
+ * >= 0 and xprod(constraint[1], cur+j*dk) <= 0. Use bilinearity
+ * of xprod. */
+ a = xprod( constraint[0], cur );
+ b = xprod( constraint[0], dk );
+ c = xprod( constraint[1], cur );
+ d = xprod( constraint[1], dk );
+
+ /* find largest integer j such that a+j*b>=0 and c+j*d<=0. This
+ * can be solved with integer arithmetic. */
+ j = INFTY;
+ if( b<0 )
+ {
+ j = floordiv( a, -b );
+ }
+ if( d>0 )
+ {
+ j = min( j, floordiv( -c, d ) );
+ }
+ pivk[i] = mod( k1 + j, n );
+foundk:
+ ;
+ } /* for i */
+
+ /* clean up: for each i, let lon[i] be the largest k such that for
+ * all i' with i<=i'<k, i'<k<=pivk[i']. */
+
+ j = pivk[n - 1];
+ pp->lon[n - 1] = j;
+ for( i = n - 2; i>=0; i-- )
+ {
+ if( cyclic( i + 1, pivk[i], j ) )
+ {
+ j = pivk[i];
+ }
+ pp->lon[i] = j;
+ }
+
+ for( i = n - 1; cyclic( mod( i + 1, n ), j, pp->lon[i] ); i-- )
+ {
+ pp->lon[i] = j;
+ }
+
+ free( pivk );
+ free( nc );
+ return 0;
+
+malloc_error:
+ free( pivk );
+ free( nc );
+ return 1;
+}
+
+
+/* ---------------------------------------------------------------------- */
+/* Stage 2: calculate the optimal polygon (Sec. 2.2.2-2.2.4). */
+
+/* Auxiliary function: calculate the penalty of an edge from i to j in
+ * the given path. This needs the "lon" and "sum*" data. */
+
+static double penalty3( privpath_t* pp, int i, int j )
+{
+ int n = pp->len;
+ point_t* pt = pp->pt;
+ sums_t* sums = pp->sums;
+
+ /* assume 0<=i<j<=n */
+ double x, y, x2, xy, y2;
+ double k;
+ double a, b, c, s;
+ double px, py, ex, ey;
+
+ int r = 0; /* rotations from i to j */
+
+ if( j>=n )
+ {
+ j -= n;
+ r += 1;
+ }
+
+ x = sums[j + 1].x - sums[i].x + r * sums[n].x;
+ y = sums[j + 1].y - sums[i].y + r * sums[n].y;
+ x2 = sums[j + 1].x2 - sums[i].x2 + r * sums[n].x2;
+ xy = sums[j + 1].xy - sums[i].xy + r * sums[n].xy;
+ y2 = sums[j + 1].y2 - sums[i].y2 + r * sums[n].y2;
+ k = j + 1 - i + r * n;
+
+ px = (pt[i].x + pt[j].x) / 2.0 - pt[0].x;
+ py = (pt[i].y + pt[j].y) / 2.0 - pt[0].y;
+ ey = (pt[j].x - pt[i].x);
+ ex = -(pt[j].y - pt[i].y);
+
+ a = ( (x2 - 2 * x * px) / k + px * px );
+ b = ( (xy - x * py - y * px) / k + px * py );
+ c = ( (y2 - 2 * y * py) / k + py * py );
+
+ s = ex * ex * a + 2 * ex * ey * b + ey * ey * c;
+
+ return sqrt( s );
+}
+
+
+/* find the optimal polygon. Fill in the m and po components. Return 1
+ * on failure with errno set, else 0. Non-cyclic version: assumes i=0
+ * is in the polygon. Fixme: ### implement cyclic version. */
+static int bestpolygon( privpath_t* pp )
+{
+ int i, j, m, k;
+ int n = pp->len;
+ double* pen = NULL; /* pen[n+1]: penalty vector */
+ int* prev = NULL; /* prev[n+1]: best path pointer vector */
+ int* clip0 = NULL; /* clip0[n]: longest segment pointer, non-cyclic */
+ int* clip1 = NULL; /* clip1[n+1]: backwards segment pointer, non-cyclic */
+ int* seg0 = NULL; /* seg0[m+1]: forward segment bounds, m<=n */
+ int* seg1 = NULL; /* seg1[m+1]: backward segment bounds, m<=n */
+ double thispen;
+ double best;
+ int c;
+
+ SAFE_MALLOC( pen, n + 1, double );
+ SAFE_MALLOC( prev, n + 1, int );
+ SAFE_MALLOC( clip0, n, int );
+ SAFE_MALLOC( clip1, n + 1, int );
+ SAFE_MALLOC( seg0, n + 1, int );
+ SAFE_MALLOC( seg1, n + 1, int );
+
+ /* calculate clipped paths */
+ for( i = 0; i<n; i++ )
+ {
+ c = mod( pp->lon[mod( i - 1, n )] - 1, n );
+ if( c == i )
+ {
+ c = mod( i + 1, n );
+ }
+ if( c < i )
+ {
+ clip0[i] = n;
+ }
+ else
+ {
+ clip0[i] = c;
+ }
+ }
+
+ /* calculate backwards path clipping, non-cyclic. j <= clip0[i] iff
+ * clip1[j] <= i, for i,j=0..n. */
+ j = 1;
+ for( i = 0; i<n; i++ )
+ {
+ while( j <= clip0[i] )
+ {
+ clip1[j] = i;
+ j++;
+ }
+ }
+
+ /* calculate seg0[j] = longest path from 0 with j segments */
+ i = 0;
+ for( j = 0; i<n; j++ )
+ {
+ seg0[j] = i;
+ i = clip0[i];
+ }
+
+ seg0[j] = n;
+ m = j;
+
+ /* calculate seg1[j] = longest path to n with m-j segments */
+ i = n;
+ for( j = m; j>0; j-- )
+ {
+ seg1[j] = i;
+ i = clip1[i];
+ }
+
+ seg1[0] = 0;
+
+ /* now find the shortest path with m segments, based on penalty3 */
+
+ /* note: the outer 2 loops jointly have at most n interations, thus
+ * the worst-case behavior here is quadratic. In practice, it is
+ * close to linear since the inner loop tends to be short. */
+ pen[0] = 0;
+ for( j = 1; j<=m; j++ )
+ {
+ for( i = seg1[j]; i<=seg0[j]; i++ )
+ {
+ best = -1;
+ for( k = seg0[j - 1]; k>=clip1[i]; k-- )
+ {
+ thispen = penalty3( pp, k, i ) + pen[k];
+ if( best < 0 || thispen < best )
+ {
+ prev[i] = k;
+ best = thispen;
+ }
+ }
+
+ pen[i] = best;
+ }
+ }
+
+ pp->m = m;
+ SAFE_MALLOC( pp->po, m, int );
+
+ /* read off shortest path */
+ for( i = n, j = m - 1; i>0; j-- )
+ {
+ i = prev[i];
+ pp->po[j] = i;
+ }
+
+ free( pen );
+ free( prev );
+ free( clip0 );
+ free( clip1 );
+ free( seg0 );
+ free( seg1 );
+ return 0;
+
+malloc_error:
+ free( pen );
+ free( prev );
+ free( clip0 );
+ free( clip1 );
+ free( seg0 );
+ free( seg1 );
+ return 1;
+}
+
+
+/* ---------------------------------------------------------------------- */
+/* Stage 3: vertex adjustment (Sec. 2.3.1). */
+
+/* Adjust vertices of optimal polygon: calculate the intersection of
+ * the two "optimal" line segments, then move it into the unit square
+ * if it lies outside. Return 1 with errno set on error; 0 on
+ * success. */
+
+static int adjust_vertices( privpath_t* pp )
+{
+ int m = pp->m;
+ int* po = pp->po;
+ int n = pp->len;
+ point_t* pt = pp->pt;
+ int x0 = pp->x0;
+ int y0 = pp->y0;
+
+ dpoint_t* ctr = NULL; /* ctr[m] */
+ dpoint_t* dir = NULL; /* dir[m] */
+ quadform_t* q = NULL; /* q[m] */
+ double v[3];
+ double d;
+ int i, j, k, l;
+ dpoint_t s;
+ int r;
+
+ SAFE_MALLOC( ctr, m, dpoint_t );
+ SAFE_MALLOC( dir, m, dpoint_t );
+ SAFE_MALLOC( q, m, quadform_t );
+
+ r = privcurve_init( &pp->curve, m );
+ if( r )
+ {
+ goto malloc_error;
+ }
+
+ /* calculate "optimal" point-slope representation for each line
+ * segment */
+ for( i = 0; i<m; i++ )
+ {
+ j = po[mod( i + 1, m )];
+ j = mod( j - po[i], n ) + po[i];
+ pointslope( pp, po[i], j, &ctr[i], &dir[i] );
+ }
+
+ /* represent each line segment as a singular quadratic form; the
+ * distance of a point (x,y) from the line segment will be
+ * (x,y,1)Q(x,y,1)^t, where Q=q[i]. */
+ for( i = 0; i<m; i++ )
+ {
+ d = sq( dir[i].x ) + sq( dir[i].y );
+ if( d == 0.0 )
+ {
+ for( j = 0; j<3; j++ )
+ {
+ for( k = 0; k<3; k++ )
+ {
+ q[i][j][k] = 0;
+ }
+ }
+ }
+ else
+ {
+ v[0] = dir[i].y;
+ v[1] = -dir[i].x;
+ v[2] = -v[1] *ctr[i].y - v[0] *ctr[i].x;
+ for( l = 0; l<3; l++ )
+ {
+ for( k = 0; k<3; k++ )
+ {
+ q[i][l][k] = v[l] *v[k] / d;
+ }
+ }
+ }
+ }
+
+ /* now calculate the "intersections" of consecutive segments.
+ * Instead of using the actual intersection, we find the point
+ * within a given unit square which minimizes the square distance to
+ * the two lines. */
+ for( i = 0; i<m; i++ )
+ {
+ quadform_t Q;
+ dpoint_t w;
+ double dx, dy;
+ double det;
+ double min, cand; /* minimum and candidate for minimum of quad. form */
+ double xmin, ymin; /* coordinates of minimum */
+ int z;
+
+ /* let s be the vertex, in coordinates relative to x0/y0 */
+ s.x = pt[po[i]].x - x0;
+ s.y = pt[po[i]].y - y0;
+
+ /* intersect segments i-1 and i */
+
+ j = mod( i - 1, m );
+
+ /* add quadratic forms */
+ for( l = 0; l<3; l++ )
+ {
+ for( k = 0; k<3; k++ )
+ {
+ Q[l][k] = q[j][l][k] + q[i][l][k];
+ }
+ }
+
+ while( 1 )
+ {
+ /* minimize the quadratic form Q on the unit square */
+ /* find intersection */
+
+#ifdef HAVE_GCC_LOOP_BUG
+ /* work around gcc bug #12243 */
+ free( NULL );
+#endif
+
+ det = Q[0][0] *Q[1][1] - Q[0][1] *Q[1][0];
+ if( det != 0.0 )
+ {
+ w.x = (-Q[0][2] *Q[1][1] + Q[1][2] *Q[0][1]) / det;
+ w.y = ( Q[0][2] *Q[1][0] - Q[1][2] *Q[0][0]) / det;
+ break;
+ }
+
+ /* matrix is singular - lines are parallel. Add another,
+ * orthogonal axis, through the center of the unit square */
+ if( Q[0][0]>Q[1][1] )
+ {
+ v[0] = -Q[0][1];
+ v[1] = Q[0][0];
+ }
+ else if( Q[1][1] )
+ {
+ v[0] = -Q[1][1];
+ v[1] = Q[1][0];
+ }
+ else
+ {
+ v[0] = 1;
+ v[1] = 0;
+ }
+ d = sq( v[0] ) + sq( v[1] );
+ v[2] = -v[1] *s.y - v[0] *s.x;
+ for( l = 0; l<3; l++ )
+ {
+ for( k = 0; k<3; k++ )
+ {
+ Q[l][k] += v[l] *v[k] / d;
+ }
+ }
+ }
+
+ dx = fabs( w.x - s.x );
+ dy = fabs( w.y - s.y );
+ if( dx <= .5 && dy <= .5 )
+ {
+ pp->curve.vertex[i].x = w.x + x0;
+ pp->curve.vertex[i].y = w.y + y0;
+ continue;
+ }
+
+ /* the minimum was not in the unit square; now minimize quadratic
+ * on boundary of square */
+ min = quadform( Q, s );
+ xmin = s.x;
+ ymin = s.y;
+
+ if( Q[0][0] == 0.0 )
+ {
+ goto fixx;
+ }
+ for( z = 0; z<2; z++ ) /* value of the y-coordinate */
+ {
+ w.y = s.y - 0.5 + z;
+ w.x = -(Q[0][1] *w.y + Q[0][2]) / Q[0][0];
+ dx = fabs( w.x - s.x );
+ cand = quadform( Q, w );
+ if( dx <= .5 && cand < min )
+ {
+ min = cand;
+ xmin = w.x;
+ ymin = w.y;
+ }
+ }
+
+fixx:
+ if( Q[1][1] == 0.0 )
+ {
+ goto corners;
+ }
+ for( z = 0; z<2; z++ ) /* value of the x-coordinate */
+ {
+ w.x = s.x - 0.5 + z;
+ w.y = -(Q[1][0] *w.x + Q[1][2]) / Q[1][1];
+ dy = fabs( w.y - s.y );
+ cand = quadform( Q, w );
+ if( dy <= .5 && cand < min )
+ {
+ min = cand;
+ xmin = w.x;
+ ymin = w.y;
+ }
+ }
+
+corners:
+ /* check four corners */
+ for( l = 0; l<2; l++ )
+ {
+ for( k = 0; k<2; k++ )
+ {
+ w.x = s.x - 0.5 + l;
+ w.y = s.y - 0.5 + k;
+ cand = quadform( Q, w );
+ if( cand < min )
+ {
+ min = cand;
+ xmin = w.x;
+ ymin = w.y;
+ }
+ }
+ }
+
+ pp->curve.vertex[i].x = xmin + x0;
+ pp->curve.vertex[i].y = ymin + y0;
+ continue;
+ }
+
+ free( ctr );
+ free( dir );
+ free( q );
+ return 0;
+
+malloc_error:
+ free( ctr );
+ free( dir );
+ free( q );
+ return 1;
+}
+
+
+/* ---------------------------------------------------------------------- */
+/* Stage 4: smoothing and corner analysis (Sec. 2.3.3) */
+
+/* Always succeeds and returns 0 */
+static int smooth( privcurve_t* curve, int sign, double alphamax )
+{
+ int m = curve->n;
+
+ int i, j, k;
+ double dd, denom, alpha;
+ dpoint_t p2, p3, p4;
+
+ if( sign == '-' )
+ {
+ /* reverse orientation of negative paths */
+ for( i = 0, j = m - 1; i<j; i++, j-- )
+ {
+ dpoint_t tmp;
+ tmp = curve->vertex[i];
+ curve->vertex[i] = curve->vertex[j];
+ curve->vertex[j] = tmp;
+ }
+ }
+
+ /* examine each vertex and find its best fit */
+ for( i = 0; i<m; i++ )
+ {
+ j = mod( i + 1, m );
+ k = mod( i + 2, m );
+ p4 = interval( 1 / 2.0, curve->vertex[k], curve->vertex[j] );
+
+ denom = ddenom( curve->vertex[i], curve->vertex[k] );
+ if( denom != 0.0 )
+ {
+ dd = dpara( curve->vertex[i], curve->vertex[j], curve->vertex[k] ) / denom;
+ dd = fabs( dd );
+ alpha = dd>1 ? (1 - 1.0 / dd) : 0;
+ alpha = alpha / 0.75;
+ }
+ else
+ {
+ alpha = 4 / 3.0;
+ }
+ curve->alpha0[j] = alpha; /* remember "original" value of alpha */
+
+ if( alpha > alphamax ) /* pointed corner */
+ {
+ curve->tag[j] = POTRACE_CORNER;
+ curve->c[j][1] = curve->vertex[j];
+ curve->c[j][2] = p4;
+ }
+ else
+ {
+ if( alpha < 0.55 )
+ {
+ alpha = 0.55;
+ }
+ else if( alpha > 1 )
+ {
+ alpha = 1;
+ }
+ p2 = interval( .5 + .5 * alpha, curve->vertex[i], curve->vertex[j] );
+ p3 = interval( .5 + .5 * alpha, curve->vertex[k], curve->vertex[j] );
+ curve->tag[j] = POTRACE_CURVETO;
+ curve->c[j][0] = p2;
+ curve->c[j][1] = p3;
+ curve->c[j][2] = p4;
+ }
+ curve->alpha[j] = alpha; /* store the "cropped" value of alpha */
+ curve->beta[j] = 0.5;
+ }
+
+ curve->alphacurve = 1;
+
+ return 0;
+}
+
+
+/* ---------------------------------------------------------------------- */
+/* Stage 5: Curve optimization (Sec. 2.4) */
+
+/* a private type for the result of opti_penalty */
+struct opti_s
+{
+ double pen; /* penalty */
+ dpoint_t c[2]; /* curve parameters */
+ double t, s; /* curve parameters */
+ double alpha; /* curve parameter */
+};
+typedef struct opti_s opti_t;
+
+/* calculate best fit from i+.5 to j+.5. Assume i<j (cyclically).
+ * Return 0 and set badness and parameters (alpha, beta), if
+ * possible. Return 1 if impossible. */
+static int opti_penalty( privpath_t* pp,
+ int i,
+ int j,
+ opti_t* res,
+ double opttolerance,
+ int* convc,
+ double* areac )
+{
+ int m = pp->curve.n;
+ int k, k1, k2, conv, i1;
+ double area, alpha, d, d1, d2;
+ dpoint_t p0, p1, p2, p3, pt;
+ double A, R, A1, A2, A3, A4;
+ double s, t;
+
+ /* check convexity, corner-freeness, and maximum bend < 179 degrees */
+
+ if( i==j ) /* sanity - a full loop can never be an opticurve */
+ {
+ return 1;
+ }
+
+ k = i;
+ i1 = mod( i + 1, m );
+ k1 = mod( k + 1, m );
+ conv = convc[k1];
+ if( conv == 0 )
+ {
+ return 1;
+ }
+ d = ddist( pp->curve.vertex[i], pp->curve.vertex[i1] );
+ for( k = k1; k!=j; k = k1 )
+ {
+ k1 = mod( k + 1, m );
+ k2 = mod( k + 2, m );
+ if( convc[k1] != conv )
+ {
+ return 1;
+ }
+ if( sign( cprod( pp->curve.vertex[i], pp->curve.vertex[i1], pp->curve.vertex[k1],
+ pp->curve.vertex[k2] ) ) != conv )
+ {
+ return 1;
+ }
+ if( iprod1( pp->curve.vertex[i], pp->curve.vertex[i1], pp->curve.vertex[k1],
+ pp->curve.vertex[k2] ) < d *
+ ddist( pp->curve.vertex[k1], pp->curve.vertex[k2] ) * COS179 )
+ {
+ return 1;
+ }
+ }
+
+ /* the curve we're working in: */
+ p0 = pp->curve.c[mod( i, m )][2];
+ p1 = pp->curve.vertex[mod( i + 1, m )];
+ p2 = pp->curve.vertex[mod( j, m )];
+ p3 = pp->curve.c[mod( j, m )][2];
+
+ /* determine its area */
+ area = areac[j] - areac[i];
+ area -= dpara( pp->curve.vertex[0], pp->curve.c[i][2], pp->curve.c[j][2] ) / 2;
+ if( i>=j )
+ {
+ area += areac[m];
+ }
+
+ /* find intersection o of p0p1 and p2p3. Let t,s such that o =
+ * interval(t,p0,p1) = interval(s,p3,p2). Let A be the area of the
+ * triangle (p0,o,p3). */
+
+ A1 = dpara( p0, p1, p2 );
+ A2 = dpara( p0, p1, p3 );
+ A3 = dpara( p0, p2, p3 );
+ /* A4 = dpara(p1, p2, p3); */
+ A4 = A1 + A3 - A2;
+
+ if( A2 == A1 ) /* this should never happen */
+ {
+ return 1;
+ }
+
+ t = A3 / (A3 - A4);
+ s = A2 / (A2 - A1);
+ A = A2 * t / 2.0;
+
+ if( A == 0.0 ) /* this should never happen */
+ {
+ return 1;
+ }
+
+ R = area / A; /* relative area */
+ alpha = 2 - sqrt( 4 - R / 0.3 ); /* overall alpha for p0-o-p3 curve */
+
+ res->c[0] = interval( t * alpha, p0, p1 );
+ res->c[1] = interval( s * alpha, p3, p2 );
+ res->alpha = alpha;
+ res->t = t;
+ res->s = s;
+
+ p1 = res->c[0];
+ p2 = res->c[1]; /* the proposed curve is now (p0,p1,p2,p3) */
+
+ res->pen = 0;
+
+ /* calculate penalty */
+ /* check tangency with edges */
+ for( k = mod( i + 1, m ); k!=j; k = k1 )
+ {
+ k1 = mod( k + 1, m );
+ t = tangent( p0, p1, p2, p3, pp->curve.vertex[k], pp->curve.vertex[k1] );
+ if( t<-.5 )
+ {
+ return 1;
+ }
+ pt = bezier( t, p0, p1, p2, p3 );
+ d = ddist( pp->curve.vertex[k], pp->curve.vertex[k1] );
+ if( d == 0.0 ) /* this should never happen */
+ {
+ return 1;
+ }
+ d1 = dpara( pp->curve.vertex[k], pp->curve.vertex[k1], pt ) / d;
+ if( fabs( d1 ) > opttolerance )
+ {
+ return 1;
+ }
+ if( iprod( pp->curve.vertex[k], pp->curve.vertex[k1],
+ pt ) < 0 || iprod( pp->curve.vertex[k1], pp->curve.vertex[k], pt ) < 0 )
+ {
+ return 1;
+ }
+ res->pen += sq( d1 );
+ }
+
+ /* check corners */
+ for( k = i; k!=j; k = k1 )
+ {
+ k1 = mod( k + 1, m );
+ t = tangent( p0, p1, p2, p3, pp->curve.c[k][2], pp->curve.c[k1][2] );
+ if( t<-.5 )
+ {
+ return 1;
+ }
+ pt = bezier( t, p0, p1, p2, p3 );
+ d = ddist( pp->curve.c[k][2], pp->curve.c[k1][2] );
+ if( d == 0.0 ) /* this should never happen */
+ {
+ return 1;
+ }
+ d1 = dpara( pp->curve.c[k][2], pp->curve.c[k1][2], pt ) / d;
+ d2 = dpara( pp->curve.c[k][2], pp->curve.c[k1][2], pp->curve.vertex[k1] ) / d;
+ d2 *= 0.75 * pp->curve.alpha[k1];
+ if( d2 < 0 )
+ {
+ d1 = -d1;
+ d2 = -d2;
+ }
+ if( d1 < d2 - opttolerance )
+ {
+ return 1;
+ }
+ if( d1 < d2 )
+ {
+ res->pen += sq( d1 - d2 );
+ }
+ }
+
+ return 0;
+}
+
+
+/* optimize the path p, replacing sequences of Bezier segments by a
+ * single segment when possible. Return 0 on success, 1 with errno set
+ * on failure. */
+static int opticurve( privpath_t* pp, double opttolerance )
+{
+ int seg_count = pp->curve.n; // segment count in pp->curve
+ int* pt = NULL; /* pt[m+1] */
+ double* pen = NULL; /* pen[m+1] */
+ int* len = NULL; /* len[m+1] */
+ opti_t* opt = NULL; /* opt[m+1] */
+ int om;
+ int j, r;
+ opti_t curve_prms;
+ dpoint_t p0;
+ int i1;
+ double area;
+ double alpha;
+ double* s = NULL;
+ double* t = NULL;
+
+ int* convc = NULL; /* conv[m]: pre-computed convexities */
+ double* areac = NULL; /* cumarea[m+1]: cache for fast area computation */
+
+ SAFE_MALLOC( pt, seg_count + 1, int );
+ SAFE_MALLOC( pen, seg_count + 1, double );
+ SAFE_MALLOC( len, seg_count + 1, int );
+ SAFE_MALLOC( opt, seg_count + 1, opti_t );
+ SAFE_MALLOC( convc, seg_count, int );
+ SAFE_MALLOC( areac, seg_count + 1, double );
+
+ /* pre-calculate convexity: +1 = right turn, -1 = left turn, 0 = corner */
+ for( int i = 0; i<seg_count; i++ )
+ {
+ if( pp->curve.tag[i] == POTRACE_CURVETO )
+ {
+ convc[i] =
+ sign( dpara( pp->curve.vertex[mod( i - 1, seg_count )],
+ pp->curve.vertex[i],
+ pp->curve.vertex[mod( i + 1, seg_count )] ) );
+ }
+ else
+ {
+ convc[i] = 0;
+ }
+ }
+
+ /* pre-calculate areas */
+ area = 0.0;
+ areac[0] = 0.0;
+ p0 = pp->curve.vertex[0];
+ for( int i = 0; i<seg_count; i++ )
+ {
+ i1 = mod( i + 1, seg_count );
+ if( pp->curve.tag[i1] == POTRACE_CURVETO )
+ {
+ alpha = pp->curve.alpha[i1];
+ area += 0.3 * alpha * (4 - alpha) * dpara( pp->curve.c[i][2],
+ pp->curve.vertex[i1],
+ pp->curve.c[i1][2] ) / 2;
+ area += dpara( p0, pp->curve.c[i][2], pp->curve.c[i1][2] ) / 2;
+ }
+ areac[i + 1] = area;
+ }
+
+ pt[0] = -1;
+ pen[0] = 0;
+ len[0] = 0;
+
+ // Avoid not initialized value for opt[j] (should not occur, but...)
+ for( j = 0; j<=seg_count; j++ )
+ {
+ opt[j].pen = 0.0; // penalty
+ opt[j].c[0].x = opt[j].c[1].x = opt[j].c[0].y = opt[j].c[1].y = 0;
+ opt[j].t = opt[j].s = opt[j].alpha = 0.0; // curve parameters
+ }
+
+
+ /* Fixme: we always start from a fixed point -- should find the best
+ * curve cyclically ### */
+
+ for( j = 1; j<=seg_count; j++ )
+ {
+ /* calculate best path from 0 to j */
+ pt[j] = j - 1;
+ pen[j] = pen[j - 1];
+ len[j] = len[j - 1] + 1;
+
+ for( int i = j - 2; i>=0; i-- )
+ {
+ r = opti_penalty( pp, i, mod( j, seg_count ), &curve_prms, opttolerance, convc, areac );
+ if( r )
+ {
+ break;
+ }
+ if( len[j] > len[i] + 1 || (len[j] == len[i] + 1 && pen[j] > pen[i] + curve_prms.pen) )
+ {
+ pt[j] = i;
+ pen[j] = pen[i] + curve_prms.pen;
+ len[j] = len[i] + 1;
+ opt[j] = curve_prms;
+ }
+ }
+ }
+
+ om = len[seg_count];
+ r = privcurve_init( &pp->ocurve, om );
+ if( r )
+ {
+ goto malloc_error;
+ }
+ SAFE_MALLOC( s, om, double );
+ SAFE_MALLOC( t, om, double );
+
+ j = seg_count;
+ for( int i = om - 1; i>=0; i-- )
+ {
+ if( pt[j]==j - 1 )
+ {
+ pp->ocurve.tag[i] = pp->curve.tag[mod( j, seg_count )];
+ pp->ocurve.c[i][0] = pp->curve.c[mod( j, seg_count )][0];
+ pp->ocurve.c[i][1] = pp->curve.c[mod( j, seg_count )][1];
+ pp->ocurve.c[i][2] = pp->curve.c[mod( j, seg_count )][2];
+ pp->ocurve.vertex[i] = pp->curve.vertex[mod( j, seg_count )];
+ pp->ocurve.alpha[i] = pp->curve.alpha[mod( j, seg_count )];
+ pp->ocurve.alpha0[i] = pp->curve.alpha0[mod( j, seg_count )];
+ pp->ocurve.beta[i] = pp->curve.beta[mod( j, seg_count )];
+ s[i] = t[i] = 1.0;
+ }
+ else
+ {
+ pp->ocurve.tag[i] = POTRACE_CURVETO;
+ pp->ocurve.c[i][0] = opt[j].c[0];
+ pp->ocurve.c[i][1] = opt[j].c[1];
+ pp->ocurve.c[i][2] = pp->curve.c[mod( j, seg_count )][2];
+ pp->ocurve.vertex[i] = interval( opt[j].s, pp->curve.c[mod( j, seg_count )][2],
+ pp->curve.vertex[mod( j, seg_count )] );
+ pp->ocurve.alpha[i] = opt[j].alpha;
+ pp->ocurve.alpha0[i] = opt[j].alpha;
+ s[i] = opt[j].s;
+ t[i] = opt[j].t;
+ }
+ j = pt[j];
+ }
+
+ /* calculate beta parameters */
+ for( int i = 0; i<om; i++ )
+ {
+ i1 = mod( i + 1, om );
+ pp->ocurve.beta[i] = s[i] / (s[i] + t[i1]);
+ }
+
+ pp->ocurve.alphacurve = 1;
+
+ free( pt );
+ free( pen );
+ free( len );
+ free( opt );
+ free( s );
+ free( t );
+ free( convc );
+ free( areac );
+ return 0;
+
+malloc_error:
+ free( pt );
+ free( pen );
+ free( len );
+ free( opt );
+ free( s );
+ free( t );
+ free( convc );
+ free( areac );
+ return 1;
+}
+
+
+/* ---------------------------------------------------------------------- */
+
+#define TRY( x ) if( x ) \
+ goto try_error
+
+/* return 0 on success, 1 on error with errno set. */
+int process_path( path_t* plist, const potrace_param_t* param, progress_t* progress )
+{
+ path_t* p;
+ double nn = 0, cn = 0;
+
+ if( progress->callback )
+ {
+ /* precompute task size for progress estimates */
+ nn = 0;
+ list_forall( p, plist ) {
+ nn += p->priv->len;
+ }
+ cn = 0;
+ }
+
+ /* call downstream function with each path */
+ list_forall( p, plist ) {
+ TRY( calc_sums( p->priv ) );
+ TRY( calc_lon( p->priv ) );
+ TRY( bestpolygon( p->priv ) );
+ TRY( adjust_vertices( p->priv ) );
+ TRY( smooth( &p->priv->curve, p->sign, param->alphamax ) );
+ if( param->opticurve )
+ {
+ TRY( opticurve( p->priv, param->opttolerance ) );
+ p->priv->fcurve = &p->priv->ocurve;
+ }
+ else
+ {
+ p->priv->fcurve = &p->priv->curve;
+ }
+ privcurve_to_curve( p->priv->fcurve, &p->curve );
+
+ if( progress->callback )
+ {
+ cn += p->priv->len;
+ progress_update( cn / nn, progress );
+ }
+ }
+
+ progress_update( 1.0, progress );
+
+ return 0;
+
+try_error:
+ return 1;
+}
generated by cgit (git 2.34.1) at 2024-12-19 09:57:11 +0000