curve: Add utilities for cusps and inflections

Add functions to find cusps and inflection points of cubics.
These will be used for intersections and in the stroker.

gsk/gskcurveintersect.c

@@ -0,0 +1,367 @@
+/*
+ * Copyright © 2020 Red Hat, Inc
+ *
+ * This library is free software; you can redistribute it and/or
+ * modify it under the terms of the GNU Lesser General Public
+ * License as published by the Free Software Foundation; either
+ * version 2.1 of the License, or (at your option) any later version.
+ *
+ * This library 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
+ * Lesser General Public License for more details.
+ *
+ * You should have received a copy of the GNU Lesser General Public
+ * License along with this library. If not, see <http://www.gnu.org/licenses/>.
+ *
+ * Authors: Matthias Clasen <mclasen@redhat.com>
+ */
+
+#include "config.h"
+
+#include <math.h>
+
+#include "gskcurveprivate.h"
+
+/* {{{ Utilities */
+
+static void
+get_tangent (const graphene_point_t *p0,
+             const graphene_point_t *p1,
+             graphene_vec2_t        *t)
+{
+  graphene_vec2_init (t, p1->x - p0->x, p1->y - p0->y);
+  graphene_vec2_normalize (t, t);
+}
+
+static inline gboolean
+acceptable (float t)
+{
+  return 0 - FLT_EPSILON <= t && t <= 1 + FLT_EPSILON;
+}
+
+static inline void
+_sincosf (float  angle,
+          float *out_s,
+          float *out_c)
+{
+#ifdef HAVE_SINCOSF
+      sincosf (angle, out_s, out_c);
+#else
+      *out_s = sinf (angle);
+      *out_c = cosf (angle);
+#endif
+}
+
+static void
+align_points (const graphene_point_t *p,
+              const graphene_point_t *a,
+              const graphene_point_t *b,
+              graphene_point_t       *q,
+              int                     n)
+{
+  graphene_vec2_t n1;
+  float angle;
+  float s, c;
+
+  get_tangent (a, b, &n1);
+  angle = - atan2 (graphene_vec2_get_y (&n1), graphene_vec2_get_x (&n1));
+  _sincosf (angle, &s, &c);
+
+  for (int i = 0; i < n; i++)
+    {
+      q[i].x = (p[i].x - a->x) * c - (p[i].y - a->y) * s;
+      q[i].y = (p[i].x - a->x) * s + (p[i].y - a->y) * c;
+    }
+}
+
+static void
+find_point_on_line (const graphene_point_t *p1,
+                    const graphene_point_t *p2,
+                    const graphene_point_t *q,
+                    float                  *t)
+{
+  if (p2->x != p1->x)
+    *t = (q->x - p1->x) / (p2->x - p1->x);
+  else
+    *t = (q->y - p1->y) / (p2->y - p1->y);
+}
+
+/* find solutions for at^2 + bt + c = 0 */
+static int
+solve_quadratic (float a, float b, float c, float t[2])
+{
+  float d;
+  int n = 0;
+
+  if (fabs (a) > 0.0001)
+    {
+      if (b*b > 4*a*c)
+        {
+          d = sqrt (b*b - 4*a*c);
+          t[n++] = (-b + d)/(2*a);
+          t[n++] = (-b - d)/(2*a);
+        }
+      else
+        {
+          t[n++] = -b / (2*a);
+        }
+    }
+  else if (fabs (b) > 0.0001)
+    {
+      t[n++] = -c / b;
+    }
+
+  return n;
+}
+
+static int
+filter_allowable (float t[3],
+                  int   n)
+{
+  float g[3];
+  int j = 0;
+
+  for (int i = 0; i < n; i++)
+    if (0 < t[i] && t[i] < 1)
+      g[j++] = t[i];
+  for (int i = 0; i < j; i++)
+    t[i] = g[i];
+  return j;
+}
+
+/* Solve P = 0 where P is
+ * P = (1-t)^2*pa + 2*t*(1-t)*pb + t^2*pc
+ */
+static int
+get_quadratic_roots (float pa, float pb, float pc, float roots[2])
+{
+  float a, b, c, d;
+  int n_roots = 0;
+
+  a = pa - 2 * pb + pc;
+  b = 2 * (pb - pa);
+  c = pa;
+
+  d = b*b - 4*a*c;
+
+  if (d > 0.0001)
+    {
+      float q = sqrt (d);
+      roots[n_roots] = (-b + q) / (2 * a);
+      if (acceptable (roots[n_roots]))
+        n_roots++;
+      roots[n_roots] = (-b - q) / (2 * a);
+      if (acceptable (roots[n_roots]))
+        n_roots++;
+    }
+  else if (fabs (d) < 0.0001)
+    {
+      roots[n_roots] = -b / (2 * a);
+      if (acceptable (roots[n_roots]))
+        n_roots++;
+    }
+
+  return n_roots;
+}
+
+static float
+cuberoot (float v)
+{
+  if (v < 0)
+    return -pow (-v, 1.f / 3);
+  return pow (v, 1.f / 3);
+}
+
+/* Solve P = 0 where P is
+ * P = (1-t)^3*pa + 3*t*(1-t)^2*pb + 3*t^2*(1-t)*pc + t^3*pd
+ */
+static int
+get_cubic_roots (float pa, float pb, float pc, float pd, float roots[3])
+{
+  float a, b, c, d;
+  float q, q2;
+  float p, p3;
+  float discriminant;
+  float u1, v1, sd;
+  int n_roots = 0;
+
+  d = -pa + 3*pb - 3*pc + pd;
+  a = 3*pa - 6*pb + 3*pc;
+  b = -3*pa + 3*pb;
+  c = pa;
+
+  if (fabs (d) < 0.0001)
+    {
+      if (fabs (a) < 0.0001)
+        {
+          if (fabs (b) < 0.0001)
+            return 0;
+
+          if (acceptable (-c / b))
+            {
+              roots[0] = -c / b;
+
+              return 1;
+            }
+
+          return 0;
+        }
+      q = sqrt (b*b - 4*a*c);
+      roots[n_roots] = (-b + q) / (2 * a);
+      if (acceptable (roots[n_roots]))
+        n_roots++;
+
+      roots[n_roots] = (-b - q) / (2 * a);
+      if (acceptable (roots[n_roots]))
+        n_roots++;
+
+      return n_roots;
+    }
+
+  a /= d;
+  b /= d;
+  c /= d;
+
+  p = (3*b - a*a)/3;
+  p3 = p/3;
+  q = (2*a*a*a - 9*a*b + 27*c)/27;
+  q2 = q/2;
+  discriminant = q2*q2 + p3*p3*p3;
+
+  if (discriminant < 0)
+    {
+      float mp3 = -p/3;
+      float mp33 = mp3*mp3*mp3;
+      float r = sqrt (mp33);
+      float t = -q / (2*r);
+      float cosphi = t < -1 ? -1 : (t > 1 ? 1 : t);
+      float phi = acos (cosphi);
+      float crtr = cuberoot (r);
+      float t1 = 2*crtr;
+
+      roots[n_roots] = t1 * cos (phi/3) - a/3;
+      if (acceptable (roots[n_roots]))
+        n_roots++;
+      roots[n_roots] = t1 * cos ((phi + 2*M_PI) / 3) - a/3;
+      if (acceptable (roots[n_roots]))
+        n_roots++;
+      roots[n_roots] = t1 * cos ((phi + 4*M_PI) / 3) - a/3;
+      if (acceptable (roots[n_roots]))
+        n_roots++;
+
+    return n_roots;
+  }
+
+  if (discriminant == 0)
+    {
+      u1 = q2 < 0 ? cuberoot (-q2) : -cuberoot (q2);
+      roots[n_roots] = 2*u1 - a/3;
+      if (acceptable (roots[n_roots]))
+        n_roots++;
+      roots[n_roots] = -u1 - a/3;
+      if (acceptable (roots[n_roots]))
+        n_roots++;
+
+      return n_roots;
+    }
+
+  sd = sqrt (discriminant);
+  u1 = cuberoot (sd - q2);
+  v1 = cuberoot (sd + q2);
+  roots[n_roots] = u1 - v1 - a/3;
+  if (acceptable (roots[n_roots]))
+    n_roots++;
+
+  return n_roots;
+}
+
+/* }}} */
+/* {{{ Cusps and inflections */
+
+/* Get the points where the curvature of curve is
+ * zero, or a maximum or minimum, inside the open
+ * interval from 0 to 1.
+ */
+int
+gsk_curve_get_curvature_points (const GskCurve *curve,
+                                float           t[3])
+{
+  const graphene_point_t *pts = curve->cubic.points;
+  graphene_point_t p[4];
+  float a, b, c, d;
+  float x, y, z;
+  int n;
+
+  if (curve->op != GSK_PATH_CUBIC)
+    return 0;
+
+  align_points (pts, &pts[0], &pts[3], p, 4);
+
+  a = p[2].x * p[1].y;
+  b = p[3].x * p[1].y;
+  c = p[1].x * p[2].y;
+  d = p[3].x * p[2].y;
+
+  x = - 3*a + 2*b + 3*c - d;
+  y = 3*a - b - 3*c;
+  z = c - a;
+
+  n = solve_quadratic (x, y, z, t);
+  return filter_allowable (t, n);
+}
+
+/* Find cusps inside the open interval from 0 to 1.
+ *
+ * According to Stone & deRose, A Geometric Characterization
+ * of Parametric Cubic curves, a necessary and sufficient
+ * condition is that the first derivative vanishes.
+ */
+int
+gsk_curve_get_cusps (const GskCurve *curve,
+                     float           t[2])
+{
+  const graphene_point_t *pts = curve->cubic.points;
+  graphene_point_t p[3];
+  float ax, bx, cx;
+  float ay, by, cy;
+  float tx[3];
+  int nx;
+  int n = 0;
+
+  if (curve->op != GSK_PATH_CUBIC)
+    return 0;
+
+  p[0].x = 3 * (pts[1].x - pts[0].x);
+  p[0].y = 3 * (pts[1].y - pts[0].y);
+  p[1].x = 3 * (pts[2].x - pts[1].x);
+  p[1].y = 3 * (pts[2].y - pts[1].y);
+  p[2].x = 3 * (pts[3].x - pts[2].x);
+  p[2].y = 3 * (pts[3].y - pts[2].y);
+
+  ax = p[0].x - 2 * p[1].x + p[2].x;
+  bx = - 2 * p[0].x + 2 * p[1].x;
+  cx = p[0].x;
+
+  nx = solve_quadratic (ax, bx, cx, tx);
+  nx = filter_allowable (tx, nx);
+
+  ay = p[0].y - 2 * p[1].y + p[2].y;
+  by = - 2 * p[0].y + 2 * p[1].y;
+  cy = p[0].y;
+
+  for (int i = 0; i < nx; i++)
+    {
+      float ti = tx[i];
+
+      if (0 < ti && ti < 1 &&
+          fabs (ay * ti * ti + by * ti + cy) < 0.001)
+        t[n++] = ti;
+    }
+
+  return n;
+}
+
+/* }}} */
+
+/* vim:set foldmethod=marker expandtab: */

gsk/gskcurveprivate.h

@@ -164,5 +164,11 @@ void                    gsk_curve_get_bounds                    (const GskCurve
 void                    gsk_curve_get_tight_bounds              (const GskCurve         *curve,
                                                                  GskBoundingBox         *bounds);
 
+int                     gsk_curve_get_curvature_points          (const GskCurve         *curve,
+                                                                 float                   t[3]);
+
+int                     gsk_curve_get_cusps                     (const GskCurve         *curve,
+                                                                 float                   t[2]);
+
 G_END_DECLS
 

gsk/meson.build

@@ -44,6 +44,7 @@ gsk_private_sources = files([
   'gskcairoblur.c',
   'gskcontour.c',
   'gskcurve.c',
+  'gskcurveintersect.c',
   'gskdebug.c',
   'gskpathdash.c',
   'gskprivate.c',