curve: Add some curve utilities

Internal apis:
- gsk_curve_get_curvature_points
- gsk_curve_get_cusps

These apis will be used in the stroker
and in path ops.

gsk/gskcurve.c

@@ -1472,6 +1472,167 @@ gsk_curve_reverse (const GskCurve *curve,
   get_class (curve->op)->reverse (curve, reverse);
 }
 
+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;
+    }
+}
+
+/* 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;
+}
+
+/* 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;
+}
+
 /* Replace a line by an equivalent quad,
  * and a quad by an equivalent cubic.
  */

gsk/gskcurveprivate.h

@@ -153,6 +153,12 @@ void                    gsk_curve_elevate                       (const GskCurve
                                                                  GskCurve               *elevated);
 
 
+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
 
 #endif /* __GSK_CURVE_PRIVATE_H__ */