shape ops work in progress
git-svn-id: http://skia.googlecode.com/svn/trunk@7031 2bbb7eff-a529-9590-31e7-b0007b416f81
diff --git a/experimental/Intersection/CubicToQuadratics.cpp b/experimental/Intersection/CubicToQuadratics.cpp
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+++ b/experimental/Intersection/CubicToQuadratics.cpp
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+/*
+http://stackoverflow.com/questions/2009160/how-do-i-convert-the-2-control-points-of-a-cubic-curve-to-the-single-control-poi
+*/
+
+/*
+Let's call the control points of the cubic Q0..Q3 and the control points of the quadratic P0..P2.
+Then for degree elevation, the equations are:
+
+Q0 = P0
+Q1 = 1/3 P0 + 2/3 P1
+Q2 = 2/3 P1 + 1/3 P2
+Q3 = P2
+In your case you have Q0..Q3 and you're solving for P0..P2. There are two ways to compute P1 from
+ the equations above:
+
+P1 = 3/2 Q1 - 1/2 Q0
+P1 = 3/2 Q2 - 1/2 Q3
+If this is a degree-elevated cubic, then both equations will give the same answer for P1. Since
+ it's likely not, your best bet is to average them. So,
+
+P1 = -1/4 Q0 + 3/4 Q1 + 3/4 Q2 - 1/4 Q3
+
+
+Cubic defined by: P1/2 - anchor points, C1/C2 control points
+|x| is the euclidean norm of x
+mid-point approx of cubic: a quad that shares the same anchors with the cubic and has the
+ control point at C = (3·C2 - P2 + 3·C1 - P1)/4
+
+Algorithm
+
+pick an absolute precision (prec)
+Compute the Tdiv as the root of (cubic) equation
+sqrt(3)/18 · |P2 - 3·C2 + 3·C1 - P1|/2 · Tdiv ^ 3 = prec
+if Tdiv < 0.5 divide the cubic at Tdiv. First segment [0..Tdiv] can be approximated with by a
+ quadratic, with a defect less than prec, by the mid-point approximation.
+ Repeat from step 2 with the second resulted segment (corresponding to 1-Tdiv)
+0.5<=Tdiv<1 - simply divide the cubic in two. The two halves can be approximated by the mid-point
+ approximation
+Tdiv>=1 - the entire cubic can be approximated by the mid-point approximation
+
+confirmed by (maybe stolen from)
+http://www.caffeineowl.com/graphics/2d/vectorial/cubic2quad01.html
+
+*/
+
+#include "CubicUtilities.h"
+#include "CurveIntersection.h"
+
+static double calcTDiv(const Cubic& cubic, double start) {
+ const double adjust = sqrt(3) / 36;
+ Cubic sub;
+ const Cubic* cPtr;
+ if (start == 0) {
+ cPtr = &cubic;
+ } else {
+ // OPTIMIZE: special-case half-split ?
+ sub_divide(cubic, start, 1, sub);
+ cPtr = ⊂
+ }
+ const Cubic& c = *cPtr;
+ double dx = c[3].x - 3 * (c[2].x - c[1].x) - c[0].x;
+ double dy = c[3].y - 3 * (c[2].y - c[1].y) - c[0].y;
+ double dist = sqrt(dx * dx + dy * dy);
+ double tDiv3 = FLT_EPSILON / (adjust * dist);
+ return cube_root(tDiv3);
+}
+
+static void demote(const Cubic& cubic, Quadratic& quad) {
+ quad[0] = cubic[0];
+ quad[1].x = (cubic[3].x - 3 * (cubic[2].x - cubic[1].x) - cubic[0].x) / 4;
+ quad[1].y = (cubic[3].y - 3 * (cubic[2].y - cubic[1].y) - cubic[0].y) / 4;
+ quad[2] = cubic[3];
+}
+
+int cubic_to_quadratics(const Cubic& cubic, SkTDArray<Quadratic>& quadratics) {
+ quadratics.setCount(1); // FIXME: every place I have setCount(), I also want setReserve()
+ Cubic reduced;
+ int order = reduceOrder(cubic, reduced, kReduceOrder_QuadraticsAllowed);
+ if (order < 3) {
+ memcpy(quadratics[0], reduced, order * sizeof(_Point));
+ return order;
+ }
+ double tDiv = calcTDiv(cubic, 0);
+ if (approximately_greater_than_one(tDiv)) {
+ demote(cubic, quadratics[0]);
+ return 2;
+ }
+ if (tDiv >= 0.5) {
+ CubicPair pair;
+ chop_at(cubic, pair, 0.5);
+ quadratics.setCount(2);
+ demote(pair.first(), quadratics[0]);
+ demote(pair.second(), quadratics[1]);
+ return 2;
+ }
+ double start = 0;
+ int index = -1;
+ // if we iterate but make little progress, check to see if the curve is flat
+ // or if the control points are tangent to each other
+ do {
+ Cubic part;
+ sub_divide(part, start, tDiv, part);
+ quadratics.append();
+ demote(part, quadratics[++index]);
+ if (tDiv == 1) {
+ break;
+ }
+ start = tDiv;
+ tDiv = calcTDiv(cubic, start);
+ if (tDiv > 1) {
+ tDiv = 1;
+ }
+ } while (true);
+ return 2;
+}