Add base types for path ops

Paths contain lines, quads, and cubics, which are
collectively curves.

To work with path intersections, intermediary curves
are constructed. For now, those intermediates use
doubles to guarantee sufficient precision.

The DVector, DPoint, DLine, DQuad, and DCubic
structs encapsulate these intermediate curves.

The DRect and DTriangle structs are created to
describe intersectable areas of interest.

The Bounds struct inherits from SkRect to create
a SkScalar-based rectangle that intersects shared
edges.

This also includes common math equalities and
debugging that the remainder of path ops builds on,
as well as a temporary top-level interface in
include/pathops/SkPathOps.h.
Review URL: https://codereview.chromium.org/12827020

git-svn-id: http://skia.googlecode.com/svn/trunk@8551 2bbb7eff-a529-9590-31e7-b0007b416f81

1 files changed, 190 insertions, 0 deletions

diff --git a/src/pathops/SkDCubicToQuads.cpp b/src/pathops/SkDCubicToQuads.cpp
new file mode 100644
index 0000000000..b8a02c4294
--- /dev/null
+++ b/src/pathops/SkDCubicToQuads.cpp
@@ -0,0 +1,190 @@
+/*
+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
+
+
+SkDCubic 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
+// maybe in turn derived from  http://www.cccg.ca/proceedings/2004/36.pdf
+// also stored at http://www.cis.usouthal.edu/~hain/general/Publications/Bezier/bezier%20cccg04%20paper.pdf
+
+*/
+
+#include "SkPathOpsCubic.h"
+#include "SkPathOpsLine.h"
+#include "SkPathOpsQuad.h"
+#include "SkReduceOrder.h"
+#include "SkTDArray.h"
+#include "TSearch.h"
+
+#define USE_CUBIC_END_POINTS 1
+
+static double calc_t_div(const SkDCubic& cubic, double precision, double start) {
+    const double adjust = sqrt(3.) / 36;
+    SkDCubic sub;
+    const SkDCubic* cPtr;
+    if (start == 0) {
+        cPtr = &cubic;
+    } else {
+        // OPTIMIZE: special-case half-split ?
+        sub = cubic.subDivide(start, 1);
+        cPtr = &sub;
+    }
+    const SkDCubic& c = *cPtr;
+    double dx = c[3].fX - 3 * (c[2].fX - c[1].fX) - c[0].fX;
+    double dy = c[3].fY - 3 * (c[2].fY - c[1].fY) - c[0].fY;
+    double dist = sqrt(dx * dx + dy * dy);
+    double tDiv3 = precision / (adjust * dist);
+    double t = SkDCubeRoot(tDiv3);
+    if (start > 0) {
+        t = start + (1 - start) * t;
+    }
+    return t;
+}
+
+SkDQuad SkDCubic::toQuad() const {
+    SkDQuad quad;
+    quad[0] = fPts[0];
+    const SkDPoint fromC1 = {(3 * fPts[1].fX - fPts[0].fX) / 2, (3 * fPts[1].fY - fPts[0].fY) / 2};
+    const SkDPoint fromC2 = {(3 * fPts[2].fX - fPts[3].fX) / 2, (3 * fPts[2].fY - fPts[3].fY) / 2};
+    quad[1].fX = (fromC1.fX + fromC2.fX) / 2;
+    quad[1].fY = (fromC1.fY + fromC2.fY) / 2;
+    quad[2] = fPts[3];
+    return quad;
+}
+
+static bool add_simple_ts(const SkDCubic& cubic, double precision, SkTDArray<double>* ts) {
+    double tDiv = calc_t_div(cubic, precision, 0);
+    if (tDiv >= 1) {
+        return true;
+    }
+    if (tDiv >= 0.5) {
+        *ts->append() = 0.5;
+        return true;
+    }
+    return false;
+}
+
+static void addTs(const SkDCubic& cubic, double precision, double start, double end,
+        SkTDArray<double>* ts) {
+    double tDiv = calc_t_div(cubic, precision, 0);
+    double parts = ceil(1.0 / tDiv);
+    for (double index = 0; index < parts; ++index) {
+        double newT = start + (index / parts) * (end - start);
+        if (newT > 0 && newT < 1) {
+            *ts->append() = newT;
+        }
+    }
+}
+
+// flavor that returns T values only, deferring computing the quads until they are needed
+// FIXME: when called from recursive intersect 2, this could take the original cubic
+// and do a more precise job when calling chop at and sub divide by computing the fractional ts.
+// it would still take the prechopped cubic for reduce order and find cubic inflections
+void SkDCubic::toQuadraticTs(double precision, SkTDArray<double>* ts) const {
+    SkReduceOrder reducer;
+    int order = reducer.reduce(*this, SkReduceOrder::kAllow_Quadratics, SkReduceOrder::kFill_Style);
+    if (order < 3) {
+        return;
+    }
+    double inflectT[5];
+    int inflections = findInflections(inflectT);
+    SkASSERT(inflections <= 2);
+    if (!endsAreExtremaInXOrY()) {
+        inflections += findMaxCurvature(&inflectT[inflections]);
+        SkASSERT(inflections <= 5);
+    }
+    QSort<double>(inflectT, &inflectT[inflections - 1]);
+    // OPTIMIZATION: is this filtering common enough that it needs to be pulled out into its
+    // own subroutine?
+    while (inflections && approximately_less_than_zero(inflectT[0])) {
+        memmove(inflectT, &inflectT[1], sizeof(inflectT[0]) * --inflections);
+    }
+    int start = 0;
+    do {
+        int next = start + 1;
+        if (next >= inflections) {
+            break;
+        }
+        if (!approximately_equal(inflectT[start], inflectT[next])) {
+            ++start;
+            continue;
+        }
+        memmove(&inflectT[start], &inflectT[next], sizeof(inflectT[0]) * (--inflections - start));
+    } while (true);
+    while (inflections && approximately_greater_than_one(inflectT[inflections - 1])) {
+        --inflections;
+    }
+    SkDCubicPair pair;
+    if (inflections == 1) {
+        pair = chopAt(inflectT[0]);
+        int orderP1 = reducer.reduce(pair.first(), SkReduceOrder::kNo_Quadratics,
+                SkReduceOrder::kFill_Style);
+        if (orderP1 < 2) {
+            --inflections;
+        } else {
+            int orderP2 = reducer.reduce(pair.second(), SkReduceOrder::kNo_Quadratics,
+                    SkReduceOrder::kFill_Style);
+            if (orderP2 < 2) {
+                --inflections;
+            }
+        }
+    }
+    if (inflections == 0 && add_simple_ts(*this, precision, ts)) {
+        return;
+    }
+    if (inflections == 1) {
+        pair = chopAt(inflectT[0]);
+        addTs(pair.first(), precision, 0, inflectT[0], ts);
+        addTs(pair.second(), precision, inflectT[0], 1, ts);
+        return;
+    }
+    if (inflections > 1) {
+        SkDCubic part = subDivide(0, inflectT[0]);
+        addTs(part, precision, 0, inflectT[0], ts);
+        int last = inflections - 1;
+        for (int idx = 0; idx < last; ++idx) {
+            part = subDivide(inflectT[idx], inflectT[idx + 1]);
+            addTs(part, precision, inflectT[idx], inflectT[idx + 1], ts);
+        }
+        part = subDivide(inflectT[last], 1);
+        addTs(part, precision, inflectT[last], 1, ts);
+        return;
+    }
+    addTs(*this, precision, 0, 1, ts);
+}