pathops version two

src/pathops/SkDCubicToQuads.cpp

 /*
 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 "SkTArray.h"
-#include "SkTSort.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, SkTArray<double, true>* ts) {
-    double tDiv = calc_t_div(cubic, precision, 0);
-    if (tDiv >= 1) {
-        return true;
-    }
-    if (tDiv >= 0.5) {
-        ts->push_back(0.5);
-        return true;
-    }
-    return false;
-}
-
-static void addTs(const SkDCubic& cubic, double precision, double start, double end,
-        SkTArray<double, true>* 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->push_back(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, SkTArray<double, true>* ts) const {
-    SkReduceOrder reducer;
-    int order = reducer.reduce(*this, SkReduceOrder::kAllow_Quadratics);
-    if (order < 3) {
-        return;
-    }
-    double inflectT[5];
-    int inflections = findInflections(inflectT);
-    SkASSERT(inflections <= 2);
-    if (!endsAreExtremaInXOrY()) {
-        inflections += findMaxCurvature(&inflectT[inflections]);
-        SkASSERT(inflections <= 5);
-    }
-    SkTQSort<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;
-    int next = 1;
-    while (next < inflections) {
-        if (!approximately_equal(inflectT[start], inflectT[next])) {
-            ++start;
-        ++next;
-            continue;
-        }
-        memmove(&inflectT[start], &inflectT[next], sizeof(inflectT[0]) * (--inflections - start));
-    }
-
-    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);
-        if (orderP1 < 2) {
-            --inflections;
-        } else {
-            int orderP2 = reducer.reduce(pair.second(), SkReduceOrder::kNo_Quadratics);
-            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);
-}