michael@0: /*
michael@0: http://stackoverflow.com/questions/2009160/how-do-i-convert-the-2-control-points-of-a-cubic-curve-to-the-single-control-poi
michael@0: */
michael@0: 
michael@0: /*
michael@0: Let's call the control points of the cubic Q0..Q3 and the control points of the quadratic P0..P2.
michael@0: Then for degree elevation, the equations are:
michael@0: 
michael@0: Q0 = P0
michael@0: Q1 = 1/3 P0 + 2/3 P1
michael@0: Q2 = 2/3 P1 + 1/3 P2
michael@0: Q3 = P2
michael@0: In your case you have Q0..Q3 and you're solving for P0..P2. There are two ways to compute P1 from
michael@0:  the equations above:
michael@0: 
michael@0: P1 = 3/2 Q1 - 1/2 Q0
michael@0: P1 = 3/2 Q2 - 1/2 Q3
michael@0: If this is a degree-elevated cubic, then both equations will give the same answer for P1. Since
michael@0:  it's likely not, your best bet is to average them. So,
michael@0: 
michael@0: P1 = -1/4 Q0 + 3/4 Q1 + 3/4 Q2 - 1/4 Q3
michael@0: 
michael@0: 
michael@0: SkDCubic defined by: P1/2 - anchor points, C1/C2 control points
michael@0: |x| is the euclidean norm of x
michael@0: mid-point approx of cubic: a quad that shares the same anchors with the cubic and has the
michael@0:  control point at C = (3·C2 - P2 + 3·C1 - P1)/4
michael@0: 
michael@0: Algorithm
michael@0: 
michael@0: pick an absolute precision (prec)
michael@0: Compute the Tdiv as the root of (cubic) equation
michael@0: sqrt(3)/18 · |P2 - 3·C2 + 3·C1 - P1|/2 · Tdiv ^ 3 = prec
michael@0: if Tdiv < 0.5 divide the cubic at Tdiv. First segment [0..Tdiv] can be approximated with by a
michael@0:  quadratic, with a defect less than prec, by the mid-point approximation.
michael@0:  Repeat from step 2 with the second resulted segment (corresponding to 1-Tdiv)
michael@0: 0.5<=Tdiv<1 - simply divide the cubic in two. The two halves can be approximated by the mid-point
michael@0:  approximation
michael@0: Tdiv>=1 - the entire cubic can be approximated by the mid-point approximation
michael@0: 
michael@0: confirmed by (maybe stolen from)
michael@0: http://www.caffeineowl.com/graphics/2d/vectorial/cubic2quad01.html
michael@0: // maybe in turn derived from  http://www.cccg.ca/proceedings/2004/36.pdf
michael@0: // also stored at http://www.cis.usouthal.edu/~hain/general/Publications/Bezier/bezier%20cccg04%20paper.pdf
michael@0: 
michael@0: */
michael@0: 
michael@0: #include "SkPathOpsCubic.h"
michael@0: #include "SkPathOpsLine.h"
michael@0: #include "SkPathOpsQuad.h"
michael@0: #include "SkReduceOrder.h"
michael@0: #include "SkTArray.h"
michael@0: #include "SkTSort.h"
michael@0: 
michael@0: #define USE_CUBIC_END_POINTS 1
michael@0: 
michael@0: static double calc_t_div(const SkDCubic& cubic, double precision, double start) {
michael@0:     const double adjust = sqrt(3.) / 36;
michael@0:     SkDCubic sub;
michael@0:     const SkDCubic* cPtr;
michael@0:     if (start == 0) {
michael@0:         cPtr = &cubic;
michael@0:     } else {
michael@0:         // OPTIMIZE: special-case half-split ?
michael@0:         sub = cubic.subDivide(start, 1);
michael@0:         cPtr = &sub;
michael@0:     }
michael@0:     const SkDCubic& c = *cPtr;
michael@0:     double dx = c[3].fX - 3 * (c[2].fX - c[1].fX) - c[0].fX;
michael@0:     double dy = c[3].fY - 3 * (c[2].fY - c[1].fY) - c[0].fY;
michael@0:     double dist = sqrt(dx * dx + dy * dy);
michael@0:     double tDiv3 = precision / (adjust * dist);
michael@0:     double t = SkDCubeRoot(tDiv3);
michael@0:     if (start > 0) {
michael@0:         t = start + (1 - start) * t;
michael@0:     }
michael@0:     return t;
michael@0: }
michael@0: 
michael@0: SkDQuad SkDCubic::toQuad() const {
michael@0:     SkDQuad quad;
michael@0:     quad[0] = fPts[0];
michael@0:     const SkDPoint fromC1 = {(3 * fPts[1].fX - fPts[0].fX) / 2, (3 * fPts[1].fY - fPts[0].fY) / 2};
michael@0:     const SkDPoint fromC2 = {(3 * fPts[2].fX - fPts[3].fX) / 2, (3 * fPts[2].fY - fPts[3].fY) / 2};
michael@0:     quad[1].fX = (fromC1.fX + fromC2.fX) / 2;
michael@0:     quad[1].fY = (fromC1.fY + fromC2.fY) / 2;
michael@0:     quad[2] = fPts[3];
michael@0:     return quad;
michael@0: }
michael@0: 
michael@0: static bool add_simple_ts(const SkDCubic& cubic, double precision, SkTArray<double, true>* ts) {
michael@0:     double tDiv = calc_t_div(cubic, precision, 0);
michael@0:     if (tDiv >= 1) {
michael@0:         return true;
michael@0:     }
michael@0:     if (tDiv >= 0.5) {
michael@0:         ts->push_back(0.5);
michael@0:         return true;
michael@0:     }
michael@0:     return false;
michael@0: }
michael@0: 
michael@0: static void addTs(const SkDCubic& cubic, double precision, double start, double end,
michael@0:         SkTArray<double, true>* ts) {
michael@0:     double tDiv = calc_t_div(cubic, precision, 0);
michael@0:     double parts = ceil(1.0 / tDiv);
michael@0:     for (double index = 0; index < parts; ++index) {
michael@0:         double newT = start + (index / parts) * (end - start);
michael@0:         if (newT > 0 && newT < 1) {
michael@0:             ts->push_back(newT);
michael@0:         }
michael@0:     }
michael@0: }
michael@0: 
michael@0: // flavor that returns T values only, deferring computing the quads until they are needed
michael@0: // FIXME: when called from recursive intersect 2, this could take the original cubic
michael@0: // and do a more precise job when calling chop at and sub divide by computing the fractional ts.
michael@0: // it would still take the prechopped cubic for reduce order and find cubic inflections
michael@0: void SkDCubic::toQuadraticTs(double precision, SkTArray<double, true>* ts) const {
michael@0:     SkReduceOrder reducer;
michael@0:     int order = reducer.reduce(*this, SkReduceOrder::kAllow_Quadratics);
michael@0:     if (order < 3) {
michael@0:         return;
michael@0:     }
michael@0:     double inflectT[5];
michael@0:     int inflections = findInflections(inflectT);
michael@0:     SkASSERT(inflections <= 2);
michael@0:     if (!endsAreExtremaInXOrY()) {
michael@0:         inflections += findMaxCurvature(&inflectT[inflections]);
michael@0:         SkASSERT(inflections <= 5);
michael@0:     }
michael@0:     SkTQSort<double>(inflectT, &inflectT[inflections - 1]);
michael@0:     // OPTIMIZATION: is this filtering common enough that it needs to be pulled out into its
michael@0:     // own subroutine?
michael@0:     while (inflections && approximately_less_than_zero(inflectT[0])) {
michael@0:         memmove(inflectT, &inflectT[1], sizeof(inflectT[0]) * --inflections);
michael@0:     }
michael@0:     int start = 0;
michael@0:     int next = 1;
michael@0:     while (next < inflections) {
michael@0:         if (!approximately_equal(inflectT[start], inflectT[next])) {
michael@0:             ++start;
michael@0:         ++next;
michael@0:             continue;
michael@0:         }
michael@0:         memmove(&inflectT[start], &inflectT[next], sizeof(inflectT[0]) * (--inflections - start));
michael@0:     }
michael@0: 
michael@0:     while (inflections && approximately_greater_than_one(inflectT[inflections - 1])) {
michael@0:         --inflections;
michael@0:     }
michael@0:     SkDCubicPair pair;
michael@0:     if (inflections == 1) {
michael@0:         pair = chopAt(inflectT[0]);
michael@0:         int orderP1 = reducer.reduce(pair.first(), SkReduceOrder::kNo_Quadratics);
michael@0:         if (orderP1 < 2) {
michael@0:             --inflections;
michael@0:         } else {
michael@0:             int orderP2 = reducer.reduce(pair.second(), SkReduceOrder::kNo_Quadratics);
michael@0:             if (orderP2 < 2) {
michael@0:                 --inflections;
michael@0:             }
michael@0:         }
michael@0:     }
michael@0:     if (inflections == 0 && add_simple_ts(*this, precision, ts)) {
michael@0:         return;
michael@0:     }
michael@0:     if (inflections == 1) {
michael@0:         pair = chopAt(inflectT[0]);
michael@0:         addTs(pair.first(), precision, 0, inflectT[0], ts);
michael@0:         addTs(pair.second(), precision, inflectT[0], 1, ts);
michael@0:         return;
michael@0:     }
michael@0:     if (inflections > 1) {
michael@0:         SkDCubic part = subDivide(0, inflectT[0]);
michael@0:         addTs(part, precision, 0, inflectT[0], ts);
michael@0:         int last = inflections - 1;
michael@0:         for (int idx = 0; idx < last; ++idx) {
michael@0:             part = subDivide(inflectT[idx], inflectT[idx + 1]);
michael@0:             addTs(part, precision, inflectT[idx], inflectT[idx + 1], ts);
michael@0:         }
michael@0:         part = subDivide(inflectT[last], 1);
michael@0:         addTs(part, precision, inflectT[last], 1, ts);
michael@0:         return;
michael@0:     }
michael@0:     addTs(*this, precision, 0, 1, ts);
michael@0: }