| /* |
| 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 = ⊂ |
| } |
| 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); |
| } |