/*
* Copyright 2012 Google Inc.
*
* Use of this source code is governed by a BSD-style license that can be
* found in the LICENSE file.
*/
#include "SkLineParameters.h"
#include "SkPathOpsCubic.h"
#include "SkPathOpsLine.h"
#include "SkPathOpsQuad.h"
#include "SkPathOpsRect.h"
#include "SkTSort.h"
const int SkDCubic::gPrecisionUnit = 256; // FIXME: test different values in test framework
// give up when changing t no longer moves point
// also, copy point rather than recompute it when it does change
double SkDCubic::binarySearch(double min, double max, double axisIntercept,
SearchAxis xAxis) const {
double t = (min + max) / 2;
double step = (t - min) / 2;
SkDPoint cubicAtT = ptAtT(t);
double calcPos = (&cubicAtT.fX)[xAxis];
double calcDist = calcPos - axisIntercept;
do {
double priorT = t - step;
SkASSERT(priorT >= min);
SkDPoint lessPt = ptAtT(priorT);
if (approximately_equal(lessPt.fX, cubicAtT.fX)
&& approximately_equal(lessPt.fY, cubicAtT.fY)) {
return -1; // binary search found no point at this axis intercept
}
double lessDist = (&lessPt.fX)[xAxis] - axisIntercept;
#if DEBUG_CUBIC_BINARY_SEARCH
SkDebugf("t=%1.9g calc=%1.9g dist=%1.9g step=%1.9g less=%1.9g\n", t, calcPos, calcDist,
step, lessDist);
#endif
double lastStep = step;
step /= 2;
if (calcDist > 0 ? calcDist > lessDist : calcDist < lessDist) {
t = priorT;
} else {
double nextT = t + lastStep;
SkASSERT(nextT <= max);
SkDPoint morePt = ptAtT(nextT);
if (approximately_equal(morePt.fX, cubicAtT.fX)
&& approximately_equal(morePt.fY, cubicAtT.fY)) {
return -1; // binary search found no point at this axis intercept
}
double moreDist = (&morePt.fX)[xAxis] - axisIntercept;
if (calcDist > 0 ? calcDist <= moreDist : calcDist >= moreDist) {
continue;
}
t = nextT;
}
SkDPoint testAtT = ptAtT(t);
cubicAtT = testAtT;
calcPos = (&cubicAtT.fX)[xAxis];
calcDist = calcPos - axisIntercept;
} while (!approximately_equal(calcPos, axisIntercept));
return t;
}
// FIXME: cache keep the bounds and/or precision with the caller?
double SkDCubic::calcPrecision() const {
SkDRect dRect;
dRect.setBounds(*this); // OPTIMIZATION: just use setRawBounds ?
double width = dRect.fRight - dRect.fLeft;
double height = dRect.fBottom - dRect.fTop;
return (width > height ? width : height) / gPrecisionUnit;
}
bool SkDCubic::clockwise() const {
double sum = (fPts[0].fX - fPts[3].fX) * (fPts[0].fY + fPts[3].fY);
for (int idx = 0; idx < 3; ++idx) {
sum += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY);
}
return sum <= 0;
}
void SkDCubic::Coefficients(const double* src, double* A, double* B, double* C, double* D) {
*A = src[6]; // d
*B = src[4] * 3; // 3*c
*C = src[2] * 3; // 3*b
*D = src[0]; // a
*A -= *D - *C + *B; // A = -a + 3*b - 3*c + d
*B += 3 * *D - 2 * *C; // B = 3*a - 6*b + 3*c
*C -= 3 * *D; // C = -3*a + 3*b
}
bool SkDCubic::controlsContainedByEnds() const {
SkDVector startTan = fPts[1] - fPts[0];
if (startTan.fX == 0 && startTan.fY == 0) {
startTan = fPts[2] - fPts[0];
}
SkDVector endTan = fPts[2] - fPts[3];
if (endTan.fX == 0 && endTan.fY == 0) {
endTan = fPts[1] - fPts[3];
}
if (startTan.dot(endTan) >= 0) {
return false;
}
SkDLine startEdge = {{fPts[0], fPts[0]}};
startEdge[1].fX -= startTan.fY;
startEdge[1].fY += startTan.fX;
SkDLine endEdge = {{fPts[3], fPts[3]}};
endEdge[1].fX -= endTan.fY;
endEdge[1].fY += endTan.fX;
double leftStart1 = startEdge.isLeft(fPts[1]);
if (leftStart1 * startEdge.isLeft(fPts[2]) < 0) {
return false;
}
double leftEnd1 = endEdge.isLeft(fPts[1]);
if (leftEnd1 * endEdge.isLeft(fPts[2]) < 0) {
return false;
}
return leftStart1 * leftEnd1 >= 0;
}
bool SkDCubic::endsAreExtremaInXOrY() const {
return (between(fPts[0].fX, fPts[1].fX, fPts[3].fX)
&& between(fPts[0].fX, fPts[2].fX, fPts[3].fX))
|| (between(fPts[0].fY, fPts[1].fY, fPts[3].fY)
&& between(fPts[0].fY, fPts[2].fY, fPts[3].fY));
}
bool SkDCubic::isLinear(int startIndex, int endIndex) const {
SkLineParameters lineParameters;
lineParameters.cubicEndPoints(*this, startIndex, endIndex);
// FIXME: maybe it's possible to avoid this and compare non-normalized
lineParameters.normalize();
double distance = lineParameters.controlPtDistance(*this, 1);
if (!approximately_zero(distance)) {
return false;
}
distance = lineParameters.controlPtDistance(*this, 2);
return approximately_zero(distance);
}
bool SkDCubic::monotonicInY() const {
return between(fPts[0].fY, fPts[1].fY, fPts[3].fY)
&& between(fPts[0].fY, fPts[2].fY, fPts[3].fY);
}
int SkDCubic::searchRoots(double extremeTs[6], int extrema, double axisIntercept,
SearchAxis xAxis, double* validRoots) const {
extrema += findInflections(&extremeTs[extrema]);
extremeTs[extrema++] = 0;
extremeTs[extrema] = 1;
SkTQSort(extremeTs, extremeTs + extrema);
int validCount = 0;
for (int index = 0; index < extrema; ) {
double min = extremeTs[index];
double max = extremeTs[++index];
if (min == max) {
continue;
}
double newT = binarySearch(min, max, axisIntercept, xAxis);
if (newT >= 0) {
validRoots[validCount++] = newT;
}
}
return validCount;
}
bool SkDCubic::serpentine() const {
#if 0 // FIXME: enabling this fixes cubicOp114 but breaks cubicOp58d and cubicOp53d
double tValues[2];
// OPTIMIZATION : another case where caching the present of cubic inflections would be useful
return findInflections(tValues) > 1;
#endif
if (!controlsContainedByEnds()) {
return false;
}
double wiggle = (fPts[0].fX - fPts[2].fX) * (fPts[0].fY + fPts[2].fY);
for (int idx = 0; idx < 2; ++idx) {
wiggle += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY);
}
double waggle = (fPts[1].fX - fPts[3].fX) * (fPts[1].fY + fPts[3].fY);
for (int idx = 1; idx < 3; ++idx) {
waggle += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY);
}
return wiggle * waggle < 0;
}
// cubic roots
static const double PI = 3.141592653589793;
// from SkGeometry.cpp (and Numeric Solutions, 5.6)
int SkDCubic::RootsValidT(double A, double B, double C, double D, double t[3]) {
double s[3];
int realRoots = RootsReal(A, B, C, D, s);
int foundRoots = SkDQuad::AddValidTs(s, realRoots, t);
return foundRoots;
}
int SkDCubic::RootsReal(double A, double B, double C, double D, double s[3]) {
#ifdef SK_DEBUG
// create a string mathematica understands
// GDB set print repe 15 # if repeated digits is a bother
// set print elements 400 # if line doesn't fit
char str[1024];
sk_bzero(str, sizeof(str));
SK_SNPRINTF(str, sizeof(str), "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]",
A, B, C, D);
SkPathOpsDebug::MathematicaIze(str, sizeof(str));
#if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA
SkDebugf("%s\n", str);
#endif
#endif
if (approximately_zero(A)
&& approximately_zero_when_compared_to(A, B)
&& approximately_zero_when_compared_to(A, C)
&& approximately_zero_when_compared_to(A, D)) { // we're just a quadratic
return SkDQuad::RootsReal(B, C, D, s);
}
if (approximately_zero_when_compared_to(D, A)
&& approximately_zero_when_compared_to(D, B)
&& approximately_zero_when_compared_to(D, C)) { // 0 is one root
int num = SkDQuad::RootsReal(A, B, C, s);
for (int i = 0; i < num; ++i) {
if (approximately_zero(s[i])) {
return num;
}
}
s[num++] = 0;
return num;
}
if (approximately_zero(A + B + C + D)) { // 1 is one root
int num = SkDQuad::RootsReal(A, A + B, -D, s);
for (int i = 0; i < num; ++i) {
if (AlmostDequalUlps(s[i], 1)) {
return num;
}
}
s[num++] = 1;
return num;
}
double a, b, c;
{
double invA = 1 / A;
a = B * invA;
b = C * invA;
c = D * invA;
}
double a2 = a * a;
double Q = (a2 - b * 3) / 9;
double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54;
double R2 = R * R;
double Q3 = Q * Q * Q;
double R2MinusQ3 = R2 - Q3;
double adiv3 = a / 3;
double r;
double* roots = s;
if (R2MinusQ3 < 0) { // we have 3 real roots
double theta = acos(R / sqrt(Q3));
double neg2RootQ = -2 * sqrt(Q);
r = neg2RootQ * cos(theta / 3) - adiv3;
*roots++ = r;
r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3;
if (!AlmostDequalUlps(s[0], r)) {
*roots++ = r;
}
r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3;
if (!AlmostDequalUlps(s[0], r) && (roots - s == 1 || !AlmostDequalUlps(s[1], r))) {
*roots++ = r;
}
} else { // we have 1 real root
double sqrtR2MinusQ3 = sqrt(R2MinusQ3);
double A = fabs(R) + sqrtR2MinusQ3;
A = SkDCubeRoot(A);
if (R > 0) {
A = -A;
}
if (A != 0) {
A += Q / A;
}
r = A - adiv3;
*roots++ = r;
if (AlmostDequalUlps((double) R2, (double) Q3)) {
r = -A / 2 - adiv3;
if (!AlmostDequalUlps(s[0], r)) {
*roots++ = r;
}
}
}
return static_cast<int>(roots - s);
}
// from http://www.cs.sunysb.edu/~qin/courses/geometry/4.pdf
// c(t) = a(1-t)^3 + 3bt(1-t)^2 + 3c(1-t)t^2 + dt^3
// c'(t) = -3a(1-t)^2 + 3b((1-t)^2 - 2t(1-t)) + 3c(2t(1-t) - t^2) + 3dt^2
// = 3(b-a)(1-t)^2 + 6(c-b)t(1-t) + 3(d-c)t^2
static double derivative_at_t(const double* src, double t) {
double one_t = 1 - t;
double a = src[0];
double b = src[2];
double c = src[4];
double d = src[6];
return 3 * ((b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t);
}
// OPTIMIZE? compute t^2, t(1-t), and (1-t)^2 and pass them to another version of derivative at t?
SkDVector SkDCubic::dxdyAtT(double t) const {
SkDVector result = { derivative_at_t(&fPts[0].fX, t), derivative_at_t(&fPts[0].fY, t) };
return result;
}
// OPTIMIZE? share code with formulate_F1DotF2
int SkDCubic::findInflections(double tValues[]) const {
double Ax = fPts[1].fX - fPts[0].fX;
double Ay = fPts[1].fY - fPts[0].fY;
double Bx = fPts[2].fX - 2 * fPts[1].fX + fPts[0].fX;
double By = fPts[2].fY - 2 * fPts[1].fY + fPts[0].fY;
double Cx = fPts[3].fX + 3 * (fPts[1].fX - fPts[2].fX) - fPts[0].fX;
double Cy = fPts[3].fY + 3 * (fPts[1].fY - fPts[2].fY) - fPts[0].fY;
return SkDQuad::RootsValidT(Bx * Cy - By * Cx, Ax * Cy - Ay * Cx, Ax * By - Ay * Bx, tValues);
}
static void formulate_F1DotF2(const double src[], double coeff[4]) {
double a = src[2] - src[0];
double b = src[4] - 2 * src[2] + src[0];
double c = src[6] + 3 * (src[2] - src[4]) - src[0];
coeff[0] = c * c;
coeff[1] = 3 * b * c;
coeff[2] = 2 * b * b + c * a;
coeff[3] = a * b;
}
/** SkDCubic'(t) = At^2 + Bt + C, where
A = 3(-a + 3(b - c) + d)
B = 6(a - 2b + c)
C = 3(b - a)
Solve for t, keeping only those that fit between 0 < t < 1
*/
int SkDCubic::FindExtrema(double a, double b, double c, double d, double tValues[2]) {
// we divide A,B,C by 3 to simplify
double A = d - a + 3*(b - c);
double B = 2*(a - b - b + c);
double C = b - a;
return SkDQuad::RootsValidT(A, B, C, tValues);
}
/* from SkGeometry.cpp
Looking for F' dot F'' == 0
A = b - a
B = c - 2b + a
C = d - 3c + 3b - a
F' = 3Ct^2 + 6Bt + 3A
F'' = 6Ct + 6B
F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
*/
int SkDCubic::findMaxCurvature(double tValues[]) const {
double coeffX[4], coeffY[4];
int i;
formulate_F1DotF2(&fPts[0].fX, coeffX);
formulate_F1DotF2(&fPts[0].fY, coeffY);
for (i = 0; i < 4; i++) {
coeffX[i] = coeffX[i] + coeffY[i];
}
return RootsValidT(coeffX[0], coeffX[1], coeffX[2], coeffX[3], tValues);
}
SkDPoint SkDCubic::top(double startT, double endT) const {
SkDCubic sub = subDivide(startT, endT);
SkDPoint topPt = sub[0];
if (topPt.fY > sub[3].fY || (topPt.fY == sub[3].fY && topPt.fX > sub[3].fX)) {
topPt = sub[3];
}
double extremeTs[2];
if (!sub.monotonicInY()) {
int roots = FindExtrema(sub[0].fY, sub[1].fY, sub[2].fY, sub[3].fY, extremeTs);
for (int index = 0; index < roots; ++index) {
double t = startT + (endT - startT) * extremeTs[index];
SkDPoint mid = ptAtT(t);
if (topPt.fY > mid.fY || (topPt.fY == mid.fY && topPt.fX > mid.fX)) {
topPt = mid;
}
}
}
return topPt;
}
SkDPoint SkDCubic::ptAtT(double t) const {
if (0 == t) {
return fPts[0];
}
if (1 == t) {
return fPts[3];
}
double one_t = 1 - t;
double one_t2 = one_t * one_t;
double a = one_t2 * one_t;
double b = 3 * one_t2 * t;
double t2 = t * t;
double c = 3 * one_t * t2;
double d = t2 * t;
SkDPoint result = {a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX + d * fPts[3].fX,
a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY + d * fPts[3].fY};
return result;
}
/*
Given a cubic c, t1, and t2, find a small cubic segment.
The new cubic is defined as points A, B, C, and D, where
s1 = 1 - t1
s2 = 1 - t2
A = c[0]*s1*s1*s1 + 3*c[1]*s1*s1*t1 + 3*c[2]*s1*t1*t1 + c[3]*t1*t1*t1
D = c[0]*s2*s2*s2 + 3*c[1]*s2*s2*t2 + 3*c[2]*s2*t2*t2 + c[3]*t2*t2*t2
We don't have B or C. So We define two equations to isolate them.
First, compute two reference T values 1/3 and 2/3 from t1 to t2:
c(at (2*t1 + t2)/3) == E
c(at (t1 + 2*t2)/3) == F
Next, compute where those values must be if we know the values of B and C:
_12 = A*2/3 + B*1/3
12_ = A*1/3 + B*2/3
_23 = B*2/3 + C*1/3
23_ = B*1/3 + C*2/3
_34 = C*2/3 + D*1/3
34_ = C*1/3 + D*2/3
_123 = (A*2/3 + B*1/3)*2/3 + (B*2/3 + C*1/3)*1/3 = A*4/9 + B*4/9 + C*1/9
123_ = (A*1/3 + B*2/3)*1/3 + (B*1/3 + C*2/3)*2/3 = A*1/9 + B*4/9 + C*4/9
_234 = (B*2/3 + C*1/3)*2/3 + (C*2/3 + D*1/3)*1/3 = B*4/9 + C*4/9 + D*1/9
234_ = (B*1/3 + C*2/3)*1/3 + (C*1/3 + D*2/3)*2/3 = B*1/9 + C*4/9 + D*4/9
_1234 = (A*4/9 + B*4/9 + C*1/9)*2/3 + (B*4/9 + C*4/9 + D*1/9)*1/3
= A*8/27 + B*12/27 + C*6/27 + D*1/27
= E
1234_ = (A*1/9 + B*4/9 + C*4/9)*1/3 + (B*1/9 + C*4/9 + D*4/9)*2/3
= A*1/27 + B*6/27 + C*12/27 + D*8/27
= F
E*27 = A*8 + B*12 + C*6 + D
F*27 = A + B*6 + C*12 + D*8
Group the known values on one side:
M = E*27 - A*8 - D = B*12 + C* 6
N = F*27 - A - D*8 = B* 6 + C*12
M*2 - N = B*18
N*2 - M = C*18
B = (M*2 - N)/18
C = (N*2 - M)/18
*/
static double interp_cubic_coords(const double* src, double t) {
double ab = SkDInterp(src[0], src[2], t);
double bc = SkDInterp(src[2], src[4], t);
double cd = SkDInterp(src[4], src[6], t);
double abc = SkDInterp(ab, bc, t);
double bcd = SkDInterp(bc, cd, t);
double abcd = SkDInterp(abc, bcd, t);
return abcd;
}
SkDCubic SkDCubic::subDivide(double t1, double t2) const {
if (t1 == 0 || t2 == 1) {
if (t1 == 0 && t2 == 1) {
return *this;
}
SkDCubicPair pair = chopAt(t1 == 0 ? t2 : t1);
SkDCubic dst = t1 == 0 ? pair.first() : pair.second();
return dst;
}
SkDCubic dst;
double ax = dst[0].fX = interp_cubic_coords(&fPts[0].fX, t1);
double ay = dst[0].fY = interp_cubic_coords(&fPts[0].fY, t1);
double ex = interp_cubic_coords(&fPts[0].fX, (t1*2+t2)/3);
double ey = interp_cubic_coords(&fPts[0].fY, (t1*2+t2)/3);
double fx = interp_cubic_coords(&fPts[0].fX, (t1+t2*2)/3);
double fy = interp_cubic_coords(&fPts[0].fY, (t1+t2*2)/3);
double dx = dst[3].fX = interp_cubic_coords(&fPts[0].fX, t2);
double dy = dst[3].fY = interp_cubic_coords(&fPts[0].fY, t2);
double mx = ex * 27 - ax * 8 - dx;
double my = ey * 27 - ay * 8 - dy;
double nx = fx * 27 - ax - dx * 8;
double ny = fy * 27 - ay - dy * 8;
/* bx = */ dst[1].fX = (mx * 2 - nx) / 18;
/* by = */ dst[1].fY = (my * 2 - ny) / 18;
/* cx = */ dst[2].fX = (nx * 2 - mx) / 18;
/* cy = */ dst[2].fY = (ny * 2 - my) / 18;
// FIXME: call align() ?
return dst;
}
void SkDCubic::align(int endIndex, int ctrlIndex, SkDPoint* dstPt) const {
if (fPts[endIndex].fX == fPts[ctrlIndex].fX) {
dstPt->fX = fPts[endIndex].fX;
}
if (fPts[endIndex].fY == fPts[ctrlIndex].fY) {
dstPt->fY = fPts[endIndex].fY;
}
}
void SkDCubic::subDivide(const SkDPoint& a, const SkDPoint& d,
double t1, double t2, SkDPoint dst[2]) const {
SkASSERT(t1 != t2);
#if 0
double ex = interp_cubic_coords(&fPts[0].fX, (t1 * 2 + t2) / 3);
double ey = interp_cubic_coords(&fPts[0].fY, (t1 * 2 + t2) / 3);
double fx = interp_cubic_coords(&fPts[0].fX, (t1 + t2 * 2) / 3);
double fy = interp_cubic_coords(&fPts[0].fY, (t1 + t2 * 2) / 3);
double mx = ex * 27 - a.fX * 8 - d.fX;
double my = ey * 27 - a.fY * 8 - d.fY;
double nx = fx * 27 - a.fX - d.fX * 8;
double ny = fy * 27 - a.fY - d.fY * 8;
/* bx = */ dst[0].fX = (mx * 2 - nx) / 18;
/* by = */ dst[0].fY = (my * 2 - ny) / 18;
/* cx = */ dst[1].fX = (nx * 2 - mx) / 18;
/* cy = */ dst[1].fY = (ny * 2 - my) / 18;
#else
// this approach assumes that the control points computed directly are accurate enough
SkDCubic sub = subDivide(t1, t2);
dst[0] = sub[1] + (a - sub[0]);
dst[1] = sub[2] + (d - sub[3]);
#endif
if (t1 == 0 || t2 == 0) {
align(0, 1, t1 == 0 ? &dst[0] : &dst[1]);
}
if (t1 == 1 || t2 == 1) {
align(3, 2, t1 == 1 ? &dst[0] : &dst[1]);
}
if (AlmostBequalUlps(dst[0].fX, a.fX)) {
dst[0].fX = a.fX;
}
if (AlmostBequalUlps(dst[0].fY, a.fY)) {
dst[0].fY = a.fY;
}
if (AlmostBequalUlps(dst[1].fX, d.fX)) {
dst[1].fX = d.fX;
}
if (AlmostBequalUlps(dst[1].fY, d.fY)) {
dst[1].fY = d.fY;
}
}
/* classic one t subdivision */
static void interp_cubic_coords(const double* src, double* dst, double t) {
double ab = SkDInterp(src[0], src[2], t);
double bc = SkDInterp(src[2], src[4], t);
double cd = SkDInterp(src[4], src[6], t);
double abc = SkDInterp(ab, bc, t);
double bcd = SkDInterp(bc, cd, t);
double abcd = SkDInterp(abc, bcd, t);
dst[0] = src[0];
dst[2] = ab;
dst[4] = abc;
dst[6] = abcd;
dst[8] = bcd;
dst[10] = cd;
dst[12] = src[6];
}
SkDCubicPair SkDCubic::chopAt(double t) const {
SkDCubicPair dst;
if (t == 0.5) {
dst.pts[0] = fPts[0];
dst.pts[1].fX = (fPts[0].fX + fPts[1].fX) / 2;
dst.pts[1].fY = (fPts[0].fY + fPts[1].fY) / 2;
dst.pts[2].fX = (fPts[0].fX + 2 * fPts[1].fX + fPts[2].fX) / 4;
dst.pts[2].fY = (fPts[0].fY + 2 * fPts[1].fY + fPts[2].fY) / 4;
dst.pts[3].fX = (fPts[0].fX + 3 * (fPts[1].fX + fPts[2].fX) + fPts[3].fX) / 8;
dst.pts[3].fY = (fPts[0].fY + 3 * (fPts[1].fY + fPts[2].fY) + fPts[3].fY) / 8;
dst.pts[4].fX = (fPts[1].fX + 2 * fPts[2].fX + fPts[3].fX) / 4;
dst.pts[4].fY = (fPts[1].fY + 2 * fPts[2].fY + fPts[3].fY) / 4;
dst.pts[5].fX = (fPts[2].fX + fPts[3].fX) / 2;
dst.pts[5].fY = (fPts[2].fY + fPts[3].fY) / 2;
dst.pts[6] = fPts[3];
return dst;
}
interp_cubic_coords(&fPts[0].fX, &dst.pts[0].fX, t);
interp_cubic_coords(&fPts[0].fY, &dst.pts[0].fY, t);
return dst;
}