static inline bool are_collinear(const Sk2f& p0, const Sk2f& p1, const Sk2f& p2,
float tolerance = kFlatnessThreshold) {
Sk2f l = p2 - p0; // Line from p0 -> p2.
// lwidth = Manhattan width of l.
Sk2f labs = l.abs();
float lwidth = labs[0] + labs[1];
// d = |p1 - p0| dot | l.y|
// |-l.x| = distance from p1 to l.
Sk2f dd = (p1 - p0) * SkNx_shuffle<1,0>(l);
float d = dd[0] - dd[1];
// We are collinear if a box with radius "tolerance", centered on p1, touches the line l.
// To decide this, we check if the distance from p1 to the line is less than the distance from
// p1 to the far corner of this imaginary box, along that same normal vector.
// The far corner of the box can be found at "p1 + sign(n) * tolerance", where n is normal to l:
//
// abs(dot(p1 - p0, n)) <= dot(sign(n) * tolerance, n)
//
// Which reduces to:
//
// abs(d) <= (n.x * sign(n.x) + n.y * sign(n.y)) * tolerance
// abs(d) <= (abs(n.x) + abs(n.y)) * tolerance
//
// Use "<=" in case l == 0.
return std::abs(d) <= lwidth * tolerance;
}
void GrCCFillGeometry::appendMonotonicConic(const Sk2f& p0, const Sk2f& p1, const Sk2f& p2,
float w) {
SkASSERT(w >= 0);
Sk2f base = p2 - p0;
Sk2f baseAbs = base.abs();
float baseWidth = baseAbs[0] + baseAbs[1];
// Find the height of the curve. Max height always occurs at T=.5 for conics.
Sk2f d = (p1 - p0) * SkNx_shuffle<1,0>(base);
float h1 = std::abs(d[1] - d[0]); // Height of p1 above the base.
float ht = h1*w, hs = 1 + w; // Height of the conic = ht/hs.
// i.e. (ht/hs <= baseWidth * kFlatnessThreshold). Use "<=" in case base == 0.
if (ht <= (baseWidth*hs) * kFlatnessThreshold) {
// We are flat. (See rationale in are_collinear.)
this->appendLine(p0, p2);
return;
}
// i.e. (w > 1 && h1 - ht/hs < baseWidth).
if (w > 1 && h1*hs - ht < baseWidth*hs) {
// If we get within 1px of p1 when w > 1, we will pick up artifacts from the implicit
// function's reflection. Chop at max height (T=.5) and draw a triangle instead.
Sk2f p1w = p1*w;
Sk2f ab = p0 + p1w;
Sk2f bc = p1w + p2;
Sk2f highpoint = (ab + bc) / (2*(1 + w));
this->appendLine(p0, highpoint);
this->appendLine(highpoint, p2);
return;
}
SkASSERT(fPoints.back() == SkPoint::Make(p0[0], p0[1]));
SkASSERT((p0 != p2).anyTrue());
p1.store(&fPoints.push_back());
p2.store(&fPoints.push_back());
fConicWeights.push_back(w);
fVerbs.push_back(Verb::kMonotonicConicTo);
++fCurrContourTallies.fConics;
}
// Finds where to chop a non-loop around its intersection point. The resulting cubic segments will
// be chopped such that a box of radius 'padRadius', centered at any point along the curve segment,
// is guaranteed to not cross the tangent lines at the intersection point (a.k.a lines L & M).
//
// 'chops' will be filled with 0, 2, or 4 T values. The segments between T0..T1 and T2..T3 must be
// drawn with quadratic splines instead of cubics.
//
// A loop intersection falls at two different T values, so this method takes Sk2f and computes the
// padding for both in SIMD.
static inline void find_chops_around_loop_intersection(float padRadius, Sk2f t2, Sk2f s2,
const Sk2f& C0, const Sk2f& C1,
ExcludedTerm skipTerm, float Cdet,
SkSTArray<4, float>* chops) {
SkASSERT(chops->empty());
SkASSERT(padRadius >= 0);
padRadius /= std::abs(Cdet); // Scale this single value rather than all of C^-1 later on.
// The parametric functions for distance from lines L & M are:
//
// l(T) = (T - Td)^2 * (T - Te)
// m(T) = (T - Td) * (T - Te)^2
//
// See "Resolution Independent Curve Rendering using Programmable Graphics Hardware",
// 4.3 Finding klmn:
//
// https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf
Sk2f T2 = t2/s2; // T2 is the double root of l(T).
Sk2f T1 = SkNx_shuffle<1,0>(T2); // T1 is the other root of l(T).
// Convert l(T), m(T) to power-basis form:
//
// | 1 1 |
// |l(T) m(T)| = |T^3 T^2 T 1| * | l2 m2 |
// | l1 m1 |
// | l0 m0 |
//
// From here on we use Sk2f with "L" names, but the second lane will be for line M.
Sk2f l2 = SkNx_fma(Sk2f(-2), T2, -T1);
Sk2f l1 = T2 * SkNx_fma(Sk2f(2), T1, T2);
Sk2f l0 = -T2*T2*T1;
// The equation for line L can be found as follows:
//
// L = C^-1 * (l excluding skipTerm)
//
// (See comments for GrPathUtils::calcCubicInverseTransposePowerBasisMatrix.)
// We are only interested in the normal to L, so only need the upper 2x2 of C^-1. And rather
// than divide by determinant(C) here, we have already performed this divide on padRadius.
Sk2f l2or1 = (ExcludedTerm::kLinearTerm == skipTerm) ? l2 : l1;
Sk2f Lx = -C0[1]*l2or1 + C1[1]; // l3 is always 1.
Sk2f Ly = C0[0]*l2or1 - C1[0];
// A box of radius "padRadius" is touching line L if "center dot L" is less than the Manhattan
// with of L. (See rationale in are_collinear.)
Sk2f Lwidth = Lx.abs() + Ly.abs();
Sk2f pad = Lwidth * padRadius;
// Is l(T=0) outside the padding around line L?
Sk2f lT0 = l0; // l(T=0) = |0 0 0 1| dot |1 l2 l1 l0| = l0
Sk2f outsideT0 = lT0.abs() - pad;
// Is l(T=1) outside the padding around line L?
Sk2f lT1 = (Sk2f(1) + l2 + l1 + l0).abs(); // l(T=1) = |1 1 1 1| dot |1 l2 l1 l0|
Sk2f outsideT1 = lT1.abs() - pad;
// Values for solving the cubic.
Sk2f p, q, qqq, discr, numRoots, D;
bool hasDiscr = false;
// Values for calculating one root (rarely needed).
Sk2f R, QQ;
bool hasOneRootVals = false;
// Values for calculating three roots.
Sk2f P, cosTheta3;
bool hasThreeRootVals = false;
// Solve for the T values where l(T) = +pad and m(T) = -pad.
for (int i = 0; i < 2; ++i) {
float T = T2[i]; // T is the point we are chopping around.
if ((T < 0 && outsideT0[i] >= 0) || (T > 1 && outsideT1[i] >= 0)) {
// The padding around T is completely out of range. No point solving for it.
continue;
}
if (!hasDiscr) {
p = Sk2f(+.5f, -.5f) * pad;
q = (1.f/3) * (T2 - T1);
qqq = q*q*q;
discr = qqq*p*2 + p*p;
numRoots = (discr < 0).thenElse(3, 1);
D = T2 - q;
hasDiscr = true;
}
if (1 == numRoots[i]) {
if (!hasOneRootVals) {
Sk2f r = qqq + p;
Sk2f s = r.abs() + discr.sqrt();
R = (r > 0).thenElse(-s, s);
QQ = q*q;
hasOneRootVals = true;
}
float A = cbrtf(R[i]);
float B = A != 0 ? QQ[i]/A : 0;
// When there is only one root, ine L chops from root..1, line M chops from 0..root.
if (1 == i) {
chops->push_back(0);
}
chops->push_back(A + B + D[i]);
if (0 == i) {
chops->push_back(1);
}
continue;
}
if (!hasThreeRootVals) {
P = q.abs() * -2;
cosTheta3 = (q >= 0).thenElse(1, -1) + p / qqq.abs();
hasThreeRootVals = true;
}
static constexpr float k2PiOver3 = 2 * SK_ScalarPI / 3;
float theta = std::acos(cosTheta3[i]) * (1.f/3);
float roots[3] = {P[i] * std::cos(theta) + D[i],
P[i] * std::cos(theta + k2PiOver3) + D[i],
P[i] * std::cos(theta - k2PiOver3) + D[i]};
// Sort the three roots.
swap_if_greater(roots[0], roots[1]);
swap_if_greater(roots[1], roots[2]);
swap_if_greater(roots[0], roots[1]);
// Line L chops around the first 2 roots, line M chops around the second 2.
chops->push_back_n(2, &roots[i]);
}
}
// Finds where to chop a non-loop around its inflection points. The resulting cubic segments will be
// chopped such that a box of radius 'padRadius', centered at any point along the curve segment, is
// guaranteed to not cross the tangent lines at the inflection points (a.k.a lines L & M).
//
// 'chops' will be filled with 0, 2, or 4 T values. The segments between T0..T1 and T2..T3 must be
// drawn with flat lines instead of cubics.
//
// A serpentine cubic has two inflection points, so this method takes Sk2f and computes the padding
// for both in SIMD.
static inline void find_chops_around_inflection_points(float padRadius, Sk2f tl, Sk2f sl,
const Sk2f& C0, const Sk2f& C1,
ExcludedTerm skipTerm, float Cdet,
SkSTArray<4, float>* chops) {
SkASSERT(chops->empty());
SkASSERT(padRadius >= 0);
padRadius /= std::abs(Cdet); // Scale this single value rather than all of C^-1 later on.
// The homogeneous parametric functions for distance from lines L & M are:
//
// l(t,s) = (t*sl - s*tl)^3
// m(t,s) = (t*sm - s*tm)^3
//
// See "Resolution Independent Curve Rendering using Programmable Graphics Hardware",
// 4.3 Finding klmn:
//
// https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf
//
// From here on we use Sk2f with "L" names, but the second lane will be for line M.
tl = (sl > 0).thenElse(tl, -tl); // Tl=tl/sl is the triple root of l(t,s). Normalize so s >= 0.
sl = sl.abs();
// Convert l(t,s), m(t,s) to power-basis form:
//
// | l3 m3 |
// |l(t,s) m(t,s)| = |t^3 t^2*s t*s^2 s^3| * | l2 m2 |
// | l1 m1 |
// | l0 m0 |
//
Sk2f l3 = sl*sl*sl;
Sk2f l2or1 = (ExcludedTerm::kLinearTerm == skipTerm) ? sl*sl*tl*-3 : sl*tl*tl*3;
// The equation for line L can be found as follows:
//
// L = C^-1 * (l excluding skipTerm)
//
// (See comments for GrPathUtils::calcCubicInverseTransposePowerBasisMatrix.)
// We are only interested in the normal to L, so only need the upper 2x2 of C^-1. And rather
// than divide by determinant(C) here, we have already performed this divide on padRadius.
Sk2f Lx = C1[1]*l3 - C0[1]*l2or1;
Sk2f Ly = -C1[0]*l3 + C0[0]*l2or1;
// A box of radius "padRadius" is touching line L if "center dot L" is less than the Manhattan
// with of L. (See rationale in are_collinear.)
Sk2f Lwidth = Lx.abs() + Ly.abs();
Sk2f pad = Lwidth * padRadius;
// Will T=(t + cbrt(pad))/s be greater than 0? No need to solve roots outside T=0..1.
Sk2f insideLeftPad = pad + tl*tl*tl;
// Will T=(t - cbrt(pad))/s be less than 1? No need to solve roots outside T=0..1.
Sk2f tms = tl - sl;
Sk2f insideRightPad = pad - tms*tms*tms;
// Solve for the T values where abs(l(T)) = pad.
if (insideLeftPad[0] > 0 && insideRightPad[0] > 0) {
float padT = cbrtf(pad[0]);
Sk2f pts = (tl[0] + Sk2f(-padT, +padT)) / sl[0];
pts.store(chops->push_back_n(2));
}
// Solve for the T values where abs(m(T)) = pad.
if (insideLeftPad[1] > 0 && insideRightPad[1] > 0) {
float padT = cbrtf(pad[1]);
Sk2f pts = (tl[1] + Sk2f(-padT, +padT)) / sl[1];
pts.store(chops->push_back_n(2));
}
}
