src/gpu/tessellate/Tessellation.cpp - skia - Git at Google

 /*
  * Copyright 2021 Google LLC.
  *
  * Use of this source code is governed by a BSD-style license that can be
  * found in the LICENSE file.
  */
 #include "src/gpu/tessellate/Tessellation.h"

 #include "include/core/SkPath.h"
 #include "include/core/SkPathTypes.h"
 #include "include/core/SkRect.h"
 #include "include/private/base/SkFloatingPoint.h"
 #include "include/private/base/SkTArray.h"
 #include "src/base/SkUtils.h"
 #include "src/base/SkVx.h"
 #include "src/core/SkGeometry.h"
 #include "src/core/SkPathPriv.h"
 #include "src/gpu/tessellate/CullTest.h"
 #include "src/gpu/tessellate/WangsFormula.h"

 using namespace skia_private;

 namespace skgpu::tess {

 namespace {

 using float2 = skvx::float2;
 using float4 = skvx::float4;

 // This value only protects us against getting stuck in infinite recursion due to fp32 precision
 // issues. Mathematically, every curve should reduce to manageable visible sections in O(log N)
 // chops, where N is the the magnitude of its control points.
 //
 // But, to define a protective upper bound, a cubic can enter or exit the viewport as many as 6
 // times. So we may need to refine the curve (via binary search chopping at T=.5) up to 6 times.
 //
 // Furthermore, chopping a cubic at T=.5 may only reduce its length by 1/8 (.5^3), so we may require
 // up to 6 chops in order to reduce the length by 1/2.
 constexpr static int kMaxChopsPerCurve = 128/*magnitude of +fp32_max - -fp32_max*/ *
                                          6/*max number of chops to reduce the length by half*/ *
                                          6/*max number of viewport boundary crosses*/;

 // Writes a new path, chopping as necessary so no verbs require more segments than
 // kMaxTessellationSegmentsPerCurve. Curves completely outside the viewport are flattened into
 // lines.
 class PathChopper {
 public:
     PathChopper(float tessellationPrecision, const SkMatrix& matrix, const SkRect& viewport)
             : fTessellationPrecision(tessellationPrecision)
             , fCullTest(viewport, matrix)
             , fVectorXform(matrix) {
         fPath.setIsVolatile(true);
     }

     SkPath path() const { return fPath; }

     void moveTo(SkPoint p) { fPath.moveTo(p); }
     void lineTo(const SkPoint p[2]) { fPath.lineTo(p[1]); }
     void close() { fPath.close(); }

     void quadTo(const SkPoint quad[3]) {
         SkASSERT(fPointStack.empty());
         // Use a heap stack to recursively chop the quad into manageable, on-screen segments.
         fPointStack.push_back_n(3, quad);
         int numChops = 0;
         while (!fPointStack.empty()) {
             const SkPoint* p = fPointStack.end() - 3;
             if (!fCullTest.areVisible3(p)) {
                 fPath.lineTo(p[2]);
             } else {
                 float n4 = wangs_formula::quadratic_p4(fTessellationPrecision, p, fVectorXform);
                 if (n4 > kMaxSegmentsPerCurve_p4 && numChops < kMaxChopsPerCurve) {
                     SkPoint chops[5];
                     SkChopQuadAtHalf(p, chops);
                     fPointStack.pop_back_n(3);
                     fPointStack.push_back_n(3, chops+2);
                     fPointStack.push_back_n(3, chops);
                     ++numChops;
                     continue;
                 }
                 fPath.quadTo(p[1], p[2]);
             }
             fPointStack.pop_back_n(3);
         }
     }

     void conicTo(const SkPoint conic[3], float weight) {
         SkASSERT(fPointStack.empty());
         SkASSERT(fWeightStack.empty());
         // Use a heap stack to recursively chop the conic into manageable, on-screen segments.
         fPointStack.push_back_n(3, conic);
         fWeightStack.push_back(weight);
         int numChops = 0;
         while (!fPointStack.empty()) {
             const SkPoint* p = fPointStack.end() - 3;
             float w = fWeightStack.back();
             if (!fCullTest.areVisible3(p)) {
                 fPath.lineTo(p[2]);
             } else {
                 float n2 = wangs_formula::conic_p2(fTessellationPrecision, p, w, fVectorXform);
                 if (n2 > kMaxSegmentsPerCurve_p2 && numChops < kMaxChopsPerCurve) {
                     SkConic chops[2];
                     if (!SkConic(p,w).chopAt(.5, chops)) {
                         SkPoint line[2] = {p[0], p[2]};
                         this->lineTo(line);
                         continue;
                     }
                     fPointStack.pop_back_n(3);
                     fWeightStack.pop_back();
                     fPointStack.push_back_n(3, chops[1].fPts);
                     fWeightStack.push_back(chops[1].fW);
                     fPointStack.push_back_n(3, chops[0].fPts);
                     fWeightStack.push_back(chops[0].fW);
                     ++numChops;
                     continue;
                 }
                 fPath.conicTo(p[1], p[2], w);
             }
             fPointStack.pop_back_n(3);
             fWeightStack.pop_back();
         }
         SkASSERT(fWeightStack.empty());
     }

     void cubicTo(const SkPoint cubic[4]) {
         SkASSERT(fPointStack.empty());
         // Use a heap stack to recursively chop the cubic into manageable, on-screen segments.
         fPointStack.push_back_n(4, cubic);
         int numChops = 0;
         while (!fPointStack.empty()) {
             SkPoint* p = fPointStack.end() - 4;
             if (!fCullTest.areVisible4(p)) {
                 fPath.lineTo(p[3]);
             } else {
                 float n4 = wangs_formula::cubic_p4(fTessellationPrecision, p, fVectorXform);
                 if (n4 > kMaxSegmentsPerCurve_p4 && numChops < kMaxChopsPerCurve) {
                     SkPoint chops[7];
                     SkChopCubicAtHalf(p, chops);
                     fPointStack.pop_back_n(4);
                     fPointStack.push_back_n(4, chops+3);
                     fPointStack.push_back_n(4, chops);
                     ++numChops;
                     continue;
                 }
                 fPath.cubicTo(p[1], p[2], p[3]);
             }
             fPointStack.pop_back_n(4);
         }
     }

 private:
     const float fTessellationPrecision;
     const CullTest fCullTest;
     const wangs_formula::VectorXform fVectorXform;
     SkPath fPath;

     // Used for stack-based recursion (instead of using the runtime stack).
     STArray<8, SkPoint> fPointStack;
     STArray<2, float> fWeightStack;
 };

 }  // namespace

 SkPath PreChopPathCurves(float tessellationPrecision,
                          const SkPath& path,
                          const SkMatrix& matrix,
                          const SkRect& viewport) {
     // If the viewport is exceptionally large, we could end up blowing out memory with an unbounded
     // number of of chops. Therefore, we require that the viewport is manageable enough that a fully
     // contained curve can be tessellated in kMaxTessellationSegmentsPerCurve or fewer. (Any larger
     // and that amount of pixels wouldn't fit in memory anyway.)
     SkASSERT(wangs_formula::worst_case_cubic(
                      tessellationPrecision,
                      viewport.width(),
                      viewport.height()) <= kMaxSegmentsPerCurve);
     PathChopper chopper(tessellationPrecision, matrix, viewport);
     for (auto [verb, p, w] : SkPathPriv::Iterate(path)) {
         switch (verb) {
             case SkPathVerb::kMove:
                 chopper.moveTo(p[0]);
                 break;
             case SkPathVerb::kLine:
                 chopper.lineTo(p);
                 break;
             case SkPathVerb::kQuad:
                 chopper.quadTo(p);
                 break;
             case SkPathVerb::kConic:
                 chopper.conicTo(p, *w);
                 break;
             case SkPathVerb::kCubic:
                 chopper.cubicTo(p);
                 break;
             case SkPathVerb::kClose:
                 chopper.close();
                 break;
         }
     }
     // Must preserve the input path's fill type (see crbug.com/1472747)
     SkPath chopped = chopper.path();
     chopped.setFillType(path.getFillType());
     return chopped;
 }

 int FindCubicConvex180Chops(const SkPoint pts[], float T[2], bool* areCusps) {
     SkASSERT(pts);
     SkASSERT(T);
     SkASSERT(areCusps);

     // If a chop falls within a distance of "kEpsilon" from 0 or 1, throw it out. Tangents become
     // unstable when we chop too close to the boundary. This works out because the tessellation
     // shaders don't allow more than 2^10 parametric segments, and they snap the beginning and
     // ending edges at 0 and 1. So if we overstep an inflection or point of 180-degree rotation by a
     // fraction of a tessellation segment, it just gets snapped.
     constexpr static float kEpsilon = 1.f / (1 << 11);
     // Floating-point representation of "1 - 2*kEpsilon".
     constexpr static uint32_t kIEEE_one_minus_2_epsilon = (127 << 23) - 2 * (1 << (24 - 11));
     // Unfortunately we don't have a way to static_assert this, but we can runtime assert that the
     // kIEEE_one_minus_2_epsilon bits are correct.
     SkASSERT(sk_bit_cast<float>(kIEEE_one_minus_2_epsilon) == 1 - 2*kEpsilon);

     float2 p0 = sk_bit_cast<float2>(pts[0]);
     float2 p1 = sk_bit_cast<float2>(pts[1]);
     float2 p2 = sk_bit_cast<float2>(pts[2]);
     float2 p3 = sk_bit_cast<float2>(pts[3]);

     // Find the cubic's power basis coefficients. These define the bezier curve as:
     //
     //                                    |T^3|
     //     Cubic(T) = x,y = |A  3B  3C| * |T^2| + P0
     //                      |.   .   .|   |T  |
     //
     // And the tangent direction (scaled by a uniform 1/3) will be:
     //
     //                                                 |T^2|
     //     Tangent_Direction(T) = dx,dy = |A  2B  C| * |T  |
     //                                    |.   .  .|   |1  |
     //
     float2 C = p1 - p0;
     float2 D = p2 - p1;
     float2 E = p3 - p0;
     float2 B = D - C;
     float2 A = -3*D + E;

     // Now find the cubic's inflection function. There are inflections where F' x F'' == 0.
     // We formulate this as a quadratic equation:  F' x F'' == aT^2 + bT + c == 0.
     // See: https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf
     // NOTE: We only need the roots, so a uniform scale factor does not affect the solution.
     float a = cross(A,B);
     float b = cross(A,C);
     float c = cross(B,C);
     float b_over_minus_2 = -.5f * b;
     float discr_over_4 = b_over_minus_2*b_over_minus_2 - a*c;

     // If -cuspThreshold <= discr_over_4 <= cuspThreshold, it means the two roots are within
     // kEpsilon of one another (in parametric space). This is close enough for our purposes to
     // consider them a single cusp.
     float cuspThreshold = a * (kEpsilon/2);
     cuspThreshold *= cuspThreshold;

     if (discr_over_4 < -cuspThreshold) {
         // The curve does not inflect or cusp. This means it might rotate more than 180 degrees
         // instead. Chop were rotation == 180 deg. (This is the 2nd root where the tangent is
         // parallel to tan0.)
         //
         //      Tangent_Direction(T) x tan0 == 0
         //      (AT^2 x tan0) + (2BT x tan0) + (C x tan0) == 0
         //      (A x C)T^2 + (2B x C)T + (C x C) == 0  [[because tan0 == P1 - P0 == C]]
         //      bT^2 + 2cT + 0 == 0  [[because A x C == b, B x C == c]]
         //      T = [0, -2c/b]
         //
         // NOTE: if C == 0, then C != tan0. But this is fine because the curve is definitely
         // convex-180 if any points are colocated, and T[0] will equal NaN which returns 0 chops.
         *areCusps = false;
         float root = sk_ieee_float_divide(c, b_over_minus_2);
         // Is "root" inside the range [kEpsilon, 1 - kEpsilon)?
         if (sk_bit_cast<uint32_t>(root - kEpsilon) < kIEEE_one_minus_2_epsilon) {
             T[0] = root;
             return 1;
         }
         return 0;
     }

     *areCusps = (discr_over_4 <= cuspThreshold);
     if (*areCusps) {
         // The two roots are close enough that we can consider them a single cusp.
         if (a != 0 || b_over_minus_2 != 0 || c != 0) {
             // Pick the average of both roots.
             float root = sk_ieee_float_divide(b_over_minus_2, a);
             // Is "root" inside the range [kEpsilon, 1 - kEpsilon)?
             if (sk_bit_cast<uint32_t>(root - kEpsilon) < kIEEE_one_minus_2_epsilon) {
                 T[0] = root;
                 return 1;
             }
             return 0;
         }

         // The curve is a flat line. The standard inflection function doesn't detect cusps from flat
         // lines. Find cusps by searching instead for points where the tangent is perpendicular to
         // tan0. This will find any cusp point.
         //
         //     dot(tan0, Tangent_Direction(T)) == 0
         //
         //                         |T^2|
         //     tan0 * |A  2B  C| * |T  | == 0
         //            |.   .  .|   |1  |
         //
         float2 tan0 = skvx::if_then_else(C != 0, C, p2 - p0);
         a = dot(tan0, A);
         b_over_minus_2 = -dot(tan0, B);
         c = dot(tan0, C);
         discr_over_4 = std::max(b_over_minus_2*b_over_minus_2 - a*c, 0.f);
     }

     // Solve our quadratic equation to find where to chop. See the quadratic formula from
     // Numerical Recipes in C.
     float q = sqrtf(discr_over_4);
     q = copysignf(q, b_over_minus_2);
     q = q + b_over_minus_2;
     float2 roots = float2{q,c} / float2{a,q};

     auto inside = (roots > kEpsilon) & (roots < (1 - kEpsilon));
     if (inside[0]) {
         if (inside[1] && roots[0] != roots[1]) {
             if (roots[0] > roots[1]) {
                 roots = skvx::shuffle<1,0>(roots);  // Sort.
             }
             roots.store(T);
             return 2;
         }
         T[0] = roots[0];
         return 1;
     }
     if (inside[1]) {
         T[0] = roots[1];
         return 1;
     }
     return 0;
 }

 }  // namespace skgpu::tess
	/*
	* Copyright 2021 Google LLC.
	*
	* Use of this source code is governed by a BSD-style license that can be
	* found in the LICENSE file.
	*/
	#include "src/gpu/tessellate/Tessellation.h"

	#include "include/core/SkPath.h"
	#include "include/core/SkPathTypes.h"
	#include "include/core/SkRect.h"
	#include "include/private/base/SkFloatingPoint.h"
	#include "include/private/base/SkTArray.h"
	#include "src/base/SkUtils.h"
	#include "src/base/SkVx.h"
	#include "src/core/SkGeometry.h"
	#include "src/core/SkPathPriv.h"
	#include "src/gpu/tessellate/CullTest.h"
	#include "src/gpu/tessellate/WangsFormula.h"

	using namespace skia_private;

	namespace skgpu::tess {

	namespace {

	using float2 = skvx::float2;
	using float4 = skvx::float4;

	// This value only protects us against getting stuck in infinite recursion due to fp32 precision
	// issues. Mathematically, every curve should reduce to manageable visible sections in O(log N)
	// chops, where N is the the magnitude of its control points.
	//
	// But, to define a protective upper bound, a cubic can enter or exit the viewport as many as 6
	// times. So we may need to refine the curve (via binary search chopping at T=.5) up to 6 times.
	//
	// Furthermore, chopping a cubic at T=.5 may only reduce its length by 1/8 (.5^3), so we may require
	// up to 6 chops in order to reduce the length by 1/2.
	constexpr static int kMaxChopsPerCurve = 128/magnitude of +fp32_max - -fp32_max/ *
	6/max number of chops to reduce the length by half/ *
	6/max number of viewport boundary crosses/;

	// Writes a new path, chopping as necessary so no verbs require more segments than
	// kMaxTessellationSegmentsPerCurve. Curves completely outside the viewport are flattened into
	// lines.
	class PathChopper {
	public:
	PathChopper(float tessellationPrecision, const SkMatrix& matrix, const SkRect& viewport)
	: fTessellationPrecision(tessellationPrecision)
	, fCullTest(viewport, matrix)
	, fVectorXform(matrix) {
	fPath.setIsVolatile(true);
	}

	SkPath path() const { return fPath; }

	void moveTo(SkPoint p) { fPath.moveTo(p); }
	void lineTo(const SkPoint p[2]) { fPath.lineTo(p[1]); }
	void close() { fPath.close(); }

	void quadTo(const SkPoint quad[3]) {
	SkASSERT(fPointStack.empty());
	// Use a heap stack to recursively chop the quad into manageable, on-screen segments.
	fPointStack.push_back_n(3, quad);
	int numChops = 0;
	while (!fPointStack.empty()) {
	const SkPoint* p = fPointStack.end() - 3;
	if (!fCullTest.areVisible3(p)) {
	fPath.lineTo(p[2]);
	} else {
	float n4 = wangs_formula::quadratic_p4(fTessellationPrecision, p, fVectorXform);
	if (n4 > kMaxSegmentsPerCurve_p4 && numChops < kMaxChopsPerCurve) {
	SkPoint chops[5];
	SkChopQuadAtHalf(p, chops);
	fPointStack.pop_back_n(3);
	fPointStack.push_back_n(3, chops+2);
	fPointStack.push_back_n(3, chops);
	++numChops;
	continue;
	}
	fPath.quadTo(p[1], p[2]);
	}
	fPointStack.pop_back_n(3);
	}
	}

	void conicTo(const SkPoint conic[3], float weight) {
	SkASSERT(fPointStack.empty());
	SkASSERT(fWeightStack.empty());
	// Use a heap stack to recursively chop the conic into manageable, on-screen segments.
	fPointStack.push_back_n(3, conic);
	fWeightStack.push_back(weight);
	int numChops = 0;
	while (!fPointStack.empty()) {
	const SkPoint* p = fPointStack.end() - 3;
	float w = fWeightStack.back();
	if (!fCullTest.areVisible3(p)) {
	fPath.lineTo(p[2]);
	} else {
	float n2 = wangs_formula::conic_p2(fTessellationPrecision, p, w, fVectorXform);
	if (n2 > kMaxSegmentsPerCurve_p2 && numChops < kMaxChopsPerCurve) {
	SkConic chops[2];
	if (!SkConic(p,w).chopAt(.5, chops)) {
	SkPoint line[2] = {p[0], p[2]};
	this->lineTo(line);
	continue;
	}
	fPointStack.pop_back_n(3);
	fWeightStack.pop_back();
	fPointStack.push_back_n(3, chops[1].fPts);
	fWeightStack.push_back(chops[1].fW);
	fPointStack.push_back_n(3, chops[0].fPts);
	fWeightStack.push_back(chops[0].fW);
	++numChops;
	continue;
	}
	fPath.conicTo(p[1], p[2], w);
	}
	fPointStack.pop_back_n(3);
	fWeightStack.pop_back();
	}
	SkASSERT(fWeightStack.empty());
	}

	void cubicTo(const SkPoint cubic[4]) {
	SkASSERT(fPointStack.empty());
	// Use a heap stack to recursively chop the cubic into manageable, on-screen segments.
	fPointStack.push_back_n(4, cubic);
	int numChops = 0;
	while (!fPointStack.empty()) {
	SkPoint* p = fPointStack.end() - 4;
	if (!fCullTest.areVisible4(p)) {
	fPath.lineTo(p[3]);
	} else {
	float n4 = wangs_formula::cubic_p4(fTessellationPrecision, p, fVectorXform);
	if (n4 > kMaxSegmentsPerCurve_p4 && numChops < kMaxChopsPerCurve) {
	SkPoint chops[7];
	SkChopCubicAtHalf(p, chops);
	fPointStack.pop_back_n(4);
	fPointStack.push_back_n(4, chops+3);
	fPointStack.push_back_n(4, chops);
	++numChops;
	continue;
	}
	fPath.cubicTo(p[1], p[2], p[3]);
	}
	fPointStack.pop_back_n(4);
	}
	}

	private:
	const float fTessellationPrecision;
	const CullTest fCullTest;
	const wangs_formula::VectorXform fVectorXform;
	SkPath fPath;

	// Used for stack-based recursion (instead of using the runtime stack).
	STArray<8, SkPoint> fPointStack;
	STArray<2, float> fWeightStack;
	};

	} // namespace

	SkPath PreChopPathCurves(float tessellationPrecision,
	const SkPath& path,
	const SkMatrix& matrix,
	const SkRect& viewport) {
	// If the viewport is exceptionally large, we could end up blowing out memory with an unbounded
	// number of of chops. Therefore, we require that the viewport is manageable enough that a fully
	// contained curve can be tessellated in kMaxTessellationSegmentsPerCurve or fewer. (Any larger
	// and that amount of pixels wouldn't fit in memory anyway.)
	SkASSERT(wangs_formula::worst_case_cubic(
	tessellationPrecision,
	viewport.width(),
	viewport.height()) <= kMaxSegmentsPerCurve);
	PathChopper chopper(tessellationPrecision, matrix, viewport);
	for (auto [verb, p, w] : SkPathPriv::Iterate(path)) {
	switch (verb) {
	case SkPathVerb::kMove:
	chopper.moveTo(p[0]);
	break;
	case SkPathVerb::kLine:
	chopper.lineTo(p);
	break;
	case SkPathVerb::kQuad:
	chopper.quadTo(p);
	break;
	case SkPathVerb::kConic:
	chopper.conicTo(p, *w);
	break;
	case SkPathVerb::kCubic:
	chopper.cubicTo(p);
	break;
	case SkPathVerb::kClose:
	chopper.close();
	break;
	}
	}
	// Must preserve the input path's fill type (see crbug.com/1472747)
	SkPath chopped = chopper.path();
	chopped.setFillType(path.getFillType());
	return chopped;
	}

	int FindCubicConvex180Chops(const SkPoint pts[], float T[2], bool* areCusps) {
	SkASSERT(pts);
	SkASSERT(T);
	SkASSERT(areCusps);

	// If a chop falls within a distance of "kEpsilon" from 0 or 1, throw it out. Tangents become
	// unstable when we chop too close to the boundary. This works out because the tessellation
	// shaders don't allow more than 2^10 parametric segments, and they snap the beginning and
	// ending edges at 0 and 1. So if we overstep an inflection or point of 180-degree rotation by a
	// fraction of a tessellation segment, it just gets snapped.
	constexpr static float kEpsilon = 1.f / (1 << 11);
	// Floating-point representation of "1 - 2*kEpsilon".
	constexpr static uint32_t kIEEE_one_minus_2_epsilon = (127 << 23) - 2 * (1 << (24 - 11));
	// Unfortunately we don't have a way to static_assert this, but we can runtime assert that the
	// kIEEE_one_minus_2_epsilon bits are correct.
	SkASSERT(sk_bit_cast<float>(kIEEE_one_minus_2_epsilon) == 1 - 2*kEpsilon);

	float2 p0 = sk_bit_cast<float2>(pts[0]);
	float2 p1 = sk_bit_cast<float2>(pts[1]);
	float2 p2 = sk_bit_cast<float2>(pts[2]);
	float2 p3 = sk_bit_cast<float2>(pts[3]);

	// Find the cubic's power basis coefficients. These define the bezier curve as:
	//
	// \|T^3\|
	// Cubic(T) = x,y = \|A 3B 3C\| * \|T^2\| + P0
	// \|. . .\| \|T \|
	//
	// And the tangent direction (scaled by a uniform 1/3) will be:
	//
	// \|T^2\|
	// Tangent_Direction(T) = dx,dy = \|A 2B C\| * \|T \|
	// \|. . .\| \|1 \|
	//
	float2 C = p1 - p0;
	float2 D = p2 - p1;
	float2 E = p3 - p0;
	float2 B = D - C;
	float2 A = -3*D + E;

	// Now find the cubic's inflection function. There are inflections where F' x F'' == 0.
	// We formulate this as a quadratic equation: F' x F'' == aT^2 + bT + c == 0.
	// See: https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf
	// NOTE: We only need the roots, so a uniform scale factor does not affect the solution.
	float a = cross(A,B);
	float b = cross(A,C);
	float c = cross(B,C);
	float b_over_minus_2 = -.5f * b;
	float discr_over_4 = b_over_minus_2b_over_minus_2 - ac;

	// If -cuspThreshold <= discr_over_4 <= cuspThreshold, it means the two roots are within
	// kEpsilon of one another (in parametric space). This is close enough for our purposes to
	// consider them a single cusp.
	float cuspThreshold = a * (kEpsilon/2);
	cuspThreshold *= cuspThreshold;

	if (discr_over_4 < -cuspThreshold) {
	// The curve does not inflect or cusp. This means it might rotate more than 180 degrees
	// instead. Chop were rotation == 180 deg. (This is the 2nd root where the tangent is
	// parallel to tan0.)
	//
	// Tangent_Direction(T) x tan0 == 0
	// (AT^2 x tan0) + (2BT x tan0) + (C x tan0) == 0
	// (A x C)T^2 + (2B x C)T + (C x C) == 0 [[because tan0 == P1 - P0 == C]]
	// bT^2 + 2cT + 0 == 0 [[because A x C == b, B x C == c]]
	// T = [0, -2c/b]
	//
	// NOTE: if C == 0, then C != tan0. But this is fine because the curve is definitely
	// convex-180 if any points are colocated, and T[0] will equal NaN which returns 0 chops.
	*areCusps = false;
	float root = sk_ieee_float_divide(c, b_over_minus_2);
	// Is "root" inside the range [kEpsilon, 1 - kEpsilon)?
	if (sk_bit_cast<uint32_t>(root - kEpsilon) < kIEEE_one_minus_2_epsilon) {
	T[0] = root;
	return 1;
	}
	return 0;
	}

	*areCusps = (discr_over_4 <= cuspThreshold);
	if (*areCusps) {
	// The two roots are close enough that we can consider them a single cusp.
	if (a != 0 \|\| b_over_minus_2 != 0 \|\| c != 0) {
	// Pick the average of both roots.
	float root = sk_ieee_float_divide(b_over_minus_2, a);
	// Is "root" inside the range [kEpsilon, 1 - kEpsilon)?
	if (sk_bit_cast<uint32_t>(root - kEpsilon) < kIEEE_one_minus_2_epsilon) {
	T[0] = root;
	return 1;
	}
	return 0;
	}

	// The curve is a flat line. The standard inflection function doesn't detect cusps from flat
	// lines. Find cusps by searching instead for points where the tangent is perpendicular to
	// tan0. This will find any cusp point.
	//
	// dot(tan0, Tangent_Direction(T)) == 0
	//
	// \|T^2\|
	// tan0 * \|A 2B C\| * \|T \| == 0
	// \|. . .\| \|1 \|
	//
	float2 tan0 = skvx::if_then_else(C != 0, C, p2 - p0);
	a = dot(tan0, A);
	b_over_minus_2 = -dot(tan0, B);
	c = dot(tan0, C);
	discr_over_4 = std::max(b_over_minus_2b_over_minus_2 - ac, 0.f);
	}

	// Solve our quadratic equation to find where to chop. See the quadratic formula from
	// Numerical Recipes in C.
	float q = sqrtf(discr_over_4);
	q = copysignf(q, b_over_minus_2);
	q = q + b_over_minus_2;
	float2 roots = float2{q,c} / float2{a,q};

	auto inside = (roots > kEpsilon) & (roots < (1 - kEpsilon));
	if (inside[0]) {
	if (inside[1] && roots[0] != roots[1]) {
	if (roots[0] > roots[1]) {
	roots = skvx::shuffle<1,0>(roots); // Sort.
	}
	roots.store(T);
	return 2;
	}
	T[0] = roots[0];
	return 1;
	}
	if (inside[1]) {
	T[0] = roots[1];
	return 1;
	}
	return 0;
	}

	} // namespace skgpu::tess