src/gpu/geometry/GrQuadUtils.cpp

/*
 * Copyright 2019 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/geometry/GrQuadUtils.h"

#include "include/core/SkRect.h"
#include "include/private/GrTypesPriv.h"
#include "include/private/SkVx.h"
#include "src/gpu/geometry/GrQuad.h"

using V4f = skvx::Vec<4, float>;
using M4f = skvx::Vec<4, int32_t>;

#define AI SK_ALWAYS_INLINE

static constexpr float kTolerance = 1e-2f;
// True/false bit masks for initializing an M4f
static constexpr int32_t kTrue    = ~0;
static constexpr int32_t kFalse   = 0;

// These rotate the points/edge values either clockwise or counterclockwise assuming tri strip
// order.
static AI V4f next_cw(const V4f& v) {
    return skvx::shuffle<2, 0, 3, 1>(v);
}

static AI V4f next_ccw(const V4f& v) {
    return skvx::shuffle<1, 3, 0, 2>(v);
}

// Replaces zero-length 'bad' edge vectors with the reversed opposite edge vector.
// e3 may be null if only 2D edges need to be corrected for.
static AI void correct_bad_edges(const M4f& bad, V4f* e1, V4f* e2, V4f* e3) {
    if (any(bad)) {
        // Want opposite edges, L B T R -> R T B L but with flipped sign to preserve winding
        *e1 = if_then_else(bad, -skvx::shuffle<3, 2, 1, 0>(*e1), *e1);
        *e2 = if_then_else(bad, -skvx::shuffle<3, 2, 1, 0>(*e2), *e2);
        if (e3) {
            *e3 = if_then_else(bad, -skvx::shuffle<3, 2, 1, 0>(*e3), *e3);
        }
    }
}

// Replace 'bad' coordinates by rotating CCW to get the next point. c3 may be null for 2D points.
static AI void correct_bad_coords(const M4f& bad, V4f* c1, V4f* c2, V4f* c3) {
    if (any(bad)) {
        *c1 = if_then_else(bad, next_ccw(*c1), *c1);
        *c2 = if_then_else(bad, next_ccw(*c2), *c2);
        if (c3) {
            *c3 = if_then_else(bad, next_ccw(*c3), *c3);
        }
    }
}

// Since the local quad may not be type kRect, this uses the opposites for each vertex when
// interpolating, and calculates new ws in addition to new xs, ys.
static void interpolate_local(float alpha, int v0, int v1, int v2, int v3,
                              float lx[4], float ly[4], float lw[4]) {
    SkASSERT(v0 >= 0 && v0 < 4);
    SkASSERT(v1 >= 0 && v1 < 4);
    SkASSERT(v2 >= 0 && v2 < 4);
    SkASSERT(v3 >= 0 && v3 < 4);

    float beta = 1.f - alpha;
    lx[v0] = alpha * lx[v0] + beta * lx[v2];
    ly[v0] = alpha * ly[v0] + beta * ly[v2];
    lw[v0] = alpha * lw[v0] + beta * lw[v2];

    lx[v1] = alpha * lx[v1] + beta * lx[v3];
    ly[v1] = alpha * ly[v1] + beta * ly[v3];
    lw[v1] = alpha * lw[v1] + beta * lw[v3];
}

// Crops v0 to v1 based on the clipDevRect. v2 is opposite of v0, v3 is opposite of v1.
// It is written to not modify coordinates if there's no intersection along the edge.
// Ideally this would have been detected earlier and the entire draw is skipped.
static bool crop_rect_edge(const SkRect& clipDevRect, int v0, int v1, int v2, int v3,
                           float x[4], float y[4], float lx[4], float ly[4], float lw[4]) {
    SkASSERT(v0 >= 0 && v0 < 4);
    SkASSERT(v1 >= 0 && v1 < 4);
    SkASSERT(v2 >= 0 && v2 < 4);
    SkASSERT(v3 >= 0 && v3 < 4);

    if (SkScalarNearlyEqual(x[v0], x[v1])) {
        // A vertical edge
        if (x[v0] < clipDevRect.fLeft && x[v2] >= clipDevRect.fLeft) {
            // Overlapping with left edge of clipDevRect
            if (lx) {
                float alpha = (x[v2] - clipDevRect.fLeft) / (x[v2] - x[v0]);
                interpolate_local(alpha, v0, v1, v2, v3, lx, ly, lw);
            }
            x[v0] = clipDevRect.fLeft;
            x[v1] = clipDevRect.fLeft;
            return true;
        } else if (x[v0] > clipDevRect.fRight && x[v2] <= clipDevRect.fRight) {
            // Overlapping with right edge of clipDevRect
            if (lx) {
                float alpha = (clipDevRect.fRight - x[v2]) / (x[v0] - x[v2]);
                interpolate_local(alpha, v0, v1, v2, v3, lx, ly, lw);
            }
            x[v0] = clipDevRect.fRight;
            x[v1] = clipDevRect.fRight;
            return true;
        }
    } else {
        // A horizontal edge
        SkASSERT(SkScalarNearlyEqual(y[v0], y[v1]));
        if (y[v0] < clipDevRect.fTop && y[v2] >= clipDevRect.fTop) {
            // Overlapping with top edge of clipDevRect
            if (lx) {
                float alpha = (y[v2] - clipDevRect.fTop) / (y[v2] - y[v0]);
                interpolate_local(alpha, v0, v1, v2, v3, lx, ly, lw);
            }
            y[v0] = clipDevRect.fTop;
            y[v1] = clipDevRect.fTop;
            return true;
        } else if (y[v0] > clipDevRect.fBottom && y[v2] <= clipDevRect.fBottom) {
            // Overlapping with bottom edge of clipDevRect
            if (lx) {
                float alpha = (clipDevRect.fBottom - y[v2]) / (y[v0] - y[v2]);
                interpolate_local(alpha, v0, v1, v2, v3, lx, ly, lw);
            }
            y[v0] = clipDevRect.fBottom;
            y[v1] = clipDevRect.fBottom;
            return true;
        }
    }

    // No overlap so don't crop it
    return false;
}

// Updates x and y to intersect with clipDevRect.  lx, ly, and lw are updated appropriately and may
// be null to skip calculations. Returns bit mask of edges that were clipped.
static GrQuadAAFlags crop_rect(const SkRect& clipDevRect, float x[4], float y[4],
                               float lx[4], float ly[4], float lw[4]) {
    GrQuadAAFlags clipEdgeFlags = GrQuadAAFlags::kNone;

    // The quad's left edge may not align with the SkRect notion of left due to 90 degree rotations
    // or mirrors. So, this processes the logical edges of the quad and clamps it to the 4 sides of
    // clipDevRect.

    // Quad's left is v0 to v1 (op. v2 and v3)
    if (crop_rect_edge(clipDevRect, 0, 1, 2, 3, x, y, lx, ly, lw)) {
        clipEdgeFlags |= GrQuadAAFlags::kLeft;
    }
    // Quad's top edge is v0 to v2 (op. v1 and v3)
    if (crop_rect_edge(clipDevRect, 0, 2, 1, 3, x, y, lx, ly, lw)) {
        clipEdgeFlags |= GrQuadAAFlags::kTop;
    }
    // Quad's right edge is v2 to v3 (op. v0 and v1)
    if (crop_rect_edge(clipDevRect, 2, 3, 0, 1, x, y, lx, ly, lw)) {
        clipEdgeFlags |= GrQuadAAFlags::kRight;
    }
    // Quad's bottom edge is v1 to v3 (op. v0 and v2)
    if (crop_rect_edge(clipDevRect, 1, 3, 0, 2, x, y, lx, ly, lw)) {
        clipEdgeFlags |= GrQuadAAFlags::kBottom;
    }

    return clipEdgeFlags;
}

// Similar to crop_rect, but assumes that both the device coordinates and optional local coordinates
// geometrically match the TL, BL, TR, BR vertex ordering, i.e. axis-aligned but not flipped, etc.
static GrQuadAAFlags crop_simple_rect(const SkRect& clipDevRect, float x[4], float y[4],
                                      float lx[4], float ly[4]) {
    GrQuadAAFlags clipEdgeFlags = GrQuadAAFlags::kNone;

    // Update local coordinates proportionately to how much the device rect edge was clipped
    const SkScalar dx = lx ? (lx[2] - lx[0]) / (x[2] - x[0]) : 0.f;
    const SkScalar dy = ly ? (ly[1] - ly[0]) / (y[1] - y[0]) : 0.f;
    if (clipDevRect.fLeft > x[0]) {
        if (lx) {
            lx[0] += (clipDevRect.fLeft - x[0]) * dx;
            lx[1] = lx[0];
        }
        x[0] = clipDevRect.fLeft;
        x[1] = clipDevRect.fLeft;
        clipEdgeFlags |= GrQuadAAFlags::kLeft;
    }
    if (clipDevRect.fTop > y[0]) {
        if (ly) {
            ly[0] += (clipDevRect.fTop - y[0]) * dy;
            ly[2] = ly[0];
        }
        y[0] = clipDevRect.fTop;
        y[2] = clipDevRect.fTop;
        clipEdgeFlags |= GrQuadAAFlags::kTop;
    }
    if (clipDevRect.fRight < x[2]) {
        if (lx) {
            lx[2] -= (x[2] - clipDevRect.fRight) * dx;
            lx[3] = lx[2];
        }
        x[2] = clipDevRect.fRight;
        x[3] = clipDevRect.fRight;
        clipEdgeFlags |= GrQuadAAFlags::kRight;
    }
    if (clipDevRect.fBottom < y[1]) {
        if (ly) {
            ly[1] -= (y[1] - clipDevRect.fBottom) * dy;
            ly[3] = ly[1];
        }
        y[1] = clipDevRect.fBottom;
        y[3] = clipDevRect.fBottom;
        clipEdgeFlags |= GrQuadAAFlags::kBottom;
    }

    return clipEdgeFlags;
}
// Consistent with GrQuad::asRect()'s return value but requires fewer operations since we don't need
// to calculate the bounds of the quad.
static bool is_simple_rect(const GrQuad& quad) {
    if (quad.quadType() != GrQuad::Type::kAxisAligned) {
        return false;
    }
    // v0 at the geometric top-left is unique, so we only need to compare x[0] < x[2] for left
    // and y[0] < y[1] for top, but add a little padding to protect against numerical precision
    // on R90 and R270 transforms tricking this check.
    return ((quad.x(0) + SK_ScalarNearlyZero) < quad.x(2)) &&
           ((quad.y(0) + SK_ScalarNearlyZero) < quad.y(1));
}

// Calculates barycentric coordinates for each point in (testX, testY) in the triangle formed by
// (x0,y0) - (x1,y1) - (x2, y2) and stores them in u, v, w.
static void barycentric_coords(float x0, float y0, float x1, float y1, float x2, float y2,
                               const V4f& testX, const V4f& testY,
                               V4f* u, V4f* v, V4f* w) {
    // Modeled after SkPathOpsQuad::pointInTriangle() but uses float instead of double, is
    // vectorized and outputs normalized barycentric coordinates instead of inside/outside test
    float v0x = x2 - x0;
    float v0y = y2 - y0;
    float v1x = x1 - x0;
    float v1y = y1 - y0;
    V4f v2x = testX - x0;
    V4f v2y = testY - y0;

    float dot00 = v0x * v0x + v0y * v0y;
    float dot01 = v0x * v1x + v0y * v1y;
    V4f   dot02 = v0x * v2x + v0y * v2y;
    float dot11 = v1x * v1x + v1y * v1y;
    V4f   dot12 = v1x * v2x + v1y * v2y;
    float invDenom = sk_ieee_float_divide(1.f, dot00 * dot11 - dot01 * dot01);
    *u = (dot11 * dot02 - dot01 * dot12) * invDenom;
    *v = (dot00 * dot12 - dot01 * dot02) * invDenom;
    *w = 1.f - *u - *v;
}

static M4f inside_triangle(const V4f& u, const V4f& v, const V4f& w) {
    return ((u >= 0.f) & (u <= 1.f)) & ((v >= 0.f) & (v <= 1.f)) & ((w >= 0.f) & (w <= 1.f));
}

namespace GrQuadUtils {

void ResolveAAType(GrAAType requestedAAType, GrQuadAAFlags requestedEdgeFlags, const GrQuad& quad,
                   GrAAType* outAAType, GrQuadAAFlags* outEdgeFlags) {
    // Most cases will keep the requested types unchanged
    *outAAType = requestedAAType;
    *outEdgeFlags = requestedEdgeFlags;

    switch (requestedAAType) {
        // When aa type is coverage, disable AA if the edge configuration doesn't actually need it
        case GrAAType::kCoverage:
            if (requestedEdgeFlags == GrQuadAAFlags::kNone) {
                // Turn off anti-aliasing
                *outAAType = GrAAType::kNone;
            } else {
                // For coverage AA, if the quad is a rect and it lines up with pixel boundaries
                // then overall aa and per-edge aa can be completely disabled
                if (quad.quadType() == GrQuad::Type::kAxisAligned && !quad.aaHasEffectOnRect()) {
                    *outAAType = GrAAType::kNone;
                    *outEdgeFlags = GrQuadAAFlags::kNone;
                }
            }
            break;
        // For no or msaa anti aliasing, override the edge flags since edge flags only make sense
        // when coverage aa is being used.
        case GrAAType::kNone:
            *outEdgeFlags = GrQuadAAFlags::kNone;
            break;
        case GrAAType::kMSAA:
            *outEdgeFlags = GrQuadAAFlags::kAll;
            break;
    }
}

bool CropToRect(const SkRect& cropRect, GrAA cropAA, GrQuadAAFlags* edgeFlags, GrQuad* quad,
                GrQuad* local) {
    SkASSERT(quad->isFinite());

    if (quad->quadType() == GrQuad::Type::kAxisAligned) {
        // crop_rect and crop_rect_simple keep the rectangles as rectangles, so the intersection
        // of the crop and quad can be calculated exactly. Some care must be taken if the quad
        // is axis-aligned but does not satisfy asRect() due to flips, etc.
        GrQuadAAFlags clippedEdges;
        if (local) {
            if (is_simple_rect(*quad) && is_simple_rect(*local)) {
                clippedEdges = crop_simple_rect(cropRect, quad->xs(), quad->ys(),
                                                local->xs(), local->ys());
            } else {
                clippedEdges = crop_rect(cropRect, quad->xs(), quad->ys(),
                                         local->xs(), local->ys(), local->ws());
            }
        } else {
            if (is_simple_rect(*quad)) {
                clippedEdges = crop_simple_rect(cropRect, quad->xs(), quad->ys(), nullptr, nullptr);
            } else {
                clippedEdges = crop_rect(cropRect, quad->xs(), quad->ys(),
                                         nullptr, nullptr, nullptr);
            }
        }

        // Apply the clipped edge updates to the original edge flags
        if (cropAA == GrAA::kYes) {
            // Turn on all edges that were clipped
            *edgeFlags |= clippedEdges;
        } else {
            // Turn off all edges that were clipped
            *edgeFlags &= ~clippedEdges;
        }
        return true;
    }

    if (local) {
        // FIXME (michaelludwig) Calculate cropped local coordinates when not kAxisAligned
        return false;
    }

    V4f devX = quad->x4f();
    V4f devY = quad->y4f();
    V4f devIW = quad->iw4f();
    // Project the 3D coordinates to 2D
    if (quad->quadType() == GrQuad::Type::kPerspective) {
        devX *= devIW;
        devY *= devIW;
    }

    V4f clipX = {cropRect.fLeft, cropRect.fLeft, cropRect.fRight, cropRect.fRight};
    V4f clipY = {cropRect.fTop, cropRect.fBottom, cropRect.fTop, cropRect.fBottom};

    // Calculate barycentric coordinates for the 4 rect corners in the 2 triangles that the quad
    // is tessellated into when drawn.
    V4f u1, v1, w1;
    barycentric_coords(devX[0], devY[0], devX[1], devY[1], devX[2], devY[2], clipX, clipY,
                       &u1, &v1, &w1);
    V4f u2, v2, w2;
    barycentric_coords(devX[1], devY[1], devX[3], devY[3], devX[2], devY[2], clipX, clipY,
                       &u2, &v2, &w2);

    // clipDevRect is completely inside this quad if each corner is in at least one of two triangles
    M4f inTri1 = inside_triangle(u1, v1, w1);
    M4f inTri2 = inside_triangle(u2, v2, w2);
    if (all(inTri1 | inTri2)) {
        // We can crop to exactly the clipDevRect.
        // FIXME (michaelludwig) - there are other ways to have determined quad covering the clip
        // rect, but the barycentric coords will be useful to derive local coordinates in the future

        // Since we are cropped to exactly clipDevRect, we have discarded any perspective and the
        // type becomes kRect. If updated locals were requested, they will incorporate perspective.
        // FIXME (michaelludwig) - once we have local coordinates handled, it may be desirable to
        // keep the draw as perspective so that the hardware does perspective interpolation instead
        // of pushing it into a local coord w and having the shader do an extra divide.
        clipX.store(quad->xs());
        clipY.store(quad->ys());
        quad->ws()[0] = 1.f;
        quad->ws()[1] = 1.f;
        quad->ws()[2] = 1.f;
        quad->ws()[3] = 1.f;
        quad->setQuadType(GrQuad::Type::kAxisAligned);

        // Update the edge flags to match the clip setting since all 4 edges have been clipped
        *edgeFlags = cropAA == GrAA::kYes ? GrQuadAAFlags::kAll : GrQuadAAFlags::kNone;

        return true;
    }

    // FIXME (michaelludwig) - use the GrQuadPerEdgeAA tessellation inset/outset math to move
    // edges to the closest clip corner they are outside of

    return false;
}

///////////////////////////////////////////////////////////////////////////////////////////////////
// TessellationHelper implementation
///////////////////////////////////////////////////////////////////////////////////////////////////

TessellationHelper::QuadMetadata TessellationHelper::getMetadata(const Vertices& vertices,
                                                                 GrQuadAAFlags aaFlags) {
    V4f dx = next_ccw(vertices.fX) - vertices.fX;
    V4f dy = next_ccw(vertices.fY) - vertices.fY;
    V4f invLengths = rsqrt(mad(dx, dx, dy * dy));

    V4f mask = aaFlags == GrQuadAAFlags::kAll ? V4f(1.f) :
            V4f{(GrQuadAAFlags::kLeft & aaFlags) ? 1.f : 0.f,
                 (GrQuadAAFlags::kBottom & aaFlags) ? 1.f : 0.f,
                 (GrQuadAAFlags::kTop & aaFlags) ? 1.f : 0.f,
                 (GrQuadAAFlags::kRight & aaFlags) ? 1.f : 0.f};
    return { dx * invLengths, dy * invLengths, invLengths, mask };
}

TessellationHelper::Edges TessellationHelper::getEdgeEquations(const QuadMetadata& metadata,
                                                               const Vertices& vertices) {
    V4f dx = metadata.fDX;
    V4f dy = metadata.fDY;
    // Correct for bad edges by copying adjacent edge information into the bad component
    correct_bad_edges(metadata.fInvLengths >= 1.f / kTolerance, &dx, &dy, nullptr);

    V4f c = mad(dx, vertices.fY, -dy * vertices.fX);
    // Make sure normals point into the shape
    V4f test = mad(dy, next_cw(vertices.fX), mad(-dx, next_cw(vertices.fY), c));
    if (any(test < -kTolerance)) {
        return {-dy, dx, -c, true};
    } else {
        return {dy, -dx, c, false};
    }
}

bool TessellationHelper::getOptimizedOutset(const QuadMetadata& metadata,
                                            bool rectilinear,
                                            V4f* outset) {
    if (rectilinear) {
        *outset = 0.5f;
        // Stay in the fast path as long as all edges are at least a pixel long (so 1/len <= 1)
        return all(metadata.fInvLengths <= 1.f);
    }

    if (any(metadata.fInvLengths >= 1.f / kTolerance)) {
        // Have an empty edge from a degenerate quad, so there's no hope
        return false;
    }

    // The distance the point needs to move is 1/2sin(theta), where theta is the angle between the
    // two edges at that point. cos(theta) is equal to dot(dxy, next_cw(dxy))
    V4f cosTheta = mad(metadata.fDX, next_cw(metadata.fDX), metadata.fDY * next_cw(metadata.fDY));
    // If the angle is too shallow between edges, go through the degenerate path, otherwise adding
    // and subtracting very large vectors in almost opposite directions leads to float errors
    if (any(abs(cosTheta) >= 0.9f)) {
        return false;
    }
    *outset = 0.5f * rsqrt(1.f - cosTheta * cosTheta); // 1/2sin(theta)

    // When outsetting or insetting, the current edge's AA adds to the length:
    //   cos(pi - theta)/2sin(theta) + cos(pi-ccw(theta))/2sin(ccw(theta))
    // Moving an adjacent edge updates the length by 1/2sin(theta|ccw(theta))
    V4f halfTanTheta = -cosTheta * (*outset); // cos(pi - theta) = -cos(theta)
    V4f edgeAdjust = metadata.fMask * (halfTanTheta + next_ccw(halfTanTheta)) +
                      next_ccw(metadata.fMask) * next_ccw(*outset) +
                      next_cw(metadata.fMask) * (*outset);
    // If either outsetting (plus edgeAdjust) or insetting (minus edgeAdjust) make edgeLen negative
    // then use the slow path
    V4f threshold = 0.1f - (1.f / metadata.fInvLengths);
    return all(edgeAdjust > threshold) && all(edgeAdjust < -threshold);
}

void TessellationHelper::outsetVertices(const V4f& outset,
                                        const QuadMetadata& metadata,
                                        Vertices* quad) {
    // The mask is rotated compared to the outsets and edge vectors, since if the edge is "on"
    // both its points need to be moved along their other edge vectors.
    auto maskedOutset = -outset * next_cw(metadata.fMask);
    auto maskedOutsetCW = outset * metadata.fMask;
    // x = x + outset * mask * next_cw(xdiff) - outset * next_cw(mask) * xdiff
    quad->fX += mad(maskedOutsetCW, next_cw(metadata.fDX), maskedOutset * metadata.fDX);
    quad->fY += mad(maskedOutsetCW, next_cw(metadata.fDY), maskedOutset * metadata.fDY);
    if (quad->fUVRCount > 0) {
        // We want to extend the texture coords by the same proportion as the positions.
        maskedOutset *= metadata.fInvLengths;
        maskedOutsetCW *= next_cw(metadata.fInvLengths);
        V4f du = next_ccw(quad->fU) - quad->fU;
        V4f dv = next_ccw(quad->fV) - quad->fV;
        quad->fU += mad(maskedOutsetCW, next_cw(du), maskedOutset * du);
        quad->fV += mad(maskedOutsetCW, next_cw(dv), maskedOutset * dv);
        if (quad->fUVRCount == 3) {
            V4f dr = next_ccw(quad->fR) - quad->fR;
            quad->fR += mad(maskedOutsetCW, next_cw(dr), maskedOutset * dr);
        }
    }
}

void TessellationHelper::outsetProjectedVertices(const V4f& x2d, const V4f& y2d,
                                                 GrQuadAAFlags aaFlags, Vertices* quad) {
    // Left to right, in device space, for each point
    V4f e1x = skvx::shuffle<2, 3, 2, 3>(quad->fX) - skvx::shuffle<0, 1, 0, 1>(quad->fX);
    V4f e1y = skvx::shuffle<2, 3, 2, 3>(quad->fY) - skvx::shuffle<0, 1, 0, 1>(quad->fY);
    V4f e1w = skvx::shuffle<2, 3, 2, 3>(quad->fW) - skvx::shuffle<0, 1, 0, 1>(quad->fW);
    correct_bad_edges(mad(e1x, e1x, e1y * e1y) < kTolerance * kTolerance, &e1x, &e1y, &e1w);

    // // Top to bottom, in device space, for each point
    V4f e2x = skvx::shuffle<1, 1, 3, 3>(quad->fX) - skvx::shuffle<0, 0, 2, 2>(quad->fX);
    V4f e2y = skvx::shuffle<1, 1, 3, 3>(quad->fY) - skvx::shuffle<0, 0, 2, 2>(quad->fY);
    V4f e2w = skvx::shuffle<1, 1, 3, 3>(quad->fW) - skvx::shuffle<0, 0, 2, 2>(quad->fW);
    correct_bad_edges(mad(e2x, e2x, e2y * e2y) < kTolerance * kTolerance, &e2x, &e2y, &e2w);

    // Can only move along e1 and e2 to reach the new 2D point, so we have
    // x2d = (x + a*e1x + b*e2x) / (w + a*e1w + b*e2w) and
    // y2d = (y + a*e1y + b*e2y) / (w + a*e1w + b*e2w) for some a, b
    // This can be rewritten to a*c1x + b*c2x + c3x = 0; a * c1y + b*c2y + c3y = 0, where
    // the cNx and cNy coefficients are:
    V4f c1x = e1w * x2d - e1x;
    V4f c1y = e1w * y2d - e1y;
    V4f c2x = e2w * x2d - e2x;
    V4f c2y = e2w * y2d - e2y;
    V4f c3x = quad->fW * x2d - quad->fX;
    V4f c3y = quad->fW * y2d - quad->fY;

    // Solve for a and b
    V4f a, b, denom;
    if (aaFlags == GrQuadAAFlags::kAll) {
        // When every edge is outset/inset, each corner can use both edge vectors
        denom = c1x * c2y - c2x * c1y;
        a = (c2x * c3y - c3x * c2y) / denom;
        b = (c3x * c1y - c1x * c3y) / denom;
    } else {
        // Force a or b to be 0 if that edge cannot be used due to non-AA
        M4f aMask = M4f{(aaFlags & GrQuadAAFlags::kLeft)   ? kTrue : kFalse,
                        (aaFlags & GrQuadAAFlags::kLeft)   ? kTrue : kFalse,
                        (aaFlags & GrQuadAAFlags::kRight)  ? kTrue : kFalse,
                        (aaFlags & GrQuadAAFlags::kRight)  ? kTrue : kFalse};
        M4f bMask = M4f{(aaFlags & GrQuadAAFlags::kTop)    ? kTrue : kFalse,
                        (aaFlags & GrQuadAAFlags::kBottom) ? kTrue : kFalse,
                        (aaFlags & GrQuadAAFlags::kTop)    ? kTrue : kFalse,
                        (aaFlags & GrQuadAAFlags::kBottom) ? kTrue : kFalse};

        // When aMask[i]&bMask[i], then a[i], b[i], denom[i] match the kAll case.
        // When aMask[i]&!bMask[i], then b[i] = 0, a[i] = -c3x/c1x or -c3y/c1y, using better denom
        // When !aMask[i]&bMask[i], then a[i] = 0, b[i] = -c3x/c2x or -c3y/c2y, ""
        // When !aMask[i]&!bMask[i], then both a[i] = 0 and b[i] = 0
        M4f useC1x = abs(c1x) > abs(c1y);
        M4f useC2x = abs(c2x) > abs(c2y);

        denom = if_then_else(aMask,
                        if_then_else(bMask,
                                c1x * c2y - c2x * c1y,            /* A & B   */
                                if_then_else(useC1x, c1x, c1y)),  /* A & !B  */
                        if_then_else(bMask,
                                if_then_else(useC2x, c2x, c2y),   /* !A & B  */
                                V4f(1.f)));                       /* !A & !B */

        a = if_then_else(aMask,
                    if_then_else(bMask,
                            c2x * c3y - c3x * c2y,                /* A & B   */
                            if_then_else(useC1x, -c3x, -c3y)),    /* A & !B  */
                    V4f(0.f)) / denom;                            /* !A      */
        b = if_then_else(bMask,
                    if_then_else(aMask,
                            c3x * c1y - c1x * c3y,                /* A & B   */
                            if_then_else(useC2x, -c3x, -c3y)),    /* !A & B  */
                    V4f(0.f)) / denom;                            /* !B      */
    }

    V4f newW = quad->fW + a * e1w + b * e2w;
    // If newW < 0, scale a and b such that the point reaches the infinity plane instead of crossing
    // This breaks orthogonality of inset/outsets, but GPUs don't handle negative Ws well so this
    // is far less visually disturbing (likely not noticeable since it's at extreme perspective).
    // The alternative correction (multiply xyw by -1) has the disadvantage of changing how local
    // coordinates would be interpolated.
    static const float kMinW = 1e-6f;
    if (any(newW < 0.f)) {
        V4f scale = if_then_else(newW < kMinW, (kMinW - quad->fW) / (newW - quad->fW), V4f(1.f));
        a *= scale;
        b *= scale;
    }

    quad->fX += a * e1x + b * e2x;
    quad->fY += a * e1y + b * e2y;
    quad->fW += a * e1w + b * e2w;
    correct_bad_coords(abs(denom) < kTolerance, &quad->fX, &quad->fY, &quad->fW);

    if (quad->fUVRCount > 0) {
        // Calculate R here so it can be corrected with U and V in case it's needed later
        V4f e1u = skvx::shuffle<2, 3, 2, 3>(quad->fU) - skvx::shuffle<0, 1, 0, 1>(quad->fU);
        V4f e1v = skvx::shuffle<2, 3, 2, 3>(quad->fV) - skvx::shuffle<0, 1, 0, 1>(quad->fV);
        V4f e1r = skvx::shuffle<2, 3, 2, 3>(quad->fR) - skvx::shuffle<0, 1, 0, 1>(quad->fR);
        correct_bad_edges(mad(e1u, e1u, e1v * e1v) < kTolerance * kTolerance, &e1u, &e1v, &e1r);

        V4f e2u = skvx::shuffle<1, 1, 3, 3>(quad->fU) - skvx::shuffle<0, 0, 2, 2>(quad->fU);
        V4f e2v = skvx::shuffle<1, 1, 3, 3>(quad->fV) - skvx::shuffle<0, 0, 2, 2>(quad->fV);
        V4f e2r = skvx::shuffle<1, 1, 3, 3>(quad->fR) - skvx::shuffle<0, 0, 2, 2>(quad->fR);
        correct_bad_edges(mad(e2u, e2u, e2v * e2v) < kTolerance * kTolerance, &e2u, &e2v, &e2r);

        quad->fU += a * e1u + b * e2u;
        quad->fV += a * e1v + b * e2v;
        if (quad->fUVRCount == 3) {
            quad->fR += a * e1r + b * e2r;
            correct_bad_coords(abs(denom) < kTolerance, &quad->fU, &quad->fV, &quad->fR);
        } else {
            correct_bad_coords(abs(denom) < kTolerance, &quad->fU, &quad->fV, nullptr);
        }
    }
}

V4f TessellationHelper::getDegenerateCoverage(const V4f& px, const V4f& py, const Edges& edges) {
    // Calculate distance of the 4 inset points (px, py) to the 4 edges
    V4f d0 = mad(edges.fA[0], px, mad(edges.fB[0], py, edges.fC[0]));
    V4f d1 = mad(edges.fA[1], px, mad(edges.fB[1], py, edges.fC[1]));
    V4f d2 = mad(edges.fA[2], px, mad(edges.fB[2], py, edges.fC[2]));
    V4f d3 = mad(edges.fA[3], px, mad(edges.fB[3], py, edges.fC[3]));

    // For each point, pretend that there's a rectangle that touches e0 and e3 on the horizontal
    // axis, so its width is "approximately" d0 + d3, and it touches e1 and e2 on the vertical axis
    // so its height is d1 + d2. Pin each of these dimensions to [0, 1] and approximate the coverage
    // at each point as clamp(d0+d3, 0, 1) x clamp(d1+d2, 0, 1). For rectilinear quads this is an
    // accurate calculation of its area clipped to an aligned pixel. For arbitrary quads it is not
    // mathematically accurate but qualitatively provides a stable value proportional to the size of
    // the shape.
    V4f w = max(0.f, min(1.f, d0 + d3));
    V4f h = max(0.f, min(1.f, d1 + d2));
    return w * h;
}

V4f TessellationHelper::computeDegenerateQuad(GrQuadAAFlags aaFlags, const V4f& mask,
                                              const Edges& edges, bool outset, Vertices* quad) {
    // Move the edge 1/2 pixel in or out depending on 'outset'.
    V4f oc = edges.fC + mask * (outset ? 0.5f : -0.5f);

    // There are 6 points that we care about to determine the final shape of the polygon, which
    // are the intersections between (e0,e2), (e1,e0), (e2,e3), (e3,e1) (corresponding to the
    // 4 corners), and (e1, e2), (e0, e3) (representing the intersections of opposite edges).
    V4f denom = edges.fA * next_cw(edges.fB) - edges.fB * next_cw(edges.fA);
    V4f px = (edges.fB * next_cw(oc) - oc * next_cw(edges.fB)) / denom;
    V4f py = (oc * next_cw(edges.fA) - edges.fA * next_cw(oc)) / denom;
    correct_bad_coords(abs(denom) < kTolerance, &px, &py, nullptr);

    // Calculate the signed distances from these 4 corners to the other two edges that did not
    // define the intersection. So p(0) is compared to e3,e1, p(1) to e3,e2 , p(2) to e0,e1, and
    // p(3) to e0,e2
    V4f dists1 = px * skvx::shuffle<3, 3, 0, 0>(edges.fA) +
                 py * skvx::shuffle<3, 3, 0, 0>(edges.fB) +
                 skvx::shuffle<3, 3, 0, 0>(oc);
    V4f dists2 = px * skvx::shuffle<1, 2, 1, 2>(edges.fA) +
                 py * skvx::shuffle<1, 2, 1, 2>(edges.fB) +
                 skvx::shuffle<1, 2, 1, 2>(oc);

    // If all the distances are >= 0, the 4 corners form a valid quadrilateral, so use them as
    // the 4 points. If any point is on the wrong side of both edges, the interior has collapsed
    // and we need to use a central point to represent it. If all four points are only on the
    // wrong side of 1 edge, one edge has crossed over another and we use a line to represent it.
    // Otherwise, use a triangle that replaces the bad points with the intersections of
    // (e1, e2) or (e0, e3) as needed.
    M4f d1v0 = dists1 < kTolerance;
    M4f d2v0 = dists2 < kTolerance;
    M4f d1And2 = d1v0 & d2v0;
    M4f d1Or2 = d1v0 | d2v0;

    V4f coverage;
    if (!any(d1Or2)) {
        // Every dists1 and dists2 >= kTolerance so it's not degenerate, use all 4 corners as-is
        // and use full coverage
        coverage = 1.f;
    } else if (any(d1And2)) {
        // A point failed against two edges, so reduce the shape to a single point, which we take as
        // the center of the original quad to ensure it is contained in the intended geometry. Since
        // it has collapsed, we know the shape cannot cover a pixel so update the coverage.
        SkPoint center = {0.25f * (quad->fX[0] + quad->fX[1] + quad->fX[2] + quad->fX[3]),
                          0.25f * (quad->fY[0] + quad->fY[1] + quad->fY[2] + quad->fY[3])};
        px = center.fX;
        py = center.fY;
        coverage = getDegenerateCoverage(px, py, edges);
    } else if (all(d1Or2)) {
        // Degenerates to a line. Compare p[2] and p[3] to edge 0. If they are on the wrong side,
        // that means edge 0 and 3 crossed, and otherwise edge 1 and 2 crossed.
        if (dists1[2] < kTolerance && dists1[3] < kTolerance) {
            // Edges 0 and 3 have crossed over, so make the line from average of (p0,p2) and (p1,p3)
            px = 0.5f * (skvx::shuffle<0, 1, 0, 1>(px) + skvx::shuffle<2, 3, 2, 3>(px));
            py = 0.5f * (skvx::shuffle<0, 1, 0, 1>(py) + skvx::shuffle<2, 3, 2, 3>(py));
        } else {
            // Edges 1 and 2 have crossed over, so make the line from average of (p0,p1) and (p2,p3)
            px = 0.5f * (skvx::shuffle<0, 0, 2, 2>(px) + skvx::shuffle<1, 1, 3, 3>(px));
            py = 0.5f * (skvx::shuffle<0, 0, 2, 2>(py) + skvx::shuffle<1, 1, 3, 3>(py));
        }
        coverage = getDegenerateCoverage(px, py, edges);
    } else {
        // This turns into a triangle. Replace corners as needed with the intersections between
        // (e0,e3) and (e1,e2), which must now be calculated
        using V2f = skvx::Vec<2, float>;
        V2f eDenom = skvx::shuffle<0, 1>(edges.fA) * skvx::shuffle<3, 2>(edges.fB) -
                      skvx::shuffle<0, 1>(edges.fB) * skvx::shuffle<3, 2>(edges.fA);
        V2f ex = (skvx::shuffle<0, 1>(edges.fB) * skvx::shuffle<3, 2>(oc) -
                   skvx::shuffle<0, 1>(oc) * skvx::shuffle<3, 2>(edges.fB)) / eDenom;
        V2f ey = (skvx::shuffle<0, 1>(oc) * skvx::shuffle<3, 2>(edges.fA) -
                   skvx::shuffle<0, 1>(edges.fA) * skvx::shuffle<3, 2>(oc)) / eDenom;

        if (SkScalarAbs(eDenom[0]) > kTolerance) {
            px = if_then_else(d1v0, V4f(ex[0]), px);
            py = if_then_else(d1v0, V4f(ey[0]), py);
        }
        if (SkScalarAbs(eDenom[1]) > kTolerance) {
            px = if_then_else(d2v0, V4f(ex[1]), px);
            py = if_then_else(d2v0, V4f(ey[1]), py);
        }

        coverage = 1.f;
    }

    outsetProjectedVertices(px, py, aaFlags, quad);
    return coverage;
}

V4f TessellationHelper::computeNestedQuadVertices(GrQuadAAFlags aaFlags, bool rectilinear,
                                                  Vertices* inner, Vertices* outer) {
    SkASSERT(inner->fUVRCount == 0 || inner->fUVRCount == 2 || inner->fUVRCount == 3);
    SkASSERT(outer->fUVRCount == inner->fUVRCount);

    QuadMetadata metadata = getMetadata(*inner, aaFlags);

    // When outsetting, we want the new edge to be .5px away from the old line, which means the
    // corners may need to be adjusted by more than .5px if the matrix had sheer. This adjustment
    // is only computed if there are no empty edges, and it may signal going through the slow path.
    V4f outset = 0.5f;
    if (getOptimizedOutset(metadata, rectilinear, &outset)) {
       // Since it's not subpixel, outsetting and insetting are trivial vector additions.
        outsetVertices(outset, metadata, outer);
        outsetVertices(-outset, metadata, inner);
        return 1.f;
    }

    // Only compute edge equations once since they are the same for inner and outer quads
    Edges edges = getEdgeEquations(metadata, *inner);

    // Calculate both outset and inset, returning the coverage reported for the inset, since the
    // outset will always have 0.0f.
    computeDegenerateQuad(aaFlags, metadata.fMask, edges, true, outer);
    return computeDegenerateQuad(aaFlags, metadata.fMask, edges, false, inner);
}

V4f TessellationHelper::computeNestedPerspQuadVertices(const GrQuadAAFlags aaFlags,
                                                       Vertices* inner,
                                                       Vertices* outer) {
    SkASSERT(inner->fUVRCount == 0 || inner->fUVRCount == 2 || inner->fUVRCount == 3);
    SkASSERT(outer->fUVRCount == inner->fUVRCount);

    // Calculate the projected 2D quad and use it to form projeccted inner/outer quads
    V4f iw = 1.0f / inner->fW;
    V4f x2d = inner->fX * iw;
    V4f y2d = inner->fY * iw;

    Vertices inner2D = { x2d, y2d, /*w*/ 1.f, 0.f, 0.f, 0.f, 0 }; // No uvr outsetting in 2D
    Vertices outer2D = inner2D;

    V4f coverage = computeNestedQuadVertices(aaFlags, /* rect */ false, &inner2D, &outer2D);

    // Now map from the 2D inset/outset back to 3D and update the local coordinates as well
    outsetProjectedVertices(inner2D.fX, inner2D.fY, aaFlags, inner);
    outsetProjectedVertices(outer2D.fX, outer2D.fY, aaFlags, outer);

    return coverage;
}

TessellationHelper::TessellationHelper(const GrQuad& deviceQuad, const GrQuad* localQuad)
        : fAAFlags(GrQuadAAFlags::kNone)
        , fCoverage(1.f)
        , fDeviceType(deviceQuad.quadType())
        , fLocalType(localQuad ? localQuad->quadType() : GrQuad::Type::kAxisAligned) {
    fOriginal.fX = deviceQuad.x4f();
    fOriginal.fY = deviceQuad.y4f();
    fOriginal.fW = deviceQuad.w4f();

    if (localQuad) {
        fOriginal.fU = localQuad->x4f();
        fOriginal.fV = localQuad->y4f();
        fOriginal.fR = localQuad->w4f();
        fOriginal.fUVRCount = fLocalType == GrQuad::Type::kPerspective ? 3 : 2;
    } else {
        fOriginal.fUVRCount = 0;
    }
}

V4f TessellationHelper::pixelCoverage() {
    // When there are no AA edges, insetting and outsetting is skipped since the original geometry
    // can just be reported directly (in which case fCoverage may be stale).
    return fAAFlags == GrQuadAAFlags::kNone ? 1.f : fCoverage;
}

void TessellationHelper::inset(GrQuadAAFlags aaFlags, GrQuad* deviceInset, GrQuad* localInset) {
    if (aaFlags != fAAFlags) {
        fAAFlags = aaFlags;
        if (aaFlags != GrQuadAAFlags::kNone) {
            this->recomputeInsetAndOutset();
        }
    }
    if (fAAFlags == GrQuadAAFlags::kNone) {
        this->setQuads(fOriginal, deviceInset, localInset);
    } else {
        this->setQuads(fInset, deviceInset, localInset);
    }
}

void TessellationHelper::outset(GrQuadAAFlags aaFlags, GrQuad* deviceOutset, GrQuad* localOutset) {
    if (aaFlags != fAAFlags) {
        fAAFlags = aaFlags;
        if (aaFlags != GrQuadAAFlags::kNone) {
            this->recomputeInsetAndOutset();
        }
    }
    if (fAAFlags == GrQuadAAFlags::kNone) {
        this->setQuads(fOriginal, deviceOutset, localOutset);
    } else {
        this->setQuads(fOutset, deviceOutset, localOutset);
    }
}

void TessellationHelper::recomputeInsetAndOutset() {
    // Start from the original geometry
    fInset = fOriginal;
    fOutset = fOriginal;

    if (fDeviceType == GrQuad::Type::kPerspective) {
        fCoverage = computeNestedPerspQuadVertices(fAAFlags, &fInset, &fOutset);
    } else {
        fCoverage = computeNestedQuadVertices(fAAFlags, fDeviceType <= GrQuad::Type::kRectilinear,
                                              &fInset, &fOutset);
    }
}

void TessellationHelper::setQuads(const Vertices& vertices,
                                  GrQuad* deviceOut, GrQuad* localOut) const {
    SkASSERT(deviceOut);
    SkASSERT(vertices.fUVRCount == 0 || localOut);

    vertices.fX.store(deviceOut->xs());
    vertices.fY.store(deviceOut->ys());
    if (fDeviceType == GrQuad::Type::kPerspective) {
        vertices.fW.store(deviceOut->ws());
    }
    deviceOut->setQuadType(fDeviceType); // This sets ws == 1 when device type != perspective

    if (vertices.fUVRCount > 0) {
        vertices.fU.store(localOut->xs());
        vertices.fV.store(localOut->ys());
        if (vertices.fUVRCount == 3) {
            vertices.fR.store(localOut->ws());
        }
        localOut->setQuadType(fLocalType);
    }
}

}; // namespace GrQuadUtils