path-kit 0.4.0

/*
 * Copyright 2006 The Android Open Source Project
 *
 * Use of this source code is governed by a BSD-style license that can be
 * found in the LICENSE file.
 */

#include "include/core/SkMatrix.h"

#include "include/core/SkPaint.h"
#include "include/core/SkPoint3.h"
#include "include/private/SkFloatBits.h"
#include "include/private/SkNx.h"
#include "include/private/SkTo.h"
#include "src/core/SkMathPriv.h"
#include "src/core/SkMatrixPriv.h"
#include "src/core/SkPathPriv.h"

#include <cstddef>
#include <utility>

namespace pk {
// In a few places, we performed the following
//      a * b + c * d + e
// as
//      a * b + (c * d + e)
//
// sdot and scross are indended to capture these compound operations into a
// function, with an eye toward considering upscaling the intermediates to
// doubles for more precision (as we do in concat and invert).
//
// However, these few lines that performed the last add before the "dot", cause
// tiny image differences, so we guard that change until we see the impact on
// chrome's layouttests.
//
#define PK_LEGACY_MATRIX_MATH_ORDER

/*      [scale-x    skew-x      trans-x]   [X]   [X']
        [skew-y     scale-y     trans-y] * [Y] = [Y']
        [persp-0    persp-1     persp-2]   [1]   [1 ]
*/

SkMatrix& SkMatrix::reset() { *this = SkMatrix(); return *this; }

SkMatrix& SkMatrix::set9(const SkScalar buffer[]) {
    memcpy(fMat, buffer, 9 * sizeof(SkScalar));
    this->setTypeMask(kUnknown_Mask);
    return *this;
}

// this aligns with the masks, so we can compute a mask from a variable 0/1
enum {
    kTranslate_Shift,
    kScale_Shift,
    kAffine_Shift,
    kPerspective_Shift,
    kRectStaysRect_Shift
};

static const int32_t kScalar1Int = 0x3f800000;

uint8_t SkMatrix::computePerspectiveTypeMask() const {
    // Benchmarking suggests that replacing this set of PkScalarAs2sCompliment
    // is a win, but replacing those below is not. We don't yet understand
    // that result.
    if (fMat[kMPersp0] != 0 || fMat[kMPersp1] != 0 || fMat[kMPersp2] != 1) {
        // If this is a perspective transform, we return true for all other
        // transform flags - this does not disable any optimizations, respects
        // the rule that the type mask must be conservative, and speeds up
        // type mask computation.
        return SkToU8(kORableMasks);
    }

    return SkToU8(kOnlyPerspectiveValid_Mask | kUnknown_Mask);
}

uint8_t SkMatrix::computeTypeMask() const {
    unsigned mask = 0;

    if (fMat[kMPersp0] != 0 || fMat[kMPersp1] != 0 || fMat[kMPersp2] != 1) {
        // Once it is determined that that this is a perspective transform,
        // all other flags are moot as far as optimizations are concerned.
        return SkToU8(kORableMasks);
    }

    if (fMat[kMTransX] != 0 || fMat[kMTransY] != 0) {
        mask |= kTranslate_Mask;
    }

    int m00 = PkScalarAs2sCompliment(fMat[SkMatrix::kMScaleX]);
    int m01 = PkScalarAs2sCompliment(fMat[SkMatrix::kMSkewX]);
    int m10 = PkScalarAs2sCompliment(fMat[SkMatrix::kMSkewY]);
    int m11 = PkScalarAs2sCompliment(fMat[SkMatrix::kMScaleY]);

    if (m01 | m10) {
        // The skew components may be scale-inducing, unless we are dealing
        // with a pure rotation.  Testing for a pure rotation is expensive,
        // so we opt for being conservative by always setting the scale bit.
        // along with affine.
        // By doing this, we are also ensuring that matrices have the same
        // type masks as their inverses.
        mask |= kAffine_Mask | kScale_Mask;

        // For rectStaysRect, in the affine case, we only need check that
        // the primary diagonal is all zeros and that the secondary diagonal
        // is all non-zero.

        // map non-zero to 1
        m01 = m01 != 0;
        m10 = m10 != 0;

        int dp0 = 0 == (m00 | m11) ;  // true if both are 0
        int ds1 = m01 & m10;        // true if both are 1

        mask |= (dp0 & ds1) << kRectStaysRect_Shift;
    } else {
        // Only test for scale explicitly if not affine, since affine sets the
        // scale bit.
        if ((m00 ^ kScalar1Int) | (m11 ^ kScalar1Int)) {
            mask |= kScale_Mask;
        }

        // Not affine, therefore we already know secondary diagonal is
        // all zeros, so we just need to check that primary diagonal is
        // all non-zero.

        // map non-zero to 1
        m00 = m00 != 0;
        m11 = m11 != 0;

        // record if the (p)rimary diagonal is all non-zero
        mask |= (m00 & m11) << kRectStaysRect_Shift;
    }

    return SkToU8(mask);
}

///////////////////////////////////////////////////////////////////////////////

bool operator==(const SkMatrix& a, const SkMatrix& b) {
    const SkScalar* PK_RESTRICT ma = a.fMat;
    const SkScalar* PK_RESTRICT mb = b.fMat;

    return  ma[0] == mb[0] && ma[1] == mb[1] && ma[2] == mb[2] &&
            ma[3] == mb[3] && ma[4] == mb[4] && ma[5] == mb[5] &&
            ma[6] == mb[6] && ma[7] == mb[7] && ma[8] == mb[8];
}

///////////////////////////////////////////////////////////////////////////////

// helper function to determine if upper-left 2x2 of matrix is degenerate
static inline bool is_degenerate_2x2(SkScalar scaleX, SkScalar skewX,
                                     SkScalar skewY,  SkScalar scaleY) {
    SkScalar perp_dot = scaleX*scaleY - skewX*skewY;
    return SkScalarNearlyZero(perp_dot, PK_ScalarNearlyZero*PK_ScalarNearlyZero);
}

static inline SkScalar sdot(SkScalar a, SkScalar b, SkScalar c, SkScalar d) {
    return a * b + c * d;
}

static inline SkScalar sdot(SkScalar a, SkScalar b, SkScalar c, SkScalar d,
                             SkScalar e, SkScalar f) {
    return a * b + c * d + e * f;
}

static inline SkScalar scross(SkScalar a, SkScalar b, SkScalar c, SkScalar d) {
    return a * b - c * d;
}

SkMatrix& SkMatrix::setTranslate(SkScalar dx, SkScalar dy) {
    *this = SkMatrix(1, 0, dx,
                     0, 1, dy,
                     0, 0, 1,
                     (dx != 0 || dy != 0) ? kTranslate_Mask | kRectStaysRect_Mask
                                          : kIdentity_Mask  | kRectStaysRect_Mask);
    return *this;
}

SkMatrix& SkMatrix::preTranslate(SkScalar dx, SkScalar dy) {
    const unsigned mask = this->getType();

    if (mask <= kTranslate_Mask) {
        fMat[kMTransX] += dx;
        fMat[kMTransY] += dy;
    } else if (mask & kPerspective_Mask) {
        SkMatrix    m;
        m.setTranslate(dx, dy);
        return this->preConcat(m);
    } else {
        fMat[kMTransX] += sdot(fMat[kMScaleX], dx, fMat[kMSkewX], dy);
        fMat[kMTransY] += sdot(fMat[kMSkewY], dx, fMat[kMScaleY], dy);
    }
    this->updateTranslateMask();
    return *this;
}

SkMatrix& SkMatrix::postTranslate(SkScalar dx, SkScalar dy) {
    if (this->hasPerspective()) {
        SkMatrix    m;
        m.setTranslate(dx, dy);
        this->postConcat(m);
    } else {
        fMat[kMTransX] += dx;
        fMat[kMTransY] += dy;
        this->updateTranslateMask();
    }
    return *this;
}

///////////////////////////////////////////////////////////////////////////////

SkMatrix& SkMatrix::setScale(SkScalar sx, SkScalar sy, SkScalar px, SkScalar py) {
    if (1 == sx && 1 == sy) {
        this->reset();
    } else {
        this->setScaleTranslate(sx, sy, px - sx * px, py - sy * py);
    }
    return *this;
}

SkMatrix& SkMatrix::setScale(SkScalar sx, SkScalar sy) {
    *this = SkMatrix(sx, 0,  0,
                     0,  sy, 0,
                     0,  0,  1,
                     (sx == 1 && sy == 1) ? kIdentity_Mask | kRectStaysRect_Mask
                                          : kScale_Mask    | kRectStaysRect_Mask);
    return *this;
}

SkMatrix& SkMatrix::preScale(SkScalar sx, SkScalar sy, SkScalar px, SkScalar py) {
    if (1 == sx && 1 == sy) {
        return *this;
    }

    SkMatrix    m;
    m.setScale(sx, sy, px, py);
    return this->preConcat(m);
}

SkMatrix& SkMatrix::preScale(SkScalar sx, SkScalar sy) {
    if (1 == sx && 1 == sy) {
        return *this;
    }

    // the assumption is that these multiplies are very cheap, and that
    // a full concat and/or just computing the matrix type is more expensive.
    // Also, the fixed-point case checks for overflow, but the float doesn't,
    // so we can get away with these blind multiplies.

    fMat[kMScaleX] *= sx;
    fMat[kMSkewY]  *= sx;
    fMat[kMPersp0] *= sx;

    fMat[kMSkewX]  *= sy;
    fMat[kMScaleY] *= sy;
    fMat[kMPersp1] *= sy;

    // Attempt to simplify our type when applying an inverse scale.
    // TODO: The persp/affine preconditions are in place to keep the mask consistent with
    //       what computeTypeMask() would produce (persp/skew always implies kScale).
    //       We should investigate whether these flag dependencies are truly needed.
    if (fMat[kMScaleX] == 1 && fMat[kMScaleY] == 1
        && !(fTypeMask & (kPerspective_Mask | kAffine_Mask))) {
        this->clearTypeMask(kScale_Mask);
    } else {
        this->orTypeMask(kScale_Mask);
    }
    return *this;
}

SkMatrix& SkMatrix::postScale(SkScalar sx, SkScalar sy, SkScalar px, SkScalar py) {
    if (1 == sx && 1 == sy) {
        return *this;
    }
    SkMatrix    m;
    m.setScale(sx, sy, px, py);
    return this->postConcat(m);
}

SkMatrix& SkMatrix::postScale(SkScalar sx, SkScalar sy) {
    if (1 == sx && 1 == sy) {
        return *this;
    }
    SkMatrix    m;
    m.setScale(sx, sy);
    return this->postConcat(m);
}

// this perhaps can go away, if we have a fract/high-precision way to
// scale matrices
bool SkMatrix::postIDiv(int divx, int divy) {
    if (divx == 0 || divy == 0) {
        return false;
    }

    const float invX = 1.f / divx;
    const float invY = 1.f / divy;

    fMat[kMScaleX] *= invX;
    fMat[kMSkewX]  *= invX;
    fMat[kMTransX] *= invX;

    fMat[kMScaleY] *= invY;
    fMat[kMSkewY]  *= invY;
    fMat[kMTransY] *= invY;

    this->setTypeMask(kUnknown_Mask);
    return true;
}

////////////////////////////////////////////////////////////////////////////////////

SkMatrix& SkMatrix::setSinCos(SkScalar sinV, SkScalar cosV, SkScalar px, SkScalar py) {
    const SkScalar oneMinusCosV = 1 - cosV;

    fMat[kMScaleX]  = cosV;
    fMat[kMSkewX]   = -sinV;
    fMat[kMTransX]  = sdot(sinV, py, oneMinusCosV, px);

    fMat[kMSkewY]   = sinV;
    fMat[kMScaleY]  = cosV;
    fMat[kMTransY]  = sdot(-sinV, px, oneMinusCosV, py);

    fMat[kMPersp0] = fMat[kMPersp1] = 0;
    fMat[kMPersp2] = 1;

    this->setTypeMask(kUnknown_Mask | kOnlyPerspectiveValid_Mask);
    return *this;
}

SkMatrix& SkMatrix::setSinCos(SkScalar sinV, SkScalar cosV) {
    fMat[kMScaleX]  = cosV;
    fMat[kMSkewX]   = -sinV;
    fMat[kMTransX]  = 0;

    fMat[kMSkewY]   = sinV;
    fMat[kMScaleY]  = cosV;
    fMat[kMTransY]  = 0;

    fMat[kMPersp0] = fMat[kMPersp1] = 0;
    fMat[kMPersp2] = 1;

    this->setTypeMask(kUnknown_Mask | kOnlyPerspectiveValid_Mask);
    return *this;
}

SkMatrix& SkMatrix::setRotate(SkScalar degrees, SkScalar px, SkScalar py) {
    SkScalar rad = PkDegreesToRadians(degrees);
    return this->setSinCos(SkScalarSinSnapToZero(rad), SkScalarCosSnapToZero(rad), px, py);
}

SkMatrix& SkMatrix::setRotate(SkScalar degrees) {
    SkScalar rad = PkDegreesToRadians(degrees);
    return this->setSinCos(SkScalarSinSnapToZero(rad), SkScalarCosSnapToZero(rad));
}

SkMatrix& SkMatrix::preRotate(SkScalar degrees, SkScalar px, SkScalar py) {
    SkMatrix    m;
    m.setRotate(degrees, px, py);
    return this->preConcat(m);
}

SkMatrix& SkMatrix::preRotate(SkScalar degrees) {
    SkMatrix    m;
    m.setRotate(degrees);
    return this->preConcat(m);
}

SkMatrix& SkMatrix::postRotate(SkScalar degrees, SkScalar px, SkScalar py) {
    SkMatrix    m;
    m.setRotate(degrees, px, py);
    return this->postConcat(m);
}

SkMatrix& SkMatrix::postRotate(SkScalar degrees) {
    SkMatrix    m;
    m.setRotate(degrees);
    return this->postConcat(m);
}

////////////////////////////////////////////////////////////////////////////////////

SkMatrix& SkMatrix::setSkew(SkScalar sx, SkScalar sy, SkScalar px, SkScalar py) {
    *this = SkMatrix(1,  sx, -sx * py,
                     sy, 1,  -sy * px,
                     0,  0,  1,
                     kUnknown_Mask | kOnlyPerspectiveValid_Mask);
    return *this;
}

SkMatrix& SkMatrix::setSkew(SkScalar sx, SkScalar sy) {
    fMat[kMScaleX]  = 1;
    fMat[kMSkewX]   = sx;
    fMat[kMTransX]  = 0;

    fMat[kMSkewY]   = sy;
    fMat[kMScaleY]  = 1;
    fMat[kMTransY]  = 0;

    fMat[kMPersp0] = fMat[kMPersp1] = 0;
    fMat[kMPersp2] = 1;

    this->setTypeMask(kUnknown_Mask | kOnlyPerspectiveValid_Mask);
    return *this;
}

SkMatrix& SkMatrix::preSkew(SkScalar sx, SkScalar sy, SkScalar px, SkScalar py) {
    SkMatrix    m;
    m.setSkew(sx, sy, px, py);
    return this->preConcat(m);
}

SkMatrix& SkMatrix::preSkew(SkScalar sx, SkScalar sy) {
    SkMatrix    m;
    m.setSkew(sx, sy);
    return this->preConcat(m);
}

SkMatrix& SkMatrix::postSkew(SkScalar sx, SkScalar sy, SkScalar px, SkScalar py) {
    SkMatrix    m;
    m.setSkew(sx, sy, px, py);
    return this->postConcat(m);
}

SkMatrix& SkMatrix::postSkew(SkScalar sx, SkScalar sy) {
    SkMatrix    m;
    m.setSkew(sx, sy);
    return this->postConcat(m);
}

///////////////////////////////////////////////////////////////////////////////

bool SkMatrix::setRectToRect(const SkRect& src, const SkRect& dst, ScaleToFit align) {
    if (src.isEmpty()) {
        this->reset();
        return false;
    }

    if (dst.isEmpty()) {
        sk_bzero(fMat, 8 * sizeof(SkScalar));
        fMat[kMPersp2] = 1;
        this->setTypeMask(kScale_Mask | kRectStaysRect_Mask);
    } else {
        SkScalar    tx, sx = dst.width() / src.width();
        SkScalar    ty, sy = dst.height() / src.height();
        bool        xLarger = false;

        if (align != kFill_ScaleToFit) {
            if (sx > sy) {
                xLarger = true;
                sx = sy;
            } else {
                sy = sx;
            }
        }

        tx = dst.fLeft - src.fLeft * sx;
        ty = dst.fTop - src.fTop * sy;
        if (align == kCenter_ScaleToFit || align == kEnd_ScaleToFit) {
            SkScalar diff;

            if (xLarger) {
                diff = dst.width() - src.width() * sy;
            } else {
                diff = dst.height() - src.height() * sy;
            }

            if (align == kCenter_ScaleToFit) {
                diff = PkScalarHalf(diff);
            }

            if (xLarger) {
                tx += diff;
            } else {
                ty += diff;
            }
        }

        this->setScaleTranslate(sx, sy, tx, ty);
    }
    return true;
}

///////////////////////////////////////////////////////////////////////////////

static inline float muladdmul(float a, float b, float c, float d) {
    return pk_double_to_float((double)a * b + (double)c * d);
}

static inline float rowcol3(const float row[], const float col[]) {
    return row[0] * col[0] + row[1] * col[3] + row[2] * col[6];
}

static bool only_scale_and_translate(unsigned mask) {
    return 0 == (mask & (SkMatrix::kAffine_Mask | SkMatrix::kPerspective_Mask));
}

SkMatrix& SkMatrix::setConcat(const SkMatrix& a, const SkMatrix& b) {
    TypeMask aType = a.getType();
    TypeMask bType = b.getType();

    if (a.isTriviallyIdentity()) {
        *this = b;
    } else if (b.isTriviallyIdentity()) {
        *this = a;
    } else if (only_scale_and_translate(aType | bType)) {
        this->setScaleTranslate(a.fMat[kMScaleX] * b.fMat[kMScaleX],
                                a.fMat[kMScaleY] * b.fMat[kMScaleY],
                                a.fMat[kMScaleX] * b.fMat[kMTransX] + a.fMat[kMTransX],
                                a.fMat[kMScaleY] * b.fMat[kMTransY] + a.fMat[kMTransY]);
    } else {
        SkMatrix tmp;

        if ((aType | bType) & kPerspective_Mask) {
            tmp.fMat[kMScaleX] = rowcol3(&a.fMat[0], &b.fMat[0]);
            tmp.fMat[kMSkewX]  = rowcol3(&a.fMat[0], &b.fMat[1]);
            tmp.fMat[kMTransX] = rowcol3(&a.fMat[0], &b.fMat[2]);
            tmp.fMat[kMSkewY]  = rowcol3(&a.fMat[3], &b.fMat[0]);
            tmp.fMat[kMScaleY] = rowcol3(&a.fMat[3], &b.fMat[1]);
            tmp.fMat[kMTransY] = rowcol3(&a.fMat[3], &b.fMat[2]);
            tmp.fMat[kMPersp0] = rowcol3(&a.fMat[6], &b.fMat[0]);
            tmp.fMat[kMPersp1] = rowcol3(&a.fMat[6], &b.fMat[1]);
            tmp.fMat[kMPersp2] = rowcol3(&a.fMat[6], &b.fMat[2]);

            tmp.setTypeMask(kUnknown_Mask);
        } else {
            tmp.fMat[kMScaleX] = muladdmul(a.fMat[kMScaleX],
                                           b.fMat[kMScaleX],
                                           a.fMat[kMSkewX],
                                           b.fMat[kMSkewY]);

            tmp.fMat[kMSkewX]  = muladdmul(a.fMat[kMScaleX],
                                           b.fMat[kMSkewX],
                                           a.fMat[kMSkewX],
                                           b.fMat[kMScaleY]);

            tmp.fMat[kMTransX] = muladdmul(a.fMat[kMScaleX],
                                           b.fMat[kMTransX],
                                           a.fMat[kMSkewX],
                                           b.fMat[kMTransY]) + a.fMat[kMTransX];

            tmp.fMat[kMSkewY]  = muladdmul(a.fMat[kMSkewY],
                                           b.fMat[kMScaleX],
                                           a.fMat[kMScaleY],
                                           b.fMat[kMSkewY]);

            tmp.fMat[kMScaleY] = muladdmul(a.fMat[kMSkewY],
                                           b.fMat[kMSkewX],
                                           a.fMat[kMScaleY],
                                           b.fMat[kMScaleY]);

            tmp.fMat[kMTransY] = muladdmul(a.fMat[kMSkewY],
                                           b.fMat[kMTransX],
                                           a.fMat[kMScaleY],
                                           b.fMat[kMTransY]) + a.fMat[kMTransY];

            tmp.fMat[kMPersp0] = 0;
            tmp.fMat[kMPersp1] = 0;
            tmp.fMat[kMPersp2] = 1;
            tmp.setTypeMask(kUnknown_Mask | kOnlyPerspectiveValid_Mask);
        }
        *this = tmp;
    }
    return *this;
}

SkMatrix& SkMatrix::preConcat(const SkMatrix& mat) {
    // check for identity first, so we don't do a needless copy of ourselves
    // to ourselves inside setConcat()
    if(!mat.isIdentity()) {
        this->setConcat(*this, mat);
    }
    return *this;
}

SkMatrix& SkMatrix::postConcat(const SkMatrix& mat) {
    // check for identity first, so we don't do a needless copy of ourselves
    // to ourselves inside setConcat()
    if (!mat.isIdentity()) {
        this->setConcat(mat, *this);
    }
    return *this;
}

///////////////////////////////////////////////////////////////////////////////

/*  Matrix inversion is very expensive, but also the place where keeping
    precision may be most important (here and matrix concat). Hence to avoid
    bitmap blitting artifacts when walking the inverse, we use doubles for
    the intermediate math, even though we know that is more expensive.
 */

static inline SkScalar scross_dscale(SkScalar a, SkScalar b,
                                     SkScalar c, SkScalar d, double scale) {
    return PkDoubleToScalar(scross(a, b, c, d) * scale);
}

static inline double dcross(double a, double b, double c, double d) {
    return a * b - c * d;
}

static inline SkScalar dcross_dscale(double a, double b,
                                     double c, double d, double scale) {
    return PkDoubleToScalar(dcross(a, b, c, d) * scale);
}

static double sk_determinant(const float mat[9], int isPerspective) {
    if (isPerspective) {
        return mat[SkMatrix::kMScaleX] *
                    dcross(mat[SkMatrix::kMScaleY], mat[SkMatrix::kMPersp2],
                           mat[SkMatrix::kMTransY], mat[SkMatrix::kMPersp1])
                    +
                    mat[SkMatrix::kMSkewX]  *
                    dcross(mat[SkMatrix::kMTransY], mat[SkMatrix::kMPersp0],
                           mat[SkMatrix::kMSkewY],  mat[SkMatrix::kMPersp2])
                    +
                    mat[SkMatrix::kMTransX] *
                    dcross(mat[SkMatrix::kMSkewY],  mat[SkMatrix::kMPersp1],
                           mat[SkMatrix::kMScaleY], mat[SkMatrix::kMPersp0]);
    } else {
        return dcross(mat[SkMatrix::kMScaleX], mat[SkMatrix::kMScaleY],
                      mat[SkMatrix::kMSkewX], mat[SkMatrix::kMSkewY]);
    }
}

static double sk_inv_determinant(const float mat[9], int isPerspective) {
    double det = sk_determinant(mat, isPerspective);

    // Since the determinant is on the order of the cube of the matrix members,
    // compare to the cube of the default nearly-zero constant (although an
    // estimate of the condition number would be better if it wasn't so expensive).
    if (SkScalarNearlyZero(pk_double_to_float(det),
                           PK_ScalarNearlyZero * PK_ScalarNearlyZero * PK_ScalarNearlyZero)) {
        return 0;
    }
    return 1.0 / det;
}

void SkMatrix::mapPoints(SkPoint dst[], const SkPoint src[], int count) const {
    this->getMapPtsProc()(*this, dst, src, count);
}

void SkMatrix::mapXY(SkScalar x, SkScalar y, SkPoint* result) const {
    this->getMapXYProc()(*this, x, y, result);
}

void SkMatrix::ComputeInv(SkScalar dst[9], const SkScalar src[9], double invDet, bool isPersp) {
    if (isPersp) {
        dst[kMScaleX] = scross_dscale(src[kMScaleY], src[kMPersp2], src[kMTransY], src[kMPersp1], invDet);
        dst[kMSkewX]  = scross_dscale(src[kMTransX], src[kMPersp1], src[kMSkewX],  src[kMPersp2], invDet);
        dst[kMTransX] = scross_dscale(src[kMSkewX],  src[kMTransY], src[kMTransX], src[kMScaleY], invDet);

        dst[kMSkewY]  = scross_dscale(src[kMTransY], src[kMPersp0], src[kMSkewY],  src[kMPersp2], invDet);
        dst[kMScaleY] = scross_dscale(src[kMScaleX], src[kMPersp2], src[kMTransX], src[kMPersp0], invDet);
        dst[kMTransY] = scross_dscale(src[kMTransX], src[kMSkewY],  src[kMScaleX], src[kMTransY], invDet);

        dst[kMPersp0] = scross_dscale(src[kMSkewY],  src[kMPersp1], src[kMScaleY], src[kMPersp0], invDet);
        dst[kMPersp1] = scross_dscale(src[kMSkewX],  src[kMPersp0], src[kMScaleX], src[kMPersp1], invDet);
        dst[kMPersp2] = scross_dscale(src[kMScaleX], src[kMScaleY], src[kMSkewX],  src[kMSkewY],  invDet);
    } else {   // not perspective
        dst[kMScaleX] = PkDoubleToScalar(src[kMScaleY] * invDet);
        dst[kMSkewX]  = PkDoubleToScalar(-src[kMSkewX] * invDet);
        dst[kMTransX] = dcross_dscale(src[kMSkewX], src[kMTransY], src[kMScaleY], src[kMTransX], invDet);

        dst[kMSkewY]  = PkDoubleToScalar(-src[kMSkewY] * invDet);
        dst[kMScaleY] = PkDoubleToScalar(src[kMScaleX] * invDet);
        dst[kMTransY] = dcross_dscale(src[kMSkewY], src[kMTransX], src[kMScaleX], src[kMTransY], invDet);

        dst[kMPersp0] = 0;
        dst[kMPersp1] = 0;
        dst[kMPersp2] = 1;
    }
}

bool SkMatrix::invertNonIdentity(SkMatrix* inv) const {
    TypeMask mask = this->getType();

    if (0 == (mask & ~(kScale_Mask | kTranslate_Mask))) {
        bool invertible = true;
        if (inv) {
            if (mask & kScale_Mask) {
                SkScalar invX = fMat[kMScaleX];
                SkScalar invY = fMat[kMScaleY];
                if (0 == invX || 0 == invY) {
                    return false;
                }
                invX = PkScalarInvert(invX);
                invY = PkScalarInvert(invY);

                // Must be careful when writing to inv, since it may be the
                // same memory as this.

                inv->fMat[kMSkewX] = inv->fMat[kMSkewY] =
                inv->fMat[kMPersp0] = inv->fMat[kMPersp1] = 0;

                inv->fMat[kMScaleX] = invX;
                inv->fMat[kMScaleY] = invY;
                inv->fMat[kMPersp2] = 1;
                inv->fMat[kMTransX] = -fMat[kMTransX] * invX;
                inv->fMat[kMTransY] = -fMat[kMTransY] * invY;

                inv->setTypeMask(mask | kRectStaysRect_Mask);
            } else {
                // translate only
                inv->setTranslate(-fMat[kMTransX], -fMat[kMTransY]);
            }
        } else {    // inv is nullptr, just check if we're invertible
            if (!fMat[kMScaleX] || !fMat[kMScaleY]) {
                invertible = false;
            }
        }
        return invertible;
    }

    int    isPersp = mask & kPerspective_Mask;
    double invDet = sk_inv_determinant(fMat, isPersp);

    if (invDet == 0) { // underflow
        return false;
    }

    bool applyingInPlace = (inv == this);

    SkMatrix* tmp = inv;

    SkMatrix storage;
    if (applyingInPlace || nullptr == tmp) {
        tmp = &storage;     // we either need to avoid trampling memory or have no memory
    }

    ComputeInv(tmp->fMat, fMat, invDet, isPersp);
    if (!tmp->isFinite()) {
        return false;
    }

    tmp->setTypeMask(fTypeMask);

    if (applyingInPlace) {
        *inv = storage; // need to copy answer back
    }

    return true;
}

///////////////////////////////////////////////////////////////////////////////

void SkMatrix::Identity_pts(const SkMatrix& m, SkPoint dst[], const SkPoint src[], int count) {
    if (dst != src && count > 0) {
        memcpy(dst, src, count * sizeof(SkPoint));
    }
}

void SkMatrix::Trans_pts(const SkMatrix& m, SkPoint dst[], const SkPoint src[], int count) {
    if (count > 0) {
        SkScalar tx = m.getTranslateX();
        SkScalar ty = m.getTranslateY();
        if (count & 1) {
            dst->fX = src->fX + tx;
            dst->fY = src->fY + ty;
            src += 1;
            dst += 1;
        }
        Sk4s trans4(tx, ty, tx, ty);
        count >>= 1;
        if (count & 1) {
            (Sk4s::Load(src) + trans4).store(dst);
            src += 2;
            dst += 2;
        }
        count >>= 1;
        for (int i = 0; i < count; ++i) {
            (Sk4s::Load(src+0) + trans4).store(dst+0);
            (Sk4s::Load(src+2) + trans4).store(dst+2);
            src += 4;
            dst += 4;
        }
    }
}

void SkMatrix::Scale_pts(const SkMatrix& m, SkPoint dst[], const SkPoint src[], int count) {
    if (count > 0) {
        SkScalar tx = m.getTranslateX();
        SkScalar ty = m.getTranslateY();
        SkScalar sx = m.getScaleX();
        SkScalar sy = m.getScaleY();
        if (count & 1) {
            dst->fX = src->fX * sx + tx;
            dst->fY = src->fY * sy + ty;
            src += 1;
            dst += 1;
        }
        Sk4s trans4(tx, ty, tx, ty);
        Sk4s scale4(sx, sy, sx, sy);
        count >>= 1;
        if (count & 1) {
            (Sk4s::Load(src) * scale4 + trans4).store(dst);
            src += 2;
            dst += 2;
        }
        count >>= 1;
        for (int i = 0; i < count; ++i) {
            (Sk4s::Load(src+0) * scale4 + trans4).store(dst+0);
            (Sk4s::Load(src+2) * scale4 + trans4).store(dst+2);
            src += 4;
            dst += 4;
        }
    }
}

void SkMatrix::Persp_pts(const SkMatrix& m, SkPoint dst[],
                         const SkPoint src[], int count) {
    if (count > 0) {
        do {
            SkScalar sy = src->fY;
            SkScalar sx = src->fX;
            src += 1;

            SkScalar x = sdot(sx, m.fMat[kMScaleX], sy, m.fMat[kMSkewX])  + m.fMat[kMTransX];
            SkScalar y = sdot(sx, m.fMat[kMSkewY],  sy, m.fMat[kMScaleY]) + m.fMat[kMTransY];
#ifdef PK_LEGACY_MATRIX_MATH_ORDER
            SkScalar z = sx * m.fMat[kMPersp0] + (sy * m.fMat[kMPersp1] + m.fMat[kMPersp2]);
#else
            SkScalar z = sdot(sx, m.fMat[kMPersp0], sy, m.fMat[kMPersp1]) + m.fMat[kMPersp2];
#endif
            if (z) {
                z = 1 / z;
            }

            dst->fY = y * z;
            dst->fX = x * z;
            dst += 1;
        } while (--count);
    }
}

void SkMatrix::Affine_vpts(const SkMatrix& m, SkPoint dst[], const SkPoint src[], int count) {
    if (count > 0) {
        SkScalar tx = m.getTranslateX();
        SkScalar ty = m.getTranslateY();
        SkScalar sx = m.getScaleX();
        SkScalar sy = m.getScaleY();
        SkScalar kx = m.getSkewX();
        SkScalar ky = m.getSkewY();
        if (count & 1) {
            dst->set(src->fX * sx + src->fY * kx + tx,
                     src->fX * ky + src->fY * sy + ty);
            src += 1;
            dst += 1;
        }
        Sk4s trans4(tx, ty, tx, ty);
        Sk4s scale4(sx, sy, sx, sy);
        Sk4s  skew4(kx, ky, kx, ky);    // applied to swizzle of src4
        count >>= 1;
        for (int i = 0; i < count; ++i) {
            Sk4s src4 = Sk4s::Load(src);
            Sk4s swz4 = SkNx_shuffle<1,0,3,2>(src4);  // y0 x0, y1 x1
            (src4 * scale4 + swz4 * skew4 + trans4).store(dst);
            src += 2;
            dst += 2;
        }
    }
}

const SkMatrix::MapPtsProc SkMatrix::gMapPtsProcs[] = {
    SkMatrix::Identity_pts, SkMatrix::Trans_pts,
    SkMatrix::Scale_pts,    SkMatrix::Scale_pts,
    SkMatrix::Affine_vpts,  SkMatrix::Affine_vpts,
    SkMatrix::Affine_vpts,  SkMatrix::Affine_vpts,
    // repeat the persp proc 8 times
    SkMatrix::Persp_pts,    SkMatrix::Persp_pts,
    SkMatrix::Persp_pts,    SkMatrix::Persp_pts,
    SkMatrix::Persp_pts,    SkMatrix::Persp_pts,
    SkMatrix::Persp_pts,    SkMatrix::Persp_pts
};

///////////////////////////////////////////////////////////////////////////////

void SkMatrixPriv::MapHomogeneousPointsWithStride(const SkMatrix& mx, SkPoint3 dst[],
                                                  size_t dstStride, const SkPoint3 src[],
                                                  size_t srcStride, int count) {
    if (count > 0) {
        if (mx.isIdentity()) {
            if (src != dst) {
                if (srcStride == sizeof(SkPoint3) && dstStride == sizeof(SkPoint3)) {
                    memcpy(dst, src, count * sizeof(SkPoint3));
                } else {
                    for (int i = 0; i < count; ++i) {
                        *dst = *src;
                        dst = reinterpret_cast<SkPoint3*>(reinterpret_cast<char*>(dst) + dstStride);
                        src = reinterpret_cast<const SkPoint3*>(reinterpret_cast<const char*>(src) +
                                                                srcStride);
                    }
                }
            }
            return;
        }
        do {
            SkScalar sx = src->fX;
            SkScalar sy = src->fY;
            SkScalar sw = src->fZ;
            src = reinterpret_cast<const SkPoint3*>(reinterpret_cast<const char*>(src) + srcStride);
            const SkScalar* mat = mx.fMat;
            typedef SkMatrix M;
            SkScalar x = sdot(sx, mat[M::kMScaleX], sy, mat[M::kMSkewX],  sw, mat[M::kMTransX]);
            SkScalar y = sdot(sx, mat[M::kMSkewY],  sy, mat[M::kMScaleY], sw, mat[M::kMTransY]);
            SkScalar w = sdot(sx, mat[M::kMPersp0], sy, mat[M::kMPersp1], sw, mat[M::kMPersp2]);

            dst->set(x, y, w);
            dst = reinterpret_cast<SkPoint3*>(reinterpret_cast<char*>(dst) + dstStride);
        } while (--count);
    }
}

void SkMatrix::mapHomogeneousPoints(SkPoint3 dst[], const SkPoint3 src[], int count) const {
    SkMatrixPriv::MapHomogeneousPointsWithStride(*this, dst, sizeof(SkPoint3), src,
                                                 sizeof(SkPoint3), count);
}

void SkMatrix::mapHomogeneousPoints(SkPoint3 dst[], const SkPoint src[], int count) const {
    if (this->isIdentity()) {
        for (int i = 0; i < count; ++i) {
            dst[i] = { src[i].fX, src[i].fY, 1 };
        }
    } else if (this->hasPerspective()) {
        for (int i = 0; i < count; ++i) {
            dst[i] = {
                fMat[0] * src[i].fX + fMat[1] * src[i].fY + fMat[2],
                fMat[3] * src[i].fX + fMat[4] * src[i].fY + fMat[5],
                fMat[6] * src[i].fX + fMat[7] * src[i].fY + fMat[8],
            };
        }
    } else {    // affine
        for (int i = 0; i < count; ++i) {
            dst[i] = {
                fMat[0] * src[i].fX + fMat[1] * src[i].fY + fMat[2],
                fMat[3] * src[i].fX + fMat[4] * src[i].fY + fMat[5],
                1,
            };
        }
    }
}

///////////////////////////////////////////////////////////////////////////////

static Sk4f sort_as_rect(const Sk4f& ltrb) {
    Sk4f rblt(ltrb[2], ltrb[3], ltrb[0], ltrb[1]);
    Sk4f min = Sk4f::Min(ltrb, rblt);
    Sk4f max = Sk4f::Max(ltrb, rblt);
    // We can extract either pair [0,1] or [2,3] from min and max and be correct, but on
    // ARM this sequence generates the fastest (a single instruction).
    return Sk4f(min[2], min[3], max[0], max[1]);
}

void SkMatrix::mapRectScaleTranslate(SkRect* dst, const SkRect& src) const {
    SkScalar sx = fMat[kMScaleX];
    SkScalar sy = fMat[kMScaleY];
    SkScalar tx = fMat[kMTransX];
    SkScalar ty = fMat[kMTransY];
    Sk4f scale(sx, sy, sx, sy);
    Sk4f trans(tx, ty, tx, ty);
    sort_as_rect(Sk4f::Load(&src.fLeft) * scale + trans).store(&dst->fLeft);
}

bool SkMatrix::mapRect(SkRect* dst, const SkRect& src) const {
    if (this->getType() <= kTranslate_Mask) {
        SkScalar tx = fMat[kMTransX];
        SkScalar ty = fMat[kMTransY];
        Sk4f trans(tx, ty, tx, ty);
        sort_as_rect(Sk4f::Load(&src.fLeft) + trans).store(&dst->fLeft);
        return true;
    }
    if (this->isScaleTranslate()) {
        this->mapRectScaleTranslate(dst, src);
        return true;
    } else {
        SkPoint quad[4];

        src.toQuad(quad);
        this->mapPoints(quad, quad, 4);
        dst->setBoundsNoCheck(quad, 4);
        return this->rectStaysRect();   // might still return true if rotated by 90, etc.
    }
}

void SkMatrix::Persp_xy(const SkMatrix& m, SkScalar sx, SkScalar sy,
                        SkPoint* pt) {
    SkScalar x = sdot(sx, m.fMat[kMScaleX], sy, m.fMat[kMSkewX])  + m.fMat[kMTransX];
    SkScalar y = sdot(sx, m.fMat[kMSkewY],  sy, m.fMat[kMScaleY]) + m.fMat[kMTransY];
    SkScalar z = sdot(sx, m.fMat[kMPersp0], sy, m.fMat[kMPersp1]) + m.fMat[kMPersp2];
    if (z) {
        z = 1 / z;
    }
    pt->fX = x * z;
    pt->fY = y * z;
}

void SkMatrix::RotTrans_xy(const SkMatrix& m, SkScalar sx, SkScalar sy,
                           SkPoint* pt) {
#ifdef PK_LEGACY_MATRIX_MATH_ORDER
    pt->fX = sx * m.fMat[kMScaleX] + (sy * m.fMat[kMSkewX]  +  m.fMat[kMTransX]);
    pt->fY = sx * m.fMat[kMSkewY]  + (sy * m.fMat[kMScaleY] + m.fMat[kMTransY]);
#else
    pt->fX = sdot(sx, m.fMat[kMScaleX], sy, m.fMat[kMSkewX])  + m.fMat[kMTransX];
    pt->fY = sdot(sx, m.fMat[kMSkewY],  sy, m.fMat[kMScaleY]) + m.fMat[kMTransY];
#endif
}

void SkMatrix::Rot_xy(const SkMatrix& m, SkScalar sx, SkScalar sy,
                      SkPoint* pt) {
#ifdef PK_LEGACY_MATRIX_MATH_ORDER
    pt->fX = sx * m.fMat[kMScaleX] + (sy * m.fMat[kMSkewX]  + m.fMat[kMTransX]);
    pt->fY = sx * m.fMat[kMSkewY]  + (sy * m.fMat[kMScaleY] + m.fMat[kMTransY]);
#else
    pt->fX = sdot(sx, m.fMat[kMScaleX], sy, m.fMat[kMSkewX])  + m.fMat[kMTransX];
    pt->fY = sdot(sx, m.fMat[kMSkewY],  sy, m.fMat[kMScaleY]) + m.fMat[kMTransY];
#endif
}

void SkMatrix::ScaleTrans_xy(const SkMatrix& m, SkScalar sx, SkScalar sy,
                             SkPoint* pt) {
    pt->fX = sx * m.fMat[kMScaleX] + m.fMat[kMTransX];
    pt->fY = sy * m.fMat[kMScaleY] + m.fMat[kMTransY];
}

void SkMatrix::Scale_xy(const SkMatrix& m, SkScalar sx, SkScalar sy,
                        SkPoint* pt) {
    pt->fX = sx * m.fMat[kMScaleX];
    pt->fY = sy * m.fMat[kMScaleY];
}

void SkMatrix::Trans_xy(const SkMatrix& m, SkScalar sx, SkScalar sy,
                        SkPoint* pt) {
    pt->fX = sx + m.fMat[kMTransX];
    pt->fY = sy + m.fMat[kMTransY];
}

void SkMatrix::Identity_xy(const SkMatrix& m, SkScalar sx, SkScalar sy,
                           SkPoint* pt) {
    pt->fX = sx;
    pt->fY = sy;
}

const SkMatrix::MapXYProc SkMatrix::gMapXYProcs[] = {
    SkMatrix::Identity_xy, SkMatrix::Trans_xy,
    SkMatrix::Scale_xy,    SkMatrix::ScaleTrans_xy,
    SkMatrix::Rot_xy,      SkMatrix::RotTrans_xy,
    SkMatrix::Rot_xy,      SkMatrix::RotTrans_xy,
    // repeat the persp proc 8 times
    SkMatrix::Persp_xy,    SkMatrix::Persp_xy,
    SkMatrix::Persp_xy,    SkMatrix::Persp_xy,
    SkMatrix::Persp_xy,    SkMatrix::Persp_xy,
    SkMatrix::Persp_xy,    SkMatrix::Persp_xy
};

static inline bool checkForZero(float x) {
    return x*x == 0;
}

bool SkMatrix::Poly2Proc(const SkPoint srcPt[], SkMatrix* dst) {
    dst->fMat[kMScaleX] = srcPt[1].fY - srcPt[0].fY;
    dst->fMat[kMSkewY]  = srcPt[0].fX - srcPt[1].fX;
    dst->fMat[kMPersp0] = 0;

    dst->fMat[kMSkewX]  = srcPt[1].fX - srcPt[0].fX;
    dst->fMat[kMScaleY] = srcPt[1].fY - srcPt[0].fY;
    dst->fMat[kMPersp1] = 0;

    dst->fMat[kMTransX] = srcPt[0].fX;
    dst->fMat[kMTransY] = srcPt[0].fY;
    dst->fMat[kMPersp2] = 1;
    dst->setTypeMask(kUnknown_Mask);
    return true;
}

bool SkMatrix::Poly3Proc(const SkPoint srcPt[], SkMatrix* dst) {
    dst->fMat[kMScaleX] = srcPt[2].fX - srcPt[0].fX;
    dst->fMat[kMSkewY]  = srcPt[2].fY - srcPt[0].fY;
    dst->fMat[kMPersp0] = 0;

    dst->fMat[kMSkewX]  = srcPt[1].fX - srcPt[0].fX;
    dst->fMat[kMScaleY] = srcPt[1].fY - srcPt[0].fY;
    dst->fMat[kMPersp1] = 0;

    dst->fMat[kMTransX] = srcPt[0].fX;
    dst->fMat[kMTransY] = srcPt[0].fY;
    dst->fMat[kMPersp2] = 1;
    dst->setTypeMask(kUnknown_Mask);
    return true;
}

bool SkMatrix::Poly4Proc(const SkPoint srcPt[], SkMatrix* dst) {
    float   a1, a2;
    float   x0, y0, x1, y1, x2, y2;

    x0 = srcPt[2].fX - srcPt[0].fX;
    y0 = srcPt[2].fY - srcPt[0].fY;
    x1 = srcPt[2].fX - srcPt[1].fX;
    y1 = srcPt[2].fY - srcPt[1].fY;
    x2 = srcPt[2].fX - srcPt[3].fX;
    y2 = srcPt[2].fY - srcPt[3].fY;

    /* check if abs(x2) > abs(y2) */
    if ( x2 > 0 ? y2 > 0 ? x2 > y2 : x2 > -y2 : y2 > 0 ? -x2 > y2 : x2 < y2) {
        float denom = sk_ieee_float_divide(x1 * y2, x2) - y1;
        if (checkForZero(denom)) {
            return false;
        }
        a1 = (((x0 - x1) * y2 / x2) - y0 + y1) / denom;
    } else {
        float denom = x1 - sk_ieee_float_divide(y1 * x2, y2);
        if (checkForZero(denom)) {
            return false;
        }
        a1 = (x0 - x1 - sk_ieee_float_divide((y0 - y1) * x2, y2)) / denom;
    }

    /* check if abs(x1) > abs(y1) */
    if ( x1 > 0 ? y1 > 0 ? x1 > y1 : x1 > -y1 : y1 > 0 ? -x1 > y1 : x1 < y1) {
        float denom = y2 - sk_ieee_float_divide(x2 * y1, x1);
        if (checkForZero(denom)) {
            return false;
        }
        a2 = (y0 - y2 - sk_ieee_float_divide((x0 - x2) * y1, x1)) / denom;
    } else {
        float denom = sk_ieee_float_divide(y2 * x1, y1) - x2;
        if (checkForZero(denom)) {
            return false;
        }
        a2 = (sk_ieee_float_divide((y0 - y2) * x1, y1) - x0 + x2) / denom;
    }

    dst->fMat[kMScaleX] = a2 * srcPt[3].fX + srcPt[3].fX - srcPt[0].fX;
    dst->fMat[kMSkewY]  = a2 * srcPt[3].fY + srcPt[3].fY - srcPt[0].fY;
    dst->fMat[kMPersp0] = a2;

    dst->fMat[kMSkewX]  = a1 * srcPt[1].fX + srcPt[1].fX - srcPt[0].fX;
    dst->fMat[kMScaleY] = a1 * srcPt[1].fY + srcPt[1].fY - srcPt[0].fY;
    dst->fMat[kMPersp1] = a1;

    dst->fMat[kMTransX] = srcPt[0].fX;
    dst->fMat[kMTransY] = srcPt[0].fY;
    dst->fMat[kMPersp2] = 1;
    dst->setTypeMask(kUnknown_Mask);
    return true;
}

enum MinMaxOrBoth {
    kMin_MinMaxOrBoth,
    kMax_MinMaxOrBoth,
    kBoth_MinMaxOrBoth
};

template <MinMaxOrBoth MIN_MAX_OR_BOTH> bool get_scale_factor(SkMatrix::TypeMask typeMask,
                                                              const SkScalar m[9],
                                                              SkScalar results[/*1 or 2*/]) {
    if (typeMask & SkMatrix::kPerspective_Mask) {
        return false;
    }
    if (SkMatrix::kIdentity_Mask == typeMask) {
        results[0] = PK_Scalar1;
        if (kBoth_MinMaxOrBoth == MIN_MAX_OR_BOTH) {
            results[1] = PK_Scalar1;
        }
        return true;
    }
    if (!(typeMask & SkMatrix::kAffine_Mask)) {
        if (kMin_MinMaxOrBoth == MIN_MAX_OR_BOTH) {
             results[0] = std::min(PkScalarAbs(m[SkMatrix::kMScaleX]),
                                  PkScalarAbs(m[SkMatrix::kMScaleY]));
        } else if (kMax_MinMaxOrBoth == MIN_MAX_OR_BOTH) {
             results[0] = std::max(PkScalarAbs(m[SkMatrix::kMScaleX]),
                                  PkScalarAbs(m[SkMatrix::kMScaleY]));
        } else {
            results[0] = PkScalarAbs(m[SkMatrix::kMScaleX]);
            results[1] = PkScalarAbs(m[SkMatrix::kMScaleY]);
             if (results[0] > results[1]) {
                 using std::swap;
                 swap(results[0], results[1]);
             }
        }
        return true;
    }
    // ignore the translation part of the matrix, just look at 2x2 portion.
    // compute singular values, take largest or smallest abs value.
    // [a b; b c] = A^T*A
    SkScalar a = sdot(m[SkMatrix::kMScaleX], m[SkMatrix::kMScaleX],
                      m[SkMatrix::kMSkewY],  m[SkMatrix::kMSkewY]);
    SkScalar b = sdot(m[SkMatrix::kMScaleX], m[SkMatrix::kMSkewX],
                      m[SkMatrix::kMScaleY], m[SkMatrix::kMSkewY]);
    SkScalar c = sdot(m[SkMatrix::kMSkewX],  m[SkMatrix::kMSkewX],
                      m[SkMatrix::kMScaleY], m[SkMatrix::kMScaleY]);
    // eigenvalues of A^T*A are the squared singular values of A.
    // characteristic equation is det((A^T*A) - l*I) = 0
    // l^2 - (a + c)l + (ac-b^2)
    // solve using quadratic equation (divisor is non-zero since l^2 has 1 coeff
    // and roots are guaranteed to be pos and real).
    SkScalar bSqd = b * b;
    // if upper left 2x2 is orthogonal save some math
    if (bSqd <= PK_ScalarNearlyZero*PK_ScalarNearlyZero) {
        if (kMin_MinMaxOrBoth == MIN_MAX_OR_BOTH) {
            results[0] = std::min(a, c);
        } else if (kMax_MinMaxOrBoth == MIN_MAX_OR_BOTH) {
            results[0] = std::max(a, c);
        } else {
            results[0] = a;
            results[1] = c;
            if (results[0] > results[1]) {
                using std::swap;
                swap(results[0], results[1]);
            }
        }
    } else {
        SkScalar aminusc = a - c;
        SkScalar apluscdiv2 = PkScalarHalf(a + c);
        SkScalar x = PkScalarHalf(PkScalarSqrt(aminusc * aminusc + 4 * bSqd));
        if (kMin_MinMaxOrBoth == MIN_MAX_OR_BOTH) {
            results[0] = apluscdiv2 - x;
        } else if (kMax_MinMaxOrBoth == MIN_MAX_OR_BOTH) {
            results[0] = apluscdiv2 + x;
        } else {
            results[0] = apluscdiv2 - x;
            results[1] = apluscdiv2 + x;
        }
    }
    if (!SkScalarIsFinite(results[0])) {
        return false;
    }
    // Due to the floating point inaccuracy, there might be an error in a, b, c
    // calculated by sdot, further deepened by subsequent arithmetic operations
    // on them. Therefore, we allow and cap the nearly-zero negative values.
    if (results[0] < 0) {
        results[0] = 0;
    }
    results[0] = PkScalarSqrt(results[0]);
    if (kBoth_MinMaxOrBoth == MIN_MAX_OR_BOTH) {
        if (!SkScalarIsFinite(results[1])) {
            return false;
        }
        if (results[1] < 0) {
            results[1] = 0;
        }
        results[1] = PkScalarSqrt(results[1]);
    }
    return true;
}

SkScalar SkMatrix::getMinScale() const {
    SkScalar factor;
    if (get_scale_factor<kMin_MinMaxOrBoth>(this->getType(), fMat, &factor)) {
        return factor;
    } else {
        return -1;
    }
}

SkScalar SkMatrix::getMaxScale() const {
    SkScalar factor;
    if (get_scale_factor<kMax_MinMaxOrBoth>(this->getType(), fMat, &factor)) {
        return factor;
    } else {
        return -1;
    }
}

bool SkMatrix::getMinMaxScales(SkScalar scaleFactors[2]) const {
    return get_scale_factor<kBoth_MinMaxOrBoth>(this->getType(), fMat, scaleFactors);
}

const SkMatrix& SkMatrix::I() {
    static constexpr SkMatrix identity;
    return identity;
}

const SkMatrix& SkMatrix::InvalidMatrix() {
    static constexpr SkMatrix invalid(PK_ScalarMax, PK_ScalarMax, PK_ScalarMax,
                                      PK_ScalarMax, PK_ScalarMax, PK_ScalarMax,
                                      PK_ScalarMax, PK_ScalarMax, PK_ScalarMax,
                                      kTranslate_Mask | kScale_Mask |
                                      kAffine_Mask | kPerspective_Mask);
    return invalid;
}

///////////////////////////////////////////////////////////////////////////////

size_t SkMatrix::writeToMemory(void* buffer) const {
    // TODO write less for simple matrices
    static const size_t sizeInMemory = 9 * sizeof(SkScalar);
    if (buffer) {
        memcpy(buffer, fMat, sizeInMemory);
    }
    return sizeInMemory;
}

size_t SkMatrix::readFromMemory(const void* buffer, size_t length) {
    static const size_t sizeInMemory = 9 * sizeof(SkScalar);
    if (length < sizeInMemory) {
        return 0;
    }
    memcpy(fMat, buffer, sizeInMemory);
    this->setTypeMask(kUnknown_Mask);
    // Figure out the type now so that we're thread-safe
    (void)this->getType();
    return sizeInMemory;
}

// A square matrix M can be decomposed (via polar decomposition) into two matrices --
// an orthogonal matrix Q and a symmetric matrix S. In turn we can decompose S into U*W*U^T,
// where U is another orthogonal matrix and W is a scale matrix. These can be recombined
// to give M = (Q*U)*W*U^T, i.e., the product of two orthogonal matrices and a scale matrix.
//
// The one wrinkle is that traditionally Q may contain a reflection -- the
// calculation has been rejiggered to put that reflection into W.
bool SkDecomposeUpper2x2(const SkMatrix& matrix,
                         SkPoint* rotation1,
                         SkPoint* scale,
                         SkPoint* rotation2) {

    SkScalar A = matrix[SkMatrix::kMScaleX];
    SkScalar B = matrix[SkMatrix::kMSkewX];
    SkScalar C = matrix[SkMatrix::kMSkewY];
    SkScalar D = matrix[SkMatrix::kMScaleY];

    if (is_degenerate_2x2(A, B, C, D)) {
        return false;
    }

    double w1, w2;
    SkScalar cos1, sin1;
    SkScalar cos2, sin2;

    // do polar decomposition (M = Q*S)
    SkScalar cosQ, sinQ;
    double Sa, Sb, Sd;
    // if M is already symmetric (i.e., M = I*S)
    if (SkScalarNearlyEqual(B, C)) {
        cosQ = 1;
        sinQ = 0;

        Sa = A;
        Sb = B;
        Sd = D;
    } else {
        cosQ = A + D;
        sinQ = C - B;
        SkScalar reciplen = PkScalarInvert(PkScalarSqrt(cosQ*cosQ + sinQ*sinQ));
        cosQ *= reciplen;
        sinQ *= reciplen;

        // S = Q^-1*M
        // we don't calc Sc since it's symmetric
        Sa = A*cosQ + C*sinQ;
        Sb = B*cosQ + D*sinQ;
        Sd = -B*sinQ + D*cosQ;
    }

    // Now we need to compute eigenvalues of S (our scale factors)
    // and eigenvectors (bases for our rotation)
    // From this, should be able to reconstruct S as U*W*U^T
    if (SkScalarNearlyZero(PkDoubleToScalar(Sb))) {
        // already diagonalized
        cos1 = 1;
        sin1 = 0;
        w1 = Sa;
        w2 = Sd;
        cos2 = cosQ;
        sin2 = sinQ;
    } else {
        double diff = Sa - Sd;
        double discriminant = sqrt(diff*diff + 4.0*Sb*Sb);
        double trace = Sa + Sd;
        if (diff > 0) {
            w1 = 0.5*(trace + discriminant);
            w2 = 0.5*(trace - discriminant);
        } else {
            w1 = 0.5*(trace - discriminant);
            w2 = 0.5*(trace + discriminant);
        }

        cos1 = PkDoubleToScalar(Sb); sin1 = PkDoubleToScalar(w1 - Sa);
        SkScalar reciplen = PkScalarInvert(PkScalarSqrt(cos1*cos1 + sin1*sin1));
        cos1 *= reciplen;
        sin1 *= reciplen;

        // rotation 2 is composition of Q and U
        cos2 = cos1*cosQ - sin1*sinQ;
        sin2 = sin1*cosQ + cos1*sinQ;

        // rotation 1 is U^T
        sin1 = -sin1;
    }

    if (scale) {
        scale->fX = PkDoubleToScalar(w1);
        scale->fY = PkDoubleToScalar(w2);
    }
    if (rotation1) {
        rotation1->fX = cos1;
        rotation1->fY = sin1;
    }
    if (rotation2) {
        rotation2->fX = cos2;
        rotation2->fY = sin2;
    }

    return true;
}

SkScalar SkMatrixPriv::DifferentialAreaScale(const SkMatrix& m, const SkPoint& p) {
    //              [m00 m01 m02]                                 [f(u,v)]
    // Assuming M = [m10 m11 m12], define the projected p'(u,v) = [g(u,v)] where
    //              [m20 m12 m22]
    //                                                        [x]     [u]
    // f(u,v) = x(u,v) / w(u,v), g(u,v) = y(u,v) / w(u,v) and [y] = M*[v]
    //                                                        [w]     [1]
    //
    // Then the differential scale factor between p = (u,v) and p' is |det J|,
    // where J is the Jacobian for p': [df/du dg/du]
    //                                 [df/dv dg/dv]
    // and df/du = (w*dx/du - x*dw/du)/w^2,   dg/du = (w*dy/du - y*dw/du)/w^2
    //     df/dv = (w*dx/dv - x*dw/dv)/w^2,   dg/dv = (w*dy/dv - y*dw/dv)/w^2
    //
    // From here, |det J| can be rewritten as |det J'/w^3|, where
    //      [x     y     w    ]   [x   y   w  ]
    // J' = [dx/du dy/du dw/du] = [m00 m10 m20]
    //      [dx/dv dy/dv dw/dv]   [m01 m11 m21]
    SkPoint3 xyw;
    m.mapHomogeneousPoints(&xyw, &p, 1);

    if (xyw.fZ < PK_ScalarNearlyZero) {
        // Reaching the discontinuity of xy/w and where the point would clip to w >= 0
        return PK_ScalarInfinity;
    }
    SkMatrix jacobian = SkMatrix::MakeAll(xyw.fX, xyw.fY, xyw.fZ,
                                          m.getScaleX(), m.getSkewY(), m.getPerspX(),
                                          m.getSkewX(), m.getScaleY(), m.getPerspY());

    double denom = 1.0 / xyw.fZ;   // 1/w
    denom = denom * denom * denom; // 1/w^3
    return PkScalarAbs(PkDoubleToScalar(sk_determinant(jacobian.fMat, true) * denom));
}
}  // namespace pk