SkMatrix.cpp

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/*
 * 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/SkPath.h"
#include "include/core/SkPoint3.h"
#include "include/core/SkRSXform.h"
#include "include/core/SkSamplingOptions.h"
#include "include/core/SkSize.h"
#include "include/core/SkString.h"
#include "include/private/base/SkDebug.h"
#include "include/private/base/SkFloatingPoint.h"
#include "include/private/base/SkMalloc.h"
#include "include/private/base/SkMath.h"
#include "include/private/base/SkTo.h"
#include "src/base/SkFloatBits.h"
#include "src/base/SkVx.h"
#include "src/core/SkMatrixPriv.h"
#include "src/core/SkMatrixUtils.h"
#include "src/core/SkSamplingPriv.h"
 
#include <algorithm>
#include <cmath>
 
void SkMatrix::doNormalizePerspective() {
    // If the bottom row of the matrix is [0, 0, not_one], we will treat the matrix as if it
    // is in perspective, even though it stills behaves like its affine. If we divide everything
    // by the not_one value, then it will behave the same, but will be treated as affine,
    // and therefore faster (e.g. clients can forward-difference calculations).
    //
    if (0 == fMat[SkMatrix::kMPersp0] && 0 == fMat[SkMatrix::kMPersp1]) {
        SkScalar p2 = fMat[SkMatrix::kMPersp2];
        if (p2 != 0 && p2 != 1) {
            double inv = 1.0 / p2;
            for (int i = 0; i < 6; ++i) {
                fMat[i] = SkDoubleToScalar(fMat[i] * inv);
            }
            fMat[SkMatrix::kMPersp2] = 1;
        }
        this->setTypeMask(kUnknown_Mask);
    }
}
 
SkMatrix& SkMatrix::reset() { *this = SkMatrix(); return *this; }
 
SkMatrix& SkMatrix::set9(const SkScalar buffer[9]) {
    memcpy(fMat, buffer, 9 * sizeof(SkScalar));
    this->setTypeMask(kUnknown_Mask);
    return *this;
}
 
SkMatrix& SkMatrix::setAffine(const SkScalar buffer[6]) {
    fMat[kMScaleX] = buffer[kAScaleX];
    fMat[kMSkewX]  = buffer[kASkewX];
    fMat[kMTransX] = buffer[kATransX];
    fMat[kMSkewY]  = buffer[kASkewY];
    fMat[kMScaleY] = buffer[kAScaleY];
    fMat[kMTransY] = buffer[kATransY];
    fMat[kMPersp0] = 0;
    fMat[kMPersp1] = 0;
    fMat[kMPersp2] = 1;
    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 SkScalarAs2sCompliment
    // 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 = SkScalarAs2sCompliment(fMat[SkMatrix::kMScaleX]);
    int m01 = SkScalarAs2sCompliment(fMat[SkMatrix::kMSkewX]);
    int m10 = SkScalarAs2sCompliment(fMat[SkMatrix::kMSkewY]);
    int m11 = SkScalarAs2sCompliment(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* SK_RESTRICT ma = a.fMat;
    const SkScalar* SK_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, SK_ScalarNearlyZero*SK_ScalarNearlyZero);
}
 
///////////////////////////////////////////////////////////////////////////////
 
bool SkMatrix::isSimilarity(SkScalar tol) const {
    // if identity or translate matrix
    TypeMask mask = this->getType();
    if (mask <= kTranslate_Mask) {
        return true;
    }
    if (mask & kPerspective_Mask) {
        return false;
    }
 
    SkScalar mx = fMat[kMScaleX];
    SkScalar my = fMat[kMScaleY];
    // if no skew, can just compare scale factors
    if (!(mask & kAffine_Mask)) {
        return !SkScalarNearlyZero(mx) && SkScalarNearlyEqual(SkScalarAbs(mx), SkScalarAbs(my));
    }
    SkScalar sx = fMat[kMSkewX];
    SkScalar sy = fMat[kMSkewY];
 
    if (is_degenerate_2x2(mx, sx, sy, my)) {
        return false;
    }
 
    // upper 2x2 is rotation/reflection + uniform scale if basis vectors
    // are 90 degree rotations of each other
    return (SkScalarNearlyEqual(mx, my, tol) && SkScalarNearlyEqual(sx, -sy, tol))
        || (SkScalarNearlyEqual(mx, -my, tol) && SkScalarNearlyEqual(sx, sy, tol));
}
 
bool SkMatrix::preservesRightAngles(SkScalar tol) const {
    TypeMask mask = this->getType();
 
    if (mask <= kTranslate_Mask) {
        // identity, translate and/or scale
        return true;
    }
    if (mask & kPerspective_Mask) {
        return false;
    }
 
    SkASSERT(mask & (kAffine_Mask | kScale_Mask));
 
    SkScalar mx = fMat[kMScaleX];
    SkScalar my = fMat[kMScaleY];
    SkScalar sx = fMat[kMSkewX];
    SkScalar sy = fMat[kMSkewY];
 
    if (is_degenerate_2x2(mx, sx, sy, my)) {
        return false;
    }
 
    // upper 2x2 is scale + rotation/reflection if basis vectors are orthogonal
    SkVector vec[2];
    vec[0].set(mx, sy);
    vec[1].set(sx, my);
 
    return SkScalarNearlyZero(vec[0].dot(vec[1]), SkScalarSquare(tol));
}
 
///////////////////////////////////////////////////////////////////////////////
 
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) {
    auto rectMask = (sx == 0 || sy == 0) ? 0 : kRectStaysRect_Mask;
    *this = SkMatrix(sx, 0,  0,
                     0,  sy, 0,
                     0,  0,  1,
                     (sx == 1 && sy == 1) ? kIdentity_Mask | rectMask
                                          : kScale_Mask    | rectMask);
    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);
        // Remove kRectStaysRect if the preScale factors were 0
        if (!sx || !sy) {
            this->clearTypeMask(kRectStaysRect_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::setRSXform(const SkRSXform& xform) {
    fMat[kMScaleX]  = xform.fSCos;
    fMat[kMSkewX]   = -xform.fSSin;
    fMat[kMTransX]  = xform.fTx;
 
    fMat[kMSkewY]   = xform.fSSin;
    fMat[kMScaleY]  = xform.fSCos;
    fMat[kMTransY]  = xform.fTy;
 
    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 = SkDegreesToRadians(degrees);
    return this->setSinCos(SkScalarSinSnapToZero(rad), SkScalarCosSnapToZero(rad), px, py);
}
 
SkMatrix& SkMatrix::setRotate(SkScalar degrees) {
    SkScalar rad = SkDegreesToRadians(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);
    } else {
        SkScalar    tx, sx = sk_ieee_float_divide(dst.width(), src.width());
        SkScalar    ty, sy = sk_ieee_float_divide(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 = SkScalarHalf(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 sk_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;
            //SkDebugf("Concat mat non-persp type: %d\n", tmp.getType());
            //SkASSERT(!(tmp.getType() & kPerspective_Mask));
            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 SkDoubleToScalar(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 SkDoubleToScalar(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(sk_double_to_float(det),
                           SK_ScalarNearlyZero * SK_ScalarNearlyZero * SK_ScalarNearlyZero)) {
        return 0;
    }
    return 1.0 / det;
}
 
void SkMatrix::SetAffineIdentity(SkScalar affine[6]) {
    affine[kAScaleX] = 1;
    affine[kASkewY] = 0;
    affine[kASkewX] = 0;
    affine[kAScaleY] = 1;
    affine[kATransX] = 0;
    affine[kATransY] = 0;
}
 
bool SkMatrix::asAffine(SkScalar affine[6]) const {
    if (this->hasPerspective()) {
        return false;
    }
    if (affine) {
        affine[kAScaleX] = this->fMat[kMScaleX];
        affine[kASkewY] = this->fMat[kMSkewY];
        affine[kASkewX] = this->fMat[kMSkewX];
        affine[kAScaleY] = this->fMat[kMScaleY];
        affine[kATransX] = this->fMat[kMTransX];
        affine[kATransY] = this->fMat[kMTransY];
    }
    return true;
}
 
void SkMatrix::mapPoints(SkPoint dst[], const SkPoint src[], int count) const {
    SkASSERT((dst && src && count > 0) || 0 == count);
    // no partial overlap
    SkASSERT(src == dst || &dst[count] <= &src[0] || &src[count] <= &dst[0]);
    this->getMapPtsProc()(*this, dst, src, count);
}
 
void SkMatrix::mapXY(SkScalar x, SkScalar y, SkPoint* result) const {
    SkASSERT(result);
    this->getMapXYProc()(*this, x, y, result);
}
 
void SkMatrix::ComputeInv(SkScalar dst[9], const SkScalar src[9], double invDet, bool isPersp) {
    SkASSERT(src != dst);
    SkASSERT(src && dst);
 
    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] = SkDoubleToScalar(src[kMScaleY] * invDet);
        dst[kMSkewX]  = SkDoubleToScalar(-src[kMSkewX] * invDet);
        dst[kMTransX] = dcross_dscale(src[kMSkewX], src[kMTransY], src[kMScaleY], src[kMTransX], invDet);
 
        dst[kMSkewY]  = SkDoubleToScalar(-src[kMSkewY] * invDet);
        dst[kMScaleY] = SkDoubleToScalar(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 {
    SkASSERT(!this->isIdentity());
 
    TypeMask mask = this->getType();
 
    if (0 == (mask & ~(kScale_Mask | kTranslate_Mask))) {
        bool invertible = true;
        if (inv) {
            if (mask & kScale_Mask) {
                SkScalar invX = sk_ieee_float_divide(1.f, fMat[kMScaleX]);
                SkScalar invY = sk_ieee_float_divide(1.f, fMat[kMScaleY]);
                // Denormalized (non-zero) scale factors will overflow when inverted, in which case
                // the inverse matrix would not be finite, so return false.
                if (!SkIsFinite(invX, invY)) {
                    return false;
                }
 
                // 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) {
    SkASSERT(m.getType() == 0);
 
    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) {
    SkASSERT(m.getType() <= SkMatrix::kTranslate_Mask);
    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;
        }
        skvx::float4 trans4(tx, ty, tx, ty);
        count >>= 1;
        if (count & 1) {
            (skvx::float4::Load(src) + trans4).store(dst);
            src += 2;
            dst += 2;
        }
        count >>= 1;
        for (int i = 0; i < count; ++i) {
            (skvx::float4::Load(src+0) + trans4).store(dst+0);
            (skvx::float4::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) {
    SkASSERT(m.getType() <= (SkMatrix::kScale_Mask | SkMatrix::kTranslate_Mask));
    if (count > 0) {
        SkScalar tx = m.getTranslateX();
        SkScalar ty = m.getTranslateY();
        SkScalar sx = m.getScaleX();
        SkScalar sy = m.getScaleY();
        skvx::float4 trans4(tx, ty, tx, ty);
        skvx::float4 scale4(sx, sy, sx, sy);
        if (count & 1) {
            skvx::float4 p(src->fX, src->fY, 0, 0);
            p = p * scale4 + trans4;
            dst->fX = p[0];
            dst->fY = p[1];
            src += 1;
            dst += 1;
        }
        count >>= 1;
        if (count & 1) {
            (skvx::float4::Load(src) * scale4 + trans4).store(dst);
            src += 2;
            dst += 2;
        }
        count >>= 1;
        for (int i = 0; i < count; ++i) {
            (skvx::float4::Load(src+0) * scale4 + trans4).store(dst+0);
            (skvx::float4::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) {
    SkASSERT(m.hasPerspective());
 
    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];
            SkScalar z = sdot(sx, m.fMat[kMPersp0], sy, m.fMat[kMPersp1]) + m.fMat[kMPersp2];
            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) {
    SkASSERT(m.getType() != SkMatrix::kPerspective_Mask);
    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();
        skvx::float4 trans4(tx, ty, tx, ty);
        skvx::float4 scale4(sx, sy, sx, sy);
        skvx::float4  skew4(kx, ky, kx, ky);    // applied to swizzle of src4
        bool trailingElement = (count & 1);
        count >>= 1;
        skvx::float4 src4;
        for (int i = 0; i < count; ++i) {
            src4 = skvx::float4::Load(src);
            skvx::float4 swz4 = skvx::shuffle<1,0,3,2>(src4);  // y0 x0, y1 x1
            (src4 * scale4 + swz4 * skew4 + trans4).store(dst);
            src += 2;
            dst += 2;
        }
        if (trailingElement) {
            // We use the same logic here to ensure that the math stays consistent throughout, even
            // though the high float2 is ignored.
            src4.lo = skvx::float2::Load(src);
            skvx::float4 swz4 = skvx::shuffle<1,0,3,2>(src4);  // y0 x0, y1 x1
            (src4 * scale4 + swz4 * skew4 + trans4).lo.store(dst);
        }
    }
}
 
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) {
    SkASSERT((dst && src && count > 0) || 0 == count);
    // no partial overlap
    SkASSERT(src == dst || &dst[count] <= &src[0] || &src[count] <= &dst[0]);
 
    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,
            };
        }
    }
}
 
///////////////////////////////////////////////////////////////////////////////
 
void SkMatrix::mapVectors(SkPoint dst[], const SkPoint src[], int count) const {
    if (this->hasPerspective()) {
        SkPoint origin;
 
        MapXYProc proc = this->getMapXYProc();
        proc(*this, 0, 0, &origin);
 
        for (int i = count - 1; i >= 0; --i) {
            SkPoint tmp;
 
            proc(*this, src[i].fX, src[i].fY, &tmp);
            dst[i].set(tmp.fX - origin.fX, tmp.fY - origin.fY);
        }
    } else {
        SkMatrix tmp = *this;
 
        tmp.fMat[kMTransX] = tmp.fMat[kMTransY] = 0;
        tmp.clearTypeMask(kTranslate_Mask);
        tmp.mapPoints(dst, src, count);
    }
}
 
static skvx::float4 sort_as_rect(const skvx::float4& ltrb) {
    skvx::float4 rblt(ltrb[2], ltrb[3], ltrb[0], ltrb[1]);
    auto min = skvx::min(ltrb, rblt);
    auto max = skvx::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 skvx::float4(min[2], min[3], max[0], max[1]);
}
 
void SkMatrix::mapRectScaleTranslate(SkRect* dst, const SkRect& src) const {
    SkASSERT(dst);
    SkASSERT(this->isScaleTranslate());
 
    SkScalar sx = fMat[kMScaleX];
    SkScalar sy = fMat[kMScaleY];
    SkScalar tx = fMat[kMTransX];
    SkScalar ty = fMat[kMTransY];
    skvx::float4 scale(sx, sy, sx, sy);
    skvx::float4 trans(tx, ty, tx, ty);
    sort_as_rect(skvx::float4::Load(&src.fLeft) * scale + trans).store(&dst->fLeft);
}
 
bool SkMatrix::mapRect(SkRect* dst, const SkRect& src, SkApplyPerspectiveClip pc) const {
    SkASSERT(dst);
 
    if (this->getType() <= kTranslate_Mask) {
        SkScalar tx = fMat[kMTransX];
        SkScalar ty = fMat[kMTransY];
        skvx::float4 trans(tx, ty, tx, ty);
        sort_as_rect(skvx::float4::Load(&src.fLeft) + trans).store(&dst->fLeft);
        return true;
    }
    if (this->isScaleTranslate()) {
        this->mapRectScaleTranslate(dst, src);
        return true;
    } else if (pc == SkApplyPerspectiveClip::kYes && this->hasPerspective()) {
        SkPath path;
        path.addRect(src);
        path.transform(*this);
        *dst = path.getBounds();
        return false;
    } 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.
    }
}
 
SkScalar SkMatrix::mapRadius(SkScalar radius) const {
    SkVector    vec[2];
 
    vec[0].set(radius, 0);
    vec[1].set(0, radius);
    this->mapVectors(vec, 2);
 
    SkScalar d0 = vec[0].length();
    SkScalar d1 = vec[1].length();
 
    // return geometric mean
    return SkScalarSqrt(d0 * d1);
}
 
///////////////////////////////////////////////////////////////////////////////
 
void SkMatrix::Persp_xy(const SkMatrix& m, SkScalar sx, SkScalar sy,
                        SkPoint* pt) {
    SkASSERT(m.hasPerspective());
 
    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) {
    SkASSERT((m.getType() & (kAffine_Mask | kPerspective_Mask)) == kAffine_Mask);
 
    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];
}
 
void SkMatrix::Rot_xy(const SkMatrix& m, SkScalar sx, SkScalar sy,
                      SkPoint* pt) {
    SkASSERT((m.getType() & (kAffine_Mask | kPerspective_Mask))== kAffine_Mask);
    SkASSERT(0 == m.fMat[kMTransX]);
    SkASSERT(0 == m.fMat[kMTransY]);
 
    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];
}
 
void SkMatrix::ScaleTrans_xy(const SkMatrix& m, SkScalar sx, SkScalar sy,
                             SkPoint* pt) {
    SkASSERT((m.getType() & (kScale_Mask | kAffine_Mask | kPerspective_Mask))
             == kScale_Mask);
 
    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) {
    SkASSERT((m.getType() & (kScale_Mask | kAffine_Mask | kPerspective_Mask))
             == kScale_Mask);
    SkASSERT(0 == m.fMat[kMTransX]);
    SkASSERT(0 == m.fMat[kMTransY]);
 
    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) {
    SkASSERT(m.getType() == kTranslate_Mask);
 
    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) {
    SkASSERT(0 == m.getType());
 
    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
};
 
///////////////////////////////////////////////////////////////////////////////
#if 0
// if its nearly zero (just made up 26, perhaps it should be bigger or smaller)
#define PerspNearlyZero(x)  SkScalarNearlyZero(x, (1.0f / (1 << 26)))
 
bool SkMatrix::isFixedStepInX() const {
  return PerspNearlyZero(fMat[kMPersp0]);
}
 
SkVector SkMatrix::fixedStepInX(SkScalar y) const {
    SkASSERT(PerspNearlyZero(fMat[kMPersp0]));
    if (PerspNearlyZero(fMat[kMPersp1]) &&
        PerspNearlyZero(fMat[kMPersp2] - 1)) {
        return SkVector::Make(fMat[kMScaleX], fMat[kMSkewY]);
    } else {
        SkScalar z = y * fMat[kMPersp1] + fMat[kMPersp2];
        return SkVector::Make(fMat[kMScaleX] / z, fMat[kMSkewY] / z);
    }
}
#endif
 
///////////////////////////////////////////////////////////////////////////////
 
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;
}
 
typedef bool (*PolyMapProc)(const SkPoint[], SkMatrix*);
 
/*  Adapted from Rob Johnson's original sample code in QuickDraw GX
*/
bool SkMatrix::setPolyToPoly(const SkPoint src[], const SkPoint dst[], int count) {
    if ((unsigned)count > 4) {
        SkDebugf("--- SkMatrix::setPolyToPoly count out of range %d\n", count);
        return false;
    }
 
    if (0 == count) {
        this->reset();
        return true;
    }
    if (1 == count) {
        this->setTranslate(dst[0].fX - src[0].fX, dst[0].fY - src[0].fY);
        return true;
    }
 
    const PolyMapProc gPolyMapProcs[] = {
        SkMatrix::Poly2Proc, SkMatrix::Poly3Proc, SkMatrix::Poly4Proc
    };
    PolyMapProc proc = gPolyMapProcs[count - 2];
 
    SkMatrix tempMap, result;
 
    if (!proc(src, &tempMap)) {
        return false;
    }
    if (!tempMap.invert(&result)) {
        return false;
    }
    if (!proc(dst, &tempMap)) {
        return false;
    }
    this->setConcat(tempMap, result);
    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] = SK_Scalar1;
        if (kBoth_MinMaxOrBoth == MIN_MAX_OR_BOTH) {
            results[1] = SK_Scalar1;
        }
        return true;
    }
    if (!(typeMask & SkMatrix::kAffine_Mask)) {
        if (kMin_MinMaxOrBoth == MIN_MAX_OR_BOTH) {
             results[0] = std::min(SkScalarAbs(m[SkMatrix::kMScaleX]),
                                   SkScalarAbs(m[SkMatrix::kMScaleY]));
        } else if (kMax_MinMaxOrBoth == MIN_MAX_OR_BOTH) {
             results[0] = std::max(SkScalarAbs(m[SkMatrix::kMScaleX]),
                                   SkScalarAbs(m[SkMatrix::kMScaleY]));
        } else {
            results[0] = SkScalarAbs(m[SkMatrix::kMScaleX]);
            results[1] = SkScalarAbs(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 <= SK_ScalarNearlyZero*SK_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 = SkScalarHalf(a + c);
        SkScalar x = SkScalarHalf(SkScalarSqrt(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 (!SkIsFinite(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] = SkScalarSqrt(results[0]);
    if (kBoth_MinMaxOrBoth == MIN_MAX_OR_BOTH) {
        if (!SkIsFinite(results[1])) {
            return false;
        }
        if (results[1] < 0) {
            results[1] = 0;
        }
        results[1] = SkScalarSqrt(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;
    SkASSERT(identity.isIdentity());
    return identity;
}
 
const SkMatrix& SkMatrix::InvalidMatrix() {
    static constexpr SkMatrix invalid(SK_ScalarMax, SK_ScalarMax, SK_ScalarMax,
                                      SK_ScalarMax, SK_ScalarMax, SK_ScalarMax,
                                      SK_ScalarMax, SK_ScalarMax, SK_ScalarMax,
                                      kTranslate_Mask | kScale_Mask |
                                      kAffine_Mask | kPerspective_Mask);
    return invalid;
}
 
bool SkMatrix::decomposeScale(SkSize* scale, SkMatrix* remaining) const {
    if (this->hasPerspective()) {
        return false;
    }
 
    const SkScalar sx = SkVector::Length(this->getScaleX(), this->getSkewY());
    const SkScalar sy = SkVector::Length(this->getSkewX(), this->getScaleY());
    if (!SkIsFinite(sx, sy) ||
        SkScalarNearlyZero(sx) || SkScalarNearlyZero(sy)) {
        return false;
    }
 
    if (scale) {
        scale->set(sx, sy);
    }
    if (remaining) {
        *remaining = *this;
        remaining->preScale(SkScalarInvert(sx), SkScalarInvert(sy));
    }
    return true;
}
 
///////////////////////////////////////////////////////////////////////////////
 
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;
}
 
void SkMatrix::dump() const {
    SkString str;
    str.appendf("[%8.4f %8.4f %8.4f][%8.4f %8.4f %8.4f][%8.4f %8.4f %8.4f]",
             fMat[0], fMat[1], fMat[2], fMat[3], fMat[4], fMat[5],
             fMat[6], fMat[7], fMat[8]);
    SkDebugf("%s\n", str.c_str());
}
 
///////////////////////////////////////////////////////////////////////////////
 
bool SkTreatAsSprite(const SkMatrix& mat, const SkISize& size, const SkSamplingOptions& sampling,
                     bool isAntiAlias) {
    if (!SkSamplingPriv::NoChangeWithIdentityMatrix(sampling)) {
        return false;
    }
 
    // Our path aa is 2-bits, and our rect aa is 8, so we could use 8,
    // but in practice 4 seems enough (still looks smooth) and allows
    // more slightly fractional cases to fall into the fast (sprite) case.
    static const unsigned kAntiAliasSubpixelBits = 4;
 
    const unsigned subpixelBits = isAntiAlias ? kAntiAliasSubpixelBits : 0;
 
    // quick reject on affine or perspective
    if (mat.getType() & ~(SkMatrix::kScale_Mask | SkMatrix::kTranslate_Mask)) {
        return false;
    }
 
    // We don't want to snap to pixels if we're asking for linear filtering with
    // a subpixel translation. (b/41322892).
    // This mirrors `tweak_sampling` in SkImageShader.cpp
    if (sampling.filter == SkFilterMode::kLinear &&
        (mat.getTranslateX() != (int)mat.getTranslateX() ||
         mat.getTranslateY() != (int)mat.getTranslateY())) {
        return false;
    }
 
    // quick success check
    if (!subpixelBits && !(mat.getType() & ~SkMatrix::kTranslate_Mask)) {
        return true;
    }
 
    // mapRect supports negative scales, so we eliminate those first
    if (mat.getScaleX() < 0 || mat.getScaleY() < 0) {
        return false;
    }
 
    SkRect dst;
    SkIRect isrc = SkIRect::MakeSize(size);
 
    {
        SkRect src;
        src.set(isrc);
        mat.mapRect(&dst, src);
    }
 
    // just apply the translate to isrc
    isrc.offset(SkScalarRoundToInt(mat.getTranslateX()),
                SkScalarRoundToInt(mat.getTranslateY()));
 
    if (subpixelBits) {
        isrc.fLeft = SkLeftShift(isrc.fLeft, subpixelBits);
        isrc.fTop = SkLeftShift(isrc.fTop, subpixelBits);
        isrc.fRight = SkLeftShift(isrc.fRight, subpixelBits);
        isrc.fBottom = SkLeftShift(isrc.fBottom, subpixelBits);
 
        const float scale = 1 << subpixelBits;
        dst.fLeft *= scale;
        dst.fTop *= scale;
        dst.fRight *= scale;
        dst.fBottom *= scale;
    }
 
    SkIRect idst;
    dst.round(&idst);
    return isrc == idst;
}
 
// 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 = SkScalarInvert(SkScalarSqrt(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(SkDoubleToScalar(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 = SkDoubleToScalar(Sb); sin1 = SkDoubleToScalar(w1 - Sa);
        SkScalar reciplen = SkScalarInvert(SkScalarSqrt(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 = SkDoubleToScalar(w1);
        scale->fY = SkDoubleToScalar(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 < SK_ScalarNearlyZero) {
        // Reaching the discontinuity of xy/w and where the point would clip to w >= 0
        return SK_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 SkScalarAbs(SkDoubleToScalar(sk_determinant(jacobian.fMat, true) * denom));
}
 
bool SkMatrixPriv::NearlyAffine(const SkMatrix& m,
                                const SkRect& bounds,
                                SkScalar tolerance) {
    if (!m.hasPerspective()) {
        return true;
    }
 
    // The idea here is that we are computing the differential area scale at each corner,
    // and comparing them with some tolerance value. If they are similar, then we can say
    // that the transformation is nearly affine.
 
    // We can map the four points simultaneously.
    SkPoint quad[4];
    bounds.toQuad(quad);
    SkPoint3 xyw[4];
    m.mapHomogeneousPoints(xyw, quad, 4);
 
    // Since the Jacobian is a 3x3 matrix, the determinant is a scalar triple product,
    // and the initial cross product is constant across all four points.
    SkPoint3 v1{m.getScaleX(), m.getSkewY(), m.getPerspX()};
    SkPoint3 v2{m.getSkewX(), m.getScaleY(), m.getPerspY()};
    SkPoint3 detCrossProd = v1.cross(v2);
 
    // Start with the calculations at P0.
    if (xyw[0].fZ < SK_ScalarNearlyZero) {
        // Reaching the discontinuity of xy/w and where the point would clip to w >= 0
        return false;
    }
 
    // Performing a dot product with the pre-w divide transformed point completes
    // the scalar triple product and the determinant calculation.
    double det = detCrossProd.dot(xyw[0]);
    // From that we can compute the differential area scale at P0.
    double denom = 1.0 / xyw[0].fZ;   // 1/w
    denom = denom * denom * denom; // 1/w^3
    SkScalar a0 = SkScalarAbs(SkDoubleToScalar(det*denom));
 
    // Now we compare P0's scale with that at the other three points
    tolerance *= tolerance; // squared tolerance since we're comparing area
    for (int i = 1; i < 4; ++i) {
        if (xyw[i].fZ < SK_ScalarNearlyZero) {
            // Reaching the discontinuity of xy/w and where the point would clip to w >= 0
            return false;
        }
 
        det = detCrossProd.dot(xyw[i]);  // completing scalar triple product
        denom = 1.0 / xyw[i].fZ;   // 1/w
        denom = denom * denom * denom; // 1/w^3
        SkScalar a = SkScalarAbs(SkDoubleToScalar(det*denom));
        if (!SkScalarNearlyEqual(a0, a, tolerance)) {
            return false;
        }
    }
 
    return true;
}
 
SkScalar SkMatrixPriv::ComputeResScaleForStroking(const SkMatrix& matrix) {
    // Not sure how to handle perspective differently, so we just don't try (yet)
    SkScalar sx = SkPoint::Length(matrix[SkMatrix::kMScaleX], matrix[SkMatrix::kMSkewY]);
    SkScalar sy = SkPoint::Length(matrix[SkMatrix::kMSkewX],  matrix[SkMatrix::kMScaleY]);
    if (SkIsFinite(sx, sy)) {
        SkScalar scale = std::max(sx, sy);
        if (scale > 0) {
            return scale;
        }
    }
    return 1;
}