TransformationMatrix.cpp   [plain text]

/*
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 * Copyright (C) 2009 Torch Mobile, Inc.
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#include "config.h"
#include "TransformationMatrix.h"

#include "AffineTransform.h"
#include "FloatRect.h"
#include "FloatQuad.h"
#include "IntRect.h"
#include "LayoutRect.h"
#include "TextStream.h"
#include <wtf/Assertions.h>
#include <wtf/MathExtras.h>

#if CPU(X86_64)
#include <emmintrin.h>
#endif

namespace WebCore {

//
// Supporting Math Functions
//
// This is a set of function from various places (attributed inline) to do things like
// inversion and decomposition of a 4x4 matrix. They are used throughout the code
//

//
// Adapted from Matrix Inversion by Richard Carling, Graphics Gems <http://tog.acm.org/GraphicsGems/index.html>.

// EULA: The Graphics Gems code is copyright-protected. In other words, you cannot claim the text of the code
// as your own and resell it. Using the code is permitted in any program, product, or library, non-commercial
// or commercial. Giving credit is not required, though is a nice gesture. The code comes as-is, and if there
// are any flaws or problems with any Gems code, nobody involved with Gems - authors, editors, publishers, or
// webmasters - are to be held responsible. Basically, don't be a jerk, and remember that anything free comes
// with no guarantee.

// A clarification about the storage of matrix elements
//
// This class uses a 2 dimensional array internally to store the elements of the matrix. The first index into
// the array refers to the column that the element lies in; the second index refers to the row.
//
// In other words, this is the layout of the matrix:
//
// | m_matrix[0][0] m_matrix[1][0] m_matrix[2][0] m_matrix[3][0] |
// | m_matrix[0][1] m_matrix[1][1] m_matrix[2][1] m_matrix[3][1] |
// | m_matrix[0][2] m_matrix[1][2] m_matrix[2][2] m_matrix[3][2] |
// | m_matrix[0][3] m_matrix[1][3] m_matrix[2][3] m_matrix[3][3] |

typedef double Vector4[4];
typedef double Vector3[3];

const double SMALL_NUMBER = 1.e-8;

// inverse(original_matrix, inverse_matrix)
//
// calculate the inverse of a 4x4 matrix
//
// -1
// A  = ___1__ adjoint A
//       det A

//  double = determinant2x2(double a, double b, double c, double d)
//
//  calculate the determinant of a 2x2 matrix.

static double determinant2x2(double a, double b, double c, double d)
{
    return a * d - b * c;
}

//  double = determinant3x3(a1, a2, a3, b1, b2, b3, c1, c2, c3)
//
//  Calculate the determinant of a 3x3 matrix
//  in the form
//
//      | a1,  b1,  c1 |
//      | a2,  b2,  c2 |
//      | a3,  b3,  c3 |

static double determinant3x3(double a1, double a2, double a3, double b1, double b2, double b3, double c1, double c2, double c3)
{
    return a1 * determinant2x2(b2, b3, c2, c3)
         - b1 * determinant2x2(a2, a3, c2, c3)
         + c1 * determinant2x2(a2, a3, b2, b3);
}

//  double = determinant4x4(matrix)
//
//  calculate the determinant of a 4x4 matrix.

static double determinant4x4(const TransformationMatrix::Matrix4& m)
{
    // Assign to individual variable names to aid selecting
    // correct elements

    double a1 = m[0][0];
    double b1 = m[0][1];
    double c1 = m[0][2];
    double d1 = m[0][3];

    double a2 = m[1][0];
    double b2 = m[1][1];
    double c2 = m[1][2];
    double d2 = m[1][3];

    double a3 = m[2][0];
    double b3 = m[2][1];
    double c3 = m[2][2];
    double d3 = m[2][3];

    double a4 = m[3][0];
    double b4 = m[3][1];
    double c4 = m[3][2];
    double d4 = m[3][3];

    return a1 * determinant3x3(b2, b3, b4, c2, c3, c4, d2, d3, d4)
         - b1 * determinant3x3(a2, a3, a4, c2, c3, c4, d2, d3, d4)
         + c1 * determinant3x3(a2, a3, a4, b2, b3, b4, d2, d3, d4)
         - d1 * determinant3x3(a2, a3, a4, b2, b3, b4, c2, c3, c4);
}

// adjoint( original_matrix, inverse_matrix )
//
//   calculate the adjoint of a 4x4 matrix
//
//    Let  a   denote the minor determinant of matrix A obtained by
//         ij
//
//    deleting the ith row and jth column from A.
//
//                  i+j
//   Let  b   = (-1)    a
//        ij            ji
//
//  The matrix B = (b  ) is the adjoint of A
//                   ij

static void adjoint(const TransformationMatrix::Matrix4& matrix, TransformationMatrix::Matrix4& result)
{
    // Assign to individual variable names to aid
    // selecting correct values
    double a1 = matrix[0][0];
    double b1 = matrix[0][1];
    double c1 = matrix[0][2];
    double d1 = matrix[0][3];

    double a2 = matrix[1][0];
    double b2 = matrix[1][1];
    double c2 = matrix[1][2];
    double d2 = matrix[1][3];

    double a3 = matrix[2][0];
    double b3 = matrix[2][1];
    double c3 = matrix[2][2];
    double d3 = matrix[2][3];

    double a4 = matrix[3][0];
    double b4 = matrix[3][1];
    double c4 = matrix[3][2];
    double d4 = matrix[3][3];

    // Row column labeling reversed since we transpose rows & columns
    result[0][0]  =   determinant3x3(b2, b3, b4, c2, c3, c4, d2, d3, d4);
    result[1][0]  = - determinant3x3(a2, a3, a4, c2, c3, c4, d2, d3, d4);
    result[2][0]  =   determinant3x3(a2, a3, a4, b2, b3, b4, d2, d3, d4);
    result[3][0]  = - determinant3x3(a2, a3, a4, b2, b3, b4, c2, c3, c4);

    result[0][1]  = - determinant3x3(b1, b3, b4, c1, c3, c4, d1, d3, d4);
    result[1][1]  =   determinant3x3(a1, a3, a4, c1, c3, c4, d1, d3, d4);
    result[2][1]  = - determinant3x3(a1, a3, a4, b1, b3, b4, d1, d3, d4);
    result[3][1]  =   determinant3x3(a1, a3, a4, b1, b3, b4, c1, c3, c4);

    result[0][2]  =   determinant3x3(b1, b2, b4, c1, c2, c4, d1, d2, d4);
    result[1][2]  = - determinant3x3(a1, a2, a4, c1, c2, c4, d1, d2, d4);
    result[2][2]  =   determinant3x3(a1, a2, a4, b1, b2, b4, d1, d2, d4);
    result[3][2]  = - determinant3x3(a1, a2, a4, b1, b2, b4, c1, c2, c4);

    result[0][3]  = - determinant3x3(b1, b2, b3, c1, c2, c3, d1, d2, d3);
    result[1][3]  =   determinant3x3(a1, a2, a3, c1, c2, c3, d1, d2, d3);
    result[2][3]  = - determinant3x3(a1, a2, a3, b1, b2, b3, d1, d2, d3);
    result[3][3]  =   determinant3x3(a1, a2, a3, b1, b2, b3, c1, c2, c3);
}

// Returns false if the matrix is not invertible
static bool inverse(const TransformationMatrix::Matrix4& matrix, TransformationMatrix::Matrix4& result)
{
    // Calculate the adjoint matrix
    adjoint(matrix, result);

    // Calculate the 4x4 determinant
    // If the determinant is zero,
    // then the inverse matrix is not unique.
    double det = determinant4x4(matrix);

    if (fabs(det) < SMALL_NUMBER)
        return false;

    // Scale the adjoint matrix to get the inverse

    for (int i = 0; i < 4; i++)
        for (int j = 0; j < 4; j++)
            result[i][j] = result[i][j] / det;

    return true;
}

// End of code adapted from Matrix Inversion by Richard Carling

// Perform a decomposition on the passed matrix, return false if unsuccessful
// From Graphics Gems: unmatrix.c

// Transpose rotation portion of matrix a, return b
static void transposeMatrix4(const TransformationMatrix::Matrix4& a, TransformationMatrix::Matrix4& b)
{
    for (int i = 0; i < 4; i++)
        for (int j = 0; j < 4; j++)
            b[i][j] = a[j][i];
}

// Multiply a homogeneous point by a matrix and return the transformed point
static void v4MulPointByMatrix(const Vector4 p, const TransformationMatrix::Matrix4& m, Vector4 result)
{
    result[0] = (p[0] * m[0][0]) + (p[1] * m[1][0]) +
                (p[2] * m[2][0]) + (p[3] * m[3][0]);
    result[1] = (p[0] * m[0][1]) + (p[1] * m[1][1]) +
                (p[2] * m[2][1]) + (p[3] * m[3][1]);
    result[2] = (p[0] * m[0][2]) + (p[1] * m[1][2]) +
                (p[2] * m[2][2]) + (p[3] * m[3][2]);
    result[3] = (p[0] * m[0][3]) + (p[1] * m[1][3]) +
                (p[2] * m[2][3]) + (p[3] * m[3][3]);
}

static double v3Length(Vector3 a)
{
    return sqrt((a[0] * a[0]) + (a[1] * a[1]) + (a[2] * a[2]));
}

static void v3Scale(Vector3 v, double desiredLength)
{
    double len = v3Length(v);
    if (len != 0) {
        double l = desiredLength / len;
        v[0] *= l;
        v[1] *= l;
        v[2] *= l;
    }
}

static double v3Dot(const Vector3 a, const Vector3 b)
{
    return (a[0] * b[0]) + (a[1] * b[1]) + (a[2] * b[2]);
}

// Make a linear combination of two vectors and return the result.
// result = (a * ascl) + (b * bscl)
static void v3Combine(const Vector3 a, const Vector3 b, Vector3 result, double ascl, double bscl)
{
    result[0] = (ascl * a[0]) + (bscl * b[0]);
    result[1] = (ascl * a[1]) + (bscl * b[1]);
    result[2] = (ascl * a[2]) + (bscl * b[2]);
}

// Return the cross product result = a cross b */
static void v3Cross(const Vector3 a, const Vector3 b, Vector3 result)
{
    result[0] = (a[1] * b[2]) - (a[2] * b[1]);
    result[1] = (a[2] * b[0]) - (a[0] * b[2]);
    result[2] = (a[0] * b[1]) - (a[1] * b[0]);
}

static bool decompose2(const TransformationMatrix::Matrix4& matrix, TransformationMatrix::Decomposed2Type& result)
{
    double row0x = matrix[0][0];
    double row0y = matrix[0][1];
    double row1x = matrix[1][0];
    double row1y = matrix[1][1];
    result.translateX = matrix[3][0];
    result.translateY = matrix[3][1];

    // Compute scaling factors.
    result.scaleX = sqrt(row0x * row0x + row0y * row0y);
    result.scaleY = sqrt(row1x * row1x + row1y * row1y);

    // If determinant is negative, one axis was flipped.
    double determinant = row0x * row1y - row0y * row1x;
    if (determinant < 0) {
        // Flip axis with minimum unit vector dot product.
        if (row0x < row1y)
            result.scaleX = -result.scaleX;
        else
            result.scaleY = -result.scaleY;
    }

    // Renormalize matrix to remove scale.
    if (result.scaleX) {
        row0x *= 1 / result.scaleX;
        row0y *= 1 / result.scaleX;
    }
    if (result.scaleY) {
        row1x *= 1 / result.scaleY;
        row1y *= 1 / result.scaleY;
    }

    // Compute rotation and renormalize matrix.
    result.angle = atan2(row0y, row0x);

    if (result.angle) {
        // Rotate(-angle) = [cos(angle), sin(angle), -sin(angle), cos(angle)]
        //                = [row0x, -row0y, row0y, row0x]
        // Thanks to the normalization above.
        double sn = -row0y;
        double cs = row0x;
        double m11 = row0x, m12 = row0y;
        double m21 = row1x, m22 = row1y;

        row0x = cs * m11 + sn * m21;
        row0y = cs * m12 + sn * m22;
        row1x = -sn * m11 + cs * m21;
        row1y = -sn * m12 + cs * m22;
    }

    result.m11 = row0x;
    result.m12 = row0y;
    result.m21 = row1x;
    result.m22 = row1y;

    // Convert into degrees because our rotation functions expect it.
    result.angle = rad2deg(result.angle);

    return true;
}

static bool decompose4(const TransformationMatrix::Matrix4& mat, TransformationMatrix::Decomposed4Type& result)
{
    TransformationMatrix::Matrix4 localMatrix;
    memcpy(localMatrix, mat, sizeof(TransformationMatrix::Matrix4));

    // Normalize the matrix.
    if (localMatrix[3][3] == 0)
        return false;

    int i, j;
    for (i = 0; i < 4; i++)
        for (j = 0; j < 4; j++)
            localMatrix[i][j] /= localMatrix[3][3];

    // perspectiveMatrix is used to solve for perspective, but it also provides
    // an easy way to test for singularity of the upper 3x3 component.
    TransformationMatrix::Matrix4 perspectiveMatrix;
    memcpy(perspectiveMatrix, localMatrix, sizeof(TransformationMatrix::Matrix4));
    for (i = 0; i < 3; i++)
        perspectiveMatrix[i][3] = 0;
    perspectiveMatrix[3][3] = 1;

    if (determinant4x4(perspectiveMatrix) == 0)
        return false;

    // First, isolate perspective. This is the messiest.
    if (localMatrix[0][3] != 0 || localMatrix[1][3] != 0 || localMatrix[2][3] != 0) {
        // rightHandSide is the right hand side of the equation.
        Vector4 rightHandSide;
        rightHandSide[0] = localMatrix[0][3];
        rightHandSide[1] = localMatrix[1][3];
        rightHandSide[2] = localMatrix[2][3];
        rightHandSide[3] = localMatrix[3][3];

        // Solve the equation by inverting perspectiveMatrix and multiplying
        // rightHandSide by the inverse. (This is the easiest way, not
        // necessarily the best.)
        TransformationMatrix::Matrix4 inversePerspectiveMatrix, transposedInversePerspectiveMatrix;
        inverse(perspectiveMatrix, inversePerspectiveMatrix);
        transposeMatrix4(inversePerspectiveMatrix, transposedInversePerspectiveMatrix);

        Vector4 perspectivePoint;
        v4MulPointByMatrix(rightHandSide, transposedInversePerspectiveMatrix, perspectivePoint);

        result.perspectiveX = perspectivePoint[0];
        result.perspectiveY = perspectivePoint[1];
        result.perspectiveZ = perspectivePoint[2];
        result.perspectiveW = perspectivePoint[3];

        // Clear the perspective partition
        localMatrix[0][3] = localMatrix[1][3] = localMatrix[2][3] = 0;
        localMatrix[3][3] = 1;
    } else {
        // No perspective.
        result.perspectiveX = result.perspectiveY = result.perspectiveZ = 0;
        result.perspectiveW = 1;
    }

    // Next take care of translation (easy).
    result.translateX = localMatrix[3][0];
    localMatrix[3][0] = 0;
    result.translateY = localMatrix[3][1];
    localMatrix[3][1] = 0;
    result.translateZ = localMatrix[3][2];
    localMatrix[3][2] = 0;

    // Vector4 type and functions need to be added to the common set.
    Vector3 row[3], pdum3;

    // Now get scale and shear.
    for (i = 0; i < 3; i++) {
        row[i][0] = localMatrix[i][0];
        row[i][1] = localMatrix[i][1];
        row[i][2] = localMatrix[i][2];
    }

    // Compute X scale factor and normalize first row.
    result.scaleX = v3Length(row[0]);
    v3Scale(row[0], 1.0);

    // Compute XY shear factor and make 2nd row orthogonal to 1st.
    result.skewXY = v3Dot(row[0], row[1]);
    v3Combine(row[1], row[0], row[1], 1.0, -result.skewXY);

    // Now, compute Y scale and normalize 2nd row.
    result.scaleY = v3Length(row[1]);
    v3Scale(row[1], 1.0);
    result.skewXY /= result.scaleY;

    // Compute XZ and YZ shears, orthogonalize 3rd row.
    result.skewXZ = v3Dot(row[0], row[2]);
    v3Combine(row[2], row[0], row[2], 1.0, -result.skewXZ);
    result.skewYZ = v3Dot(row[1], row[2]);
    v3Combine(row[2], row[1], row[2], 1.0, -result.skewYZ);

    // Next, get Z scale and normalize 3rd row.
    result.scaleZ = v3Length(row[2]);
    v3Scale(row[2], 1.0);
    result.skewXZ /= result.scaleZ;
    result.skewYZ /= result.scaleZ;

    // At this point, the matrix (in rows[]) is orthonormal.
    // Check for a coordinate system flip. If the determinant
    // is -1, then negate the matrix and the scaling factors.
    v3Cross(row[1], row[2], pdum3);
    if (v3Dot(row[0], pdum3) < 0) {

        result.scaleX *= -1;
        result.scaleY *= -1;
        result.scaleZ *= -1;

        for (i = 0; i < 3; i++) {
            row[i][0] *= -1;
            row[i][1] *= -1;
            row[i][2] *= -1;
        }
    }

    // Now, get the rotations out, as described in the gem.

    // FIXME - Add the ability to return either quaternions (which are
    // easier to recompose with) or Euler angles (rx, ry, rz), which
    // are easier for authors to deal with. The latter will only be useful
    // when we fix https://bugs.webkit.org/show_bug.cgi?id=23799, so I
    // will leave the Euler angle code here for now.

    // ret.rotateY = asin(-row[0][2]);
    // if (cos(ret.rotateY) != 0) {
    //     ret.rotateX = atan2(row[1][2], row[2][2]);
    //     ret.rotateZ = atan2(row[0][1], row[0][0]);
    // } else {
    //     ret.rotateX = atan2(-row[2][0], row[1][1]);
    //     ret.rotateZ = 0;
    // }

    double s, t, x, y, z, w;

    t = row[0][0] + row[1][1] + row[2][2] + 1.0;

    if (t > 1e-4) {
        s = 0.5 / sqrt(t);
        w = 0.25 / s;
        x = (row[2][1] - row[1][2]) * s;
        y = (row[0][2] - row[2][0]) * s;
        z = (row[1][0] - row[0][1]) * s;
    } else if (row[0][0] > row[1][1] && row[0][0] > row[2][2]) {
        s = sqrt(1.0 + row[0][0] - row[1][1] - row[2][2]) * 2.0; // S = 4 * qx.
        x = 0.25 * s;
        y = (row[0][1] + row[1][0]) / s;
        z = (row[0][2] + row[2][0]) / s;
        w = (row[2][1] - row[1][2]) / s;
    } else if (row[1][1] > row[2][2]) {
        s = sqrt(1.0 + row[1][1] - row[0][0] - row[2][2]) * 2.0; // S = 4 * qy.
        x = (row[0][1] + row[1][0]) / s;
        y = 0.25 * s;
        z = (row[1][2] + row[2][1]) / s;
        w = (row[0][2] - row[2][0]) / s;
    } else {
        s = sqrt(1.0 + row[2][2] - row[0][0] - row[1][1]) * 2.0; // S = 4 * qz.
        x = (row[0][2] + row[2][0]) / s;
        y = (row[1][2] + row[2][1]) / s;
        z = 0.25 * s;
        w = (row[1][0] - row[0][1]) / s;
    }

    result.quaternionX = x;
    result.quaternionY = y;
    result.quaternionZ = z;
    result.quaternionW = w;

    return true;
}

// Perform a spherical linear interpolation between the two
// passed quaternions with 0 <= t <= 1.
static void slerp(double qa[4], const double qb[4], double t)
{
    double ax, ay, az, aw;
    double bx, by, bz, bw;
    double cx, cy, cz, cw;
    double angle;
    double th, invth, scale, invscale;

    ax = qa[0]; ay = qa[1]; az = qa[2]; aw = qa[3];
    bx = qb[0]; by = qb[1]; bz = qb[2]; bw = qb[3];

    angle = ax * bx + ay * by + az * bz + aw * bw;

    if (angle < 0.0) {
        ax = -ax; ay = -ay;
        az = -az; aw = -aw;
        angle = -angle;
    }

    if (angle + 1.0 > .05) {
        if (1.0 - angle >= .05) {
            th = acos (angle);
            invth = 1.0 / sin (th);
            scale = sin (th * (1.0 - t)) * invth;
            invscale = sin (th * t) * invth;
        } else {
            scale = 1.0 - t;
            invscale = t;
        }
    } else {
        bx = -ay;
        by = ax;
        bz = -aw;
        bw = az;
        scale = sin(piDouble * (.5 - t));
        invscale = sin (piDouble * t);
    }

    cx = ax * scale + bx * invscale;
    cy = ay * scale + by * invscale;
    cz = az * scale + bz * invscale;
    cw = aw * scale + bw * invscale;

    qa[0] = cx; qa[1] = cy; qa[2] = cz; qa[3] = cw;
}

// End of Supporting Math Functions

TransformationMatrix::TransformationMatrix(const AffineTransform& t)
{
    setMatrix(t.a(), t.b(), t.c(), t.d(), t.e(), t.f());
}

TransformationMatrix& TransformationMatrix::scale(double s)
{
    return scaleNonUniform(s, s);
}

TransformationMatrix& TransformationMatrix::rotateFromVector(double x, double y)
{
    return rotate(rad2deg(atan2(y, x)));
}

TransformationMatrix& TransformationMatrix::flipX()
{
    return scaleNonUniform(-1.0, 1.0);
}

TransformationMatrix& TransformationMatrix::flipY()
{
    return scaleNonUniform(1.0, -1.0);
}

FloatPoint TransformationMatrix::projectPoint(const FloatPoint& p, bool* clamped) const
{
    // This is basically raytracing. We have a point in the destination
    // plane with z=0, and we cast a ray parallel to the z-axis from that
    // point to find the z-position at which it intersects the z=0 plane
    // with the transform applied. Once we have that point we apply the
    // inverse transform to find the corresponding point in the source
    // space.
    //
    // Given a plane with normal Pn, and a ray starting at point R0 and
    // with direction defined by the vector Rd, we can find the
    // intersection point as a distance d from R0 in units of Rd by:
    //
    // d = -dot (Pn', R0) / dot (Pn', Rd)
    if (clamped)
        *clamped = false;

    if (m33() == 0) {
        // In this case, the projection plane is parallel to the ray we are trying to
        // trace, and there is no well-defined value for the projection.
        return FloatPoint();
    }

    double x = p.x();
    double y = p.y();
    double z = -(m13() * x + m23() * y + m43()) / m33();

    // FIXME: use multVecMatrix()
    double outX = x * m11() + y * m21() + z * m31() + m41();
    double outY = x * m12() + y * m22() + z * m32() + m42();

    double w = x * m14() + y * m24() + z * m34() + m44();
    if (w <= 0) {
        // Using int max causes overflow when other code uses the projected point. To
        // represent infinity yet reduce the risk of overflow, we use a large but
        // not-too-large number here when clamping.
        const int largeNumber = 100000000 / kFixedPointDenominator;
        outX = copysign(largeNumber, outX);
        outY = copysign(largeNumber, outY);
        if (clamped)
            *clamped = true;
    } else if (w != 1) {
        outX /= w;
        outY /= w;
    }

    return FloatPoint(static_cast<float>(outX), static_cast<float>(outY));
}

FloatQuad TransformationMatrix::projectQuad(const FloatQuad& q, bool* clamped) const
{
    FloatQuad projectedQuad;

    bool clamped1 = false;
    bool clamped2 = false;
    bool clamped3 = false;
    bool clamped4 = false;

    projectedQuad.setP1(projectPoint(q.p1(), &clamped1));
    projectedQuad.setP2(projectPoint(q.p2(), &clamped2));
    projectedQuad.setP3(projectPoint(q.p3(), &clamped3));
    projectedQuad.setP4(projectPoint(q.p4(), &clamped4));

    if (clamped)
        *clamped = clamped1 || clamped2 || clamped3 || clamped4;

    // If all points on the quad had w < 0, then the entire quad would not be visible to the projected surface.
    bool everythingWasClipped = clamped1 && clamped2 && clamped3 && clamped4;
    if (everythingWasClipped)
        return FloatQuad();

    return projectedQuad;
}

static float clampEdgeValue(float f)
{
    ASSERT(!std::isnan(f));
    return std::min<float>(std::max<float>(f, -LayoutUnit::max() / 2), LayoutUnit::max() / 2);
}

LayoutRect TransformationMatrix::clampedBoundsOfProjectedQuad(const FloatQuad& q) const
{
    FloatRect mappedQuadBounds = projectQuad(q).boundingBox();

    float left = clampEdgeValue(floorf(mappedQuadBounds.x()));
    float top = clampEdgeValue(floorf(mappedQuadBounds.y()));

    float right;
    if (std::isinf(mappedQuadBounds.x()) && std::isinf(mappedQuadBounds.width()))
        right = LayoutUnit::max() / 2;
    else
        right = clampEdgeValue(ceilf(mappedQuadBounds.maxX()));

    float bottom;
    if (std::isinf(mappedQuadBounds.y()) && std::isinf(mappedQuadBounds.height()))
        bottom = LayoutUnit::max() / 2;
    else
        bottom = clampEdgeValue(ceilf(mappedQuadBounds.maxY()));

    return LayoutRect(LayoutUnit::clamp(left), LayoutUnit::clamp(top),  LayoutUnit::clamp(right - left), LayoutUnit::clamp(bottom - top));
}

FloatPoint TransformationMatrix::mapPoint(const FloatPoint& p) const
{
    if (isIdentityOrTranslation())
        return FloatPoint(p.x() + static_cast<float>(m_matrix[3][0]), p.y() + static_cast<float>(m_matrix[3][1]));

    return internalMapPoint(p);
}

FloatPoint3D TransformationMatrix::mapPoint(const FloatPoint3D& p) const
{
    if (isIdentityOrTranslation())
        return FloatPoint3D(p.x() + static_cast<float>(m_matrix[3][0]),
                            p.y() + static_cast<float>(m_matrix[3][1]),
                            p.z() + static_cast<float>(m_matrix[3][2]));

    return internalMapPoint(p);
}

IntRect TransformationMatrix::mapRect(const IntRect &rect) const
{
    return enclosingIntRect(mapRect(FloatRect(rect)));
}

LayoutRect TransformationMatrix::mapRect(const LayoutRect& r) const
{
    return enclosingLayoutRect(mapRect(FloatRect(r)));
}

FloatRect TransformationMatrix::mapRect(const FloatRect& r) const
{
    if (isIdentityOrTranslation()) {
        FloatRect mappedRect(r);
        mappedRect.move(static_cast<float>(m_matrix[3][0]), static_cast<float>(m_matrix[3][1]));
        return mappedRect;
    }

    FloatQuad result;

    float maxX = r.maxX();
    float maxY = r.maxY();
    result.setP1(internalMapPoint(FloatPoint(r.x(), r.y())));
    result.setP2(internalMapPoint(FloatPoint(maxX, r.y())));
    result.setP3(internalMapPoint(FloatPoint(maxX, maxY)));
    result.setP4(internalMapPoint(FloatPoint(r.x(), maxY)));

    return result.boundingBox();
}

FloatQuad TransformationMatrix::mapQuad(const FloatQuad& q) const
{
    if (isIdentityOrTranslation()) {
        FloatQuad mappedQuad(q);
        mappedQuad.move(static_cast<float>(m_matrix[3][0]), static_cast<float>(m_matrix[3][1]));
        return mappedQuad;
    }

    FloatQuad result;
    result.setP1(internalMapPoint(q.p1()));
    result.setP2(internalMapPoint(q.p2()));
    result.setP3(internalMapPoint(q.p3()));
    result.setP4(internalMapPoint(q.p4()));
    return result;
}

TransformationMatrix& TransformationMatrix::scaleNonUniform(double sx, double sy)
{
    m_matrix[0][0] *= sx;
    m_matrix[0][1] *= sx;
    m_matrix[0][2] *= sx;
    m_matrix[0][3] *= sx;

    m_matrix[1][0] *= sy;
    m_matrix[1][1] *= sy;
    m_matrix[1][2] *= sy;
    m_matrix[1][3] *= sy;
    return *this;
}

TransformationMatrix& TransformationMatrix::scale3d(double sx, double sy, double sz)
{
    scaleNonUniform(sx, sy);

    m_matrix[2][0] *= sz;
    m_matrix[2][1] *= sz;
    m_matrix[2][2] *= sz;
    m_matrix[2][3] *= sz;
    return *this;
}

TransformationMatrix& TransformationMatrix::rotate3d(double x, double y, double z, double angle)
{
    // Normalize the axis of rotation
    double length = sqrt(x * x + y * y + z * z);
    if (length == 0) {
        // A direction vector that cannot be normalized, such as [0, 0, 0], will cause the rotation to not be applied.
        return *this;
    } else if (length != 1) {
        x /= length;
        y /= length;
        z /= length;
    }

    // Angles are in degrees. Switch to radians.
    angle = deg2rad(angle);

    double sinTheta = sin(angle);
    double cosTheta = cos(angle);

    TransformationMatrix mat;

    // Optimize cases where the axis is along a major axis
    if (x == 1.0 && y == 0.0 && z == 0.0) {
        mat.m_matrix[0][0] = 1.0;
        mat.m_matrix[0][1] = 0.0;
        mat.m_matrix[0][2] = 0.0;
        mat.m_matrix[1][0] = 0.0;
        mat.m_matrix[1][1] = cosTheta;
        mat.m_matrix[1][2] = sinTheta;
        mat.m_matrix[2][0] = 0.0;
        mat.m_matrix[2][1] = -sinTheta;
        mat.m_matrix[2][2] = cosTheta;
        mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0;
        mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0;
        mat.m_matrix[3][3] = 1.0;
    } else if (x == 0.0 && y == 1.0 && z == 0.0) {
        mat.m_matrix[0][0] = cosTheta;
        mat.m_matrix[0][1] = 0.0;
        mat.m_matrix[0][2] = -sinTheta;
        mat.m_matrix[1][0] = 0.0;
        mat.m_matrix[1][1] = 1.0;
        mat.m_matrix[1][2] = 0.0;
        mat.m_matrix[2][0] = sinTheta;
        mat.m_matrix[2][1] = 0.0;
        mat.m_matrix[2][2] = cosTheta;
        mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0;
        mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0;
        mat.m_matrix[3][3] = 1.0;
    } else if (x == 0.0 && y == 0.0 && z == 1.0) {
        mat.m_matrix[0][0] = cosTheta;
        mat.m_matrix[0][1] = sinTheta;
        mat.m_matrix[0][2] = 0.0;
        mat.m_matrix[1][0] = -sinTheta;
        mat.m_matrix[1][1] = cosTheta;
        mat.m_matrix[1][2] = 0.0;
        mat.m_matrix[2][0] = 0.0;
        mat.m_matrix[2][1] = 0.0;
        mat.m_matrix[2][2] = 1.0;
        mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0;
        mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0;
        mat.m_matrix[3][3] = 1.0;
    } else {
        // This case is the rotation about an arbitrary unit vector.
        //
        // Formula is adapted from Wikipedia article on Rotation matrix,
        // http://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle
        //
        // An alternate resource with the same matrix: http://www.fastgraph.com/makegames/3drotation/
        //
        double oneMinusCosTheta = 1 - cosTheta;
        mat.m_matrix[0][0] = cosTheta + x * x * oneMinusCosTheta;
        mat.m_matrix[0][1] = y * x * oneMinusCosTheta + z * sinTheta;
        mat.m_matrix[0][2] = z * x * oneMinusCosTheta - y * sinTheta;
        mat.m_matrix[1][0] = x * y * oneMinusCosTheta - z * sinTheta;
        mat.m_matrix[1][1] = cosTheta + y * y * oneMinusCosTheta;
        mat.m_matrix[1][2] = z * y * oneMinusCosTheta + x * sinTheta;
        mat.m_matrix[2][0] = x * z * oneMinusCosTheta + y * sinTheta;
        mat.m_matrix[2][1] = y * z * oneMinusCosTheta - x * sinTheta;
        mat.m_matrix[2][2] = cosTheta + z * z * oneMinusCosTheta;
        mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0;
        mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0;
        mat.m_matrix[3][3] = 1.0;
    }
    multiply(mat);
    return *this;
}

TransformationMatrix& TransformationMatrix::rotate3d(double rx, double ry, double rz)
{
    // Angles are in degrees. Switch to radians.
    rx = deg2rad(rx);
    ry = deg2rad(ry);
    rz = deg2rad(rz);

    TransformationMatrix mat;

    double sinTheta = sin(rz);
    double cosTheta = cos(rz);

    mat.m_matrix[0][0] = cosTheta;
    mat.m_matrix[0][1] = sinTheta;
    mat.m_matrix[0][2] = 0.0;
    mat.m_matrix[1][0] = -sinTheta;
    mat.m_matrix[1][1] = cosTheta;
    mat.m_matrix[1][2] = 0.0;
    mat.m_matrix[2][0] = 0.0;
    mat.m_matrix[2][1] = 0.0;
    mat.m_matrix[2][2] = 1.0;
    mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0;
    mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0;
    mat.m_matrix[3][3] = 1.0;

    TransformationMatrix rmat(mat);

    sinTheta = sin(ry);
    cosTheta = cos(ry);

    mat.m_matrix[0][0] = cosTheta;
    mat.m_matrix[0][1] = 0.0;
    mat.m_matrix[0][2] = -sinTheta;
    mat.m_matrix[1][0] = 0.0;
    mat.m_matrix[1][1] = 1.0;
    mat.m_matrix[1][2] = 0.0;
    mat.m_matrix[2][0] = sinTheta;
    mat.m_matrix[2][1] = 0.0;
    mat.m_matrix[2][2] = cosTheta;
    mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0;
    mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0;
    mat.m_matrix[3][3] = 1.0;

    rmat.multiply(mat);

    sinTheta = sin(rx);
    cosTheta = cos(rx);

    mat.m_matrix[0][0] = 1.0;
    mat.m_matrix[0][1] = 0.0;
    mat.m_matrix[0][2] = 0.0;
    mat.m_matrix[1][0] = 0.0;
    mat.m_matrix[1][1] = cosTheta;
    mat.m_matrix[1][2] = sinTheta;
    mat.m_matrix[2][0] = 0.0;
    mat.m_matrix[2][1] = -sinTheta;
    mat.m_matrix[2][2] = cosTheta;
    mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0;
    mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0;
    mat.m_matrix[3][3] = 1.0;

    rmat.multiply(mat);

    multiply(rmat);
    return *this;
}

TransformationMatrix& TransformationMatrix::translate(double tx, double ty)
{
    m_matrix[3][0] += tx * m_matrix[0][0] + ty * m_matrix[1][0];
    m_matrix[3][1] += tx * m_matrix[0][1] + ty * m_matrix[1][1];
    m_matrix[3][2] += tx * m_matrix[0][2] + ty * m_matrix[1][2];
    m_matrix[3][3] += tx * m_matrix[0][3] + ty * m_matrix[1][3];
    return *this;
}

TransformationMatrix& TransformationMatrix::translate3d(double tx, double ty, double tz)
{
    m_matrix[3][0] += tx * m_matrix[0][0] + ty * m_matrix[1][0] + tz * m_matrix[2][0];
    m_matrix[3][1] += tx * m_matrix[0][1] + ty * m_matrix[1][1] + tz * m_matrix[2][1];
    m_matrix[3][2] += tx * m_matrix[0][2] + ty * m_matrix[1][2] + tz * m_matrix[2][2];
    m_matrix[3][3] += tx * m_matrix[0][3] + ty * m_matrix[1][3] + tz * m_matrix[2][3];
    return *this;
}

TransformationMatrix& TransformationMatrix::translateRight(double tx, double ty)
{
    if (tx != 0) {
        m_matrix[0][0] +=  m_matrix[0][3] * tx;
        m_matrix[1][0] +=  m_matrix[1][3] * tx;
        m_matrix[2][0] +=  m_matrix[2][3] * tx;
        m_matrix[3][0] +=  m_matrix[3][3] * tx;
    }

    if (ty != 0) {
        m_matrix[0][1] +=  m_matrix[0][3] * ty;
        m_matrix[1][1] +=  m_matrix[1][3] * ty;
        m_matrix[2][1] +=  m_matrix[2][3] * ty;
        m_matrix[3][1] +=  m_matrix[3][3] * ty;
    }

    return *this;
}

TransformationMatrix& TransformationMatrix::translateRight3d(double tx, double ty, double tz)
{
    translateRight(tx, ty);
    if (tz != 0) {
        m_matrix[0][2] +=  m_matrix[0][3] * tz;
        m_matrix[1][2] +=  m_matrix[1][3] * tz;
        m_matrix[2][2] +=  m_matrix[2][3] * tz;
        m_matrix[3][2] +=  m_matrix[3][3] * tz;
    }

    return *this;
}

TransformationMatrix& TransformationMatrix::skew(double sx, double sy)
{
    // angles are in degrees. Switch to radians
    sx = deg2rad(sx);
    sy = deg2rad(sy);

    TransformationMatrix mat;
    mat.m_matrix[0][1] = tan(sy); // note that the y shear goes in the first row
    mat.m_matrix[1][0] = tan(sx); // and the x shear in the second row

    multiply(mat);
    return *this;
}

TransformationMatrix& TransformationMatrix::applyPerspective(double p)
{
    TransformationMatrix mat;
    if (p != 0)
        mat.m_matrix[2][3] = -1/p;

    multiply(mat);
    return *this;
}

TransformationMatrix TransformationMatrix::rectToRect(const FloatRect& from, const FloatRect& to)
{
    ASSERT(!from.isEmpty());
    return TransformationMatrix(to.width() / from.width(),
                                0, 0,
                                to.height() / from.height(),
                                to.x() - from.x(),
                                to.y() - from.y());
}

// this = mat * this.
TransformationMatrix& TransformationMatrix::multiply(const TransformationMatrix& mat)
{
#if CPU(ARM64)
    double* leftMatrix = &(m_matrix[0][0]);
    const double* rightMatrix = &(mat.m_matrix[0][0]);
    asm volatile (
        // First, load the leftMatrix completely in memory. The leftMatrix is in v16-v23.
        "mov   x4, %[leftMatrix]\n\t"
        "ld1   {v16.2d, v17.2d, v18.2d, v19.2d}, [%[leftMatrix]], #64\n\t"
        "ld1   {v20.2d, v21.2d, v22.2d, v23.2d}, [%[leftMatrix]]\n\t"

        // First row.
        "ld4r  {v24.2d, v25.2d, v26.2d, v27.2d}, [%[rightMatrix]], #32\n\t"
        "fmul  v28.2d, v24.2d, v16.2d\n\t"
        "fmul  v29.2d, v24.2d, v17.2d\n\t"
        "fmla  v28.2d, v25.2d, v18.2d\n\t"
        "fmla  v29.2d, v25.2d, v19.2d\n\t"
        "fmla  v28.2d, v26.2d, v20.2d\n\t"
        "fmla  v29.2d, v26.2d, v21.2d\n\t"
        "fmla  v28.2d, v27.2d, v22.2d\n\t"
        "fmla  v29.2d, v27.2d, v23.2d\n\t"

        "ld4r  {v0.2d, v1.2d, v2.2d, v3.2d}, [%[rightMatrix]], #32\n\t"
        "st1  {v28.2d, v29.2d}, [x4], #32\n\t"

        // Second row.
        "fmul  v30.2d, v0.2d, v16.2d\n\t"
        "fmul  v31.2d, v0.2d, v17.2d\n\t"
        "fmla  v30.2d, v1.2d, v18.2d\n\t"
        "fmla  v31.2d, v1.2d, v19.2d\n\t"
        "fmla  v30.2d, v2.2d, v20.2d\n\t"
        "fmla  v31.2d, v2.2d, v21.2d\n\t"
        "fmla  v30.2d, v3.2d, v22.2d\n\t"
        "fmla  v31.2d, v3.2d, v23.2d\n\t"

        "ld4r  {v24.2d, v25.2d, v26.2d, v27.2d}, [%[rightMatrix]], #32\n\t"
        "st1   {v30.2d, v31.2d}, [x4], #32\n\t"

        // Third row.
        "fmul  v28.2d, v24.2d, v16.2d\n\t"
        "fmul  v29.2d, v24.2d, v17.2d\n\t"
        "fmla  v28.2d, v25.2d, v18.2d\n\t"
        "fmla  v29.2d, v25.2d, v19.2d\n\t"
        "fmla  v28.2d, v26.2d, v20.2d\n\t"
        "fmla  v29.2d, v26.2d, v21.2d\n\t"
        "fmla  v28.2d, v27.2d, v22.2d\n\t"
        "fmla  v29.2d, v27.2d, v23.2d\n\t"

        "ld4r  {v0.2d, v1.2d, v2.2d, v3.2d}, [%[rightMatrix]], #32\n\t"
        "st1   {v28.2d, v29.2d}, [x4], #32\n\t"

        // Fourth row.
        "fmul  v30.2d, v0.2d, v16.2d\n\t"
        "fmul  v31.2d, v0.2d, v17.2d\n\t"
        "fmla  v30.2d, v1.2d, v18.2d\n\t"
        "fmla  v31.2d, v1.2d, v19.2d\n\t"
        "fmla  v30.2d, v2.2d, v20.2d\n\t"
        "fmla  v31.2d, v2.2d, v21.2d\n\t"
        "fmla  v30.2d, v3.2d, v22.2d\n\t"
        "fmla  v31.2d, v3.2d, v23.2d\n\t"

        "st1  {v30.2d, v31.2d}, [x4]\n\t"

        : [leftMatrix]"+r"(leftMatrix), [rightMatrix]"+r"(rightMatrix)
        :
        : "memory", "x4", "v0", "v1", "v2", "v3", "v16", "v17", "v18", "v19", "v20", "v21", "v22", "v23", "v24", "v25", "v26", "v27", "v28", "v29", "v30", "v31");
#elif CPU(APPLE_ARMV7S)
    double* leftMatrix = &(m_matrix[0][0]);
    const double* rightMatrix = &(mat.m_matrix[0][0]);
    asm volatile (// First row of leftMatrix.
        "mov        r3, %[leftMatrix]\n\t"
        "vld1.64    { d16-d19 }, [%[leftMatrix], :128]!\n\t"
        "vld1.64    { d0-d3}, [%[rightMatrix], :128]!\n\t"
        "vmul.f64   d4, d0, d16\n\t"
        "vld1.64    { d20-d23 }, [%[leftMatrix], :128]!\n\t"
        "vmla.f64   d4, d1, d20\n\t"
        "vld1.64    { d24-d27 }, [%[leftMatrix], :128]!\n\t"
        "vmla.f64   d4, d2, d24\n\t"
        "vld1.64    { d28-d31 }, [%[leftMatrix], :128]!\n\t"
        "vmla.f64   d4, d3, d28\n\t"

        "vmul.f64   d5, d0, d17\n\t"
        "vmla.f64   d5, d1, d21\n\t"
        "vmla.f64   d5, d2, d25\n\t"
        "vmla.f64   d5, d3, d29\n\t"

        "vmul.f64   d6, d0, d18\n\t"
        "vmla.f64   d6, d1, d22\n\t"
        "vmla.f64   d6, d2, d26\n\t"
        "vmla.f64   d6, d3, d30\n\t"

        "vmul.f64   d7, d0, d19\n\t"
        "vmla.f64   d7, d1, d23\n\t"
        "vmla.f64   d7, d2, d27\n\t"
        "vmla.f64   d7, d3, d31\n\t"
        "vld1.64    { d0-d3}, [%[rightMatrix], :128]!\n\t"
        "vst1.64    { d4-d7 }, [r3, :128]!\n\t"

        // Second row of leftMatrix.
        "vmul.f64   d4, d0, d16\n\t"
        "vmla.f64   d4, d1, d20\n\t"
        "vmla.f64   d4, d2, d24\n\t"
        "vmla.f64   d4, d3, d28\n\t"

        "vmul.f64   d5, d0, d17\n\t"
        "vmla.f64   d5, d1, d21\n\t"
        "vmla.f64   d5, d2, d25\n\t"
        "vmla.f64   d5, d3, d29\n\t"

        "vmul.f64   d6, d0, d18\n\t"
        "vmla.f64   d6, d1, d22\n\t"
        "vmla.f64   d6, d2, d26\n\t"
        "vmla.f64   d6, d3, d30\n\t"

        "vmul.f64   d7, d0, d19\n\t"
        "vmla.f64   d7, d1, d23\n\t"
        "vmla.f64   d7, d2, d27\n\t"
        "vmla.f64   d7, d3, d31\n\t"
        "vld1.64    { d0-d3}, [%[rightMatrix], :128]!\n\t"
        "vst1.64    { d4-d7 }, [r3, :128]!\n\t"

        // Third row of leftMatrix.
        "vmul.f64   d4, d0, d16\n\t"
        "vmla.f64   d4, d1, d20\n\t"
        "vmla.f64   d4, d2, d24\n\t"
        "vmla.f64   d4, d3, d28\n\t"

        "vmul.f64   d5, d0, d17\n\t"
        "vmla.f64   d5, d1, d21\n\t"
        "vmla.f64   d5, d2, d25\n\t"
        "vmla.f64   d5, d3, d29\n\t"

        "vmul.f64   d6, d0, d18\n\t"
        "vmla.f64   d6, d1, d22\n\t"
        "vmla.f64   d6, d2, d26\n\t"
        "vmla.f64   d6, d3, d30\n\t"

        "vmul.f64   d7, d0, d19\n\t"
        "vmla.f64   d7, d1, d23\n\t"
        "vmla.f64   d7, d2, d27\n\t"
        "vmla.f64   d7, d3, d31\n\t"
        "vld1.64    { d0-d3}, [%[rightMatrix], :128]\n\t"
        "vst1.64    { d4-d7 }, [r3, :128]!\n\t"

        // Fourth and last row of leftMatrix.
        "vmul.f64   d4, d0, d16\n\t"
        "vmla.f64   d4, d1, d20\n\t"
        "vmla.f64   d4, d2, d24\n\t"
        "vmla.f64   d4, d3, d28\n\t"

        "vmul.f64   d5, d0, d17\n\t"
        "vmla.f64   d5, d1, d21\n\t"
        "vmla.f64   d5, d2, d25\n\t"
        "vmla.f64   d5, d3, d29\n\t"

        "vmul.f64   d6, d0, d18\n\t"
        "vmla.f64   d6, d1, d22\n\t"
        "vmla.f64   d6, d2, d26\n\t"
        "vmla.f64   d6, d3, d30\n\t"

        "vmul.f64   d7, d0, d19\n\t"
        "vmla.f64   d7, d1, d23\n\t"
        "vmla.f64   d7, d2, d27\n\t"
        "vmla.f64   d7, d3, d31\n\t"
        "vst1.64    { d4-d7 }, [r3, :128]\n\t"
        : [leftMatrix]"+r"(leftMatrix), [rightMatrix]"+r"(rightMatrix)
        :
        : "memory", "r3", "d0", "d1", "d2", "d3", "d4", "d5", "d6", "d7", "d16", "d17", "d18", "d19", "d20", "d21", "d22", "d23", "d24", "d25", "d26", "d27", "d28", "d29", "d30", "d31");
#elif CPU(ARM_VFP) && PLATFORM(IOS)

#define MATRIX_MULTIPLY_ONE_LINE \
    "vldmia.64  %[rightMatrix]!, { d0-d3}\n\t" \
    "vmul.f64   d4, d0, d16\n\t" \
    "vmla.f64   d4, d1, d20\n\t" \
    "vmla.f64   d4, d2, d24\n\t" \
    "vmla.f64   d4, d3, d28\n\t" \
    \
    "vmul.f64   d5, d0, d17\n\t" \
    "vmla.f64   d5, d1, d21\n\t" \
    "vmla.f64   d5, d2, d25\n\t" \
    "vmla.f64   d5, d3, d29\n\t" \
    \
    "vmul.f64   d6, d0, d18\n\t" \
    "vmla.f64   d6, d1, d22\n\t" \
    "vmla.f64   d6, d2, d26\n\t" \
    "vmla.f64   d6, d3, d30\n\t" \
    \
    "vmul.f64   d7, d0, d19\n\t" \
    "vmla.f64   d7, d1, d23\n\t" \
    "vmla.f64   d7, d2, d27\n\t" \
    "vmla.f64   d7, d3, d31\n\t" \
    "vstmia.64  %[leftMatrix]!, { d4-d7 }\n\t"

    double* leftMatrix = &(m_matrix[0][0]);
    const double* rightMatrix = &(mat.m_matrix[0][0]);
    // We load the full m_matrix at once in d16-d31.
    asm volatile("vldmia.64  %[leftMatrix], { d16-d31 }\n\t"
                 :
                 : [leftMatrix]"r"(leftMatrix)
                 : "d16", "d17", "d18", "d19", "d20", "d21", "d22", "d23", "d24", "d25", "d26", "d27", "d28", "d29", "d30", "d31");
    for (unsigned i = 0; i < 4; ++i) {
        asm volatile(MATRIX_MULTIPLY_ONE_LINE
                     : [leftMatrix]"+r"(leftMatrix), [rightMatrix]"+r"(rightMatrix)
                     :
                     : "memory", "d0", "d1", "d2", "d3", "d4", "d5", "d6", "d7");
    }
#undef MATRIX_MULTIPLY_ONE_LINE

#elif defined(TRANSFORMATION_MATRIX_USE_X86_64_SSE2)
    // x86_64 has 16 XMM registers which is enough to do the multiplication fully in registers.
    __m128d matrixBlockA = _mm_load_pd(&(m_matrix[0][0]));
    __m128d matrixBlockC = _mm_load_pd(&(m_matrix[1][0]));
    __m128d matrixBlockE = _mm_load_pd(&(m_matrix[2][0]));
    __m128d matrixBlockG = _mm_load_pd(&(m_matrix[3][0]));

    // First row.
    __m128d otherMatrixFirstParam = _mm_set1_pd(mat.m_matrix[0][0]);
    __m128d otherMatrixSecondParam = _mm_set1_pd(mat.m_matrix[0][1]);
    __m128d otherMatrixThirdParam = _mm_set1_pd(mat.m_matrix[0][2]);
    __m128d otherMatrixFourthParam = _mm_set1_pd(mat.m_matrix[0][3]);

    // output00 and output01.
    __m128d accumulator = _mm_mul_pd(matrixBlockA, otherMatrixFirstParam);
    __m128d temp1 = _mm_mul_pd(matrixBlockC, otherMatrixSecondParam);
    __m128d temp2 = _mm_mul_pd(matrixBlockE, otherMatrixThirdParam);
    __m128d temp3 = _mm_mul_pd(matrixBlockG, otherMatrixFourthParam);

    __m128d matrixBlockB = _mm_load_pd(&(m_matrix[0][2]));
    __m128d matrixBlockD = _mm_load_pd(&(m_matrix[1][2]));
    __m128d matrixBlockF = _mm_load_pd(&(m_matrix[2][2]));
    __m128d matrixBlockH = _mm_load_pd(&(m_matrix[3][2]));

    accumulator = _mm_add_pd(accumulator, temp1);
    accumulator = _mm_add_pd(accumulator, temp2);
    accumulator = _mm_add_pd(accumulator, temp3);
    _mm_store_pd(&m_matrix[0][0], accumulator);

    // output02 and output03.
    accumulator = _mm_mul_pd(matrixBlockB, otherMatrixFirstParam);
    temp1 = _mm_mul_pd(matrixBlockD, otherMatrixSecondParam);
    temp2 = _mm_mul_pd(matrixBlockF, otherMatrixThirdParam);
    temp3 = _mm_mul_pd(matrixBlockH, otherMatrixFourthParam);

    accumulator = _mm_add_pd(accumulator, temp1);
    accumulator = _mm_add_pd(accumulator, temp2);
    accumulator = _mm_add_pd(accumulator, temp3);
    _mm_store_pd(&m_matrix[0][2], accumulator);

    // Second row.
    otherMatrixFirstParam = _mm_set1_pd(mat.m_matrix[1][0]);
    otherMatrixSecondParam = _mm_set1_pd(mat.m_matrix[1][1]);
    otherMatrixThirdParam = _mm_set1_pd(mat.m_matrix[1][2]);
    otherMatrixFourthParam = _mm_set1_pd(mat.m_matrix[1][3]);

    // output10 and output11.
    accumulator = _mm_mul_pd(matrixBlockA, otherMatrixFirstParam);
    temp1 = _mm_mul_pd(matrixBlockC, otherMatrixSecondParam);
    temp2 = _mm_mul_pd(matrixBlockE, otherMatrixThirdParam);
    temp3 = _mm_mul_pd(matrixBlockG, otherMatrixFourthParam);

    accumulator = _mm_add_pd(accumulator, temp1);
    accumulator = _mm_add_pd(accumulator, temp2);
    accumulator = _mm_add_pd(accumulator, temp3);
    _mm_store_pd(&m_matrix[1][0], accumulator);

    // output12 and output13.
    accumulator = _mm_mul_pd(matrixBlockB, otherMatrixFirstParam);
    temp1 = _mm_mul_pd(matrixBlockD, otherMatrixSecondParam);
    temp2 = _mm_mul_pd(matrixBlockF, otherMatrixThirdParam);
    temp3 = _mm_mul_pd(matrixBlockH, otherMatrixFourthParam);

    accumulator = _mm_add_pd(accumulator, temp1);
    accumulator = _mm_add_pd(accumulator, temp2);
    accumulator = _mm_add_pd(accumulator, temp3);
    _mm_store_pd(&m_matrix[1][2], accumulator);

    // Third row.
    otherMatrixFirstParam = _mm_set1_pd(mat.m_matrix[2][0]);
    otherMatrixSecondParam = _mm_set1_pd(mat.m_matrix[2][1]);
    otherMatrixThirdParam = _mm_set1_pd(mat.m_matrix[2][2]);
    otherMatrixFourthParam = _mm_set1_pd(mat.m_matrix[2][3]);

    // output20 and output21.
    accumulator = _mm_mul_pd(matrixBlockA, otherMatrixFirstParam);
    temp1 = _mm_mul_pd(matrixBlockC, otherMatrixSecondParam);
    temp2 = _mm_mul_pd(matrixBlockE, otherMatrixThirdParam);
    temp3 = _mm_mul_pd(matrixBlockG, otherMatrixFourthParam);

    accumulator = _mm_add_pd(accumulator, temp1);
    accumulator = _mm_add_pd(accumulator, temp2);
    accumulator = _mm_add_pd(accumulator, temp3);
    _mm_store_pd(&m_matrix[2][0], accumulator);

    // output22 and output23.
    accumulator = _mm_mul_pd(matrixBlockB, otherMatrixFirstParam);
    temp1 = _mm_mul_pd(matrixBlockD, otherMatrixSecondParam);
    temp2 = _mm_mul_pd(matrixBlockF, otherMatrixThirdParam);
    temp3 = _mm_mul_pd(matrixBlockH, otherMatrixFourthParam);

    accumulator = _mm_add_pd(accumulator, temp1);
    accumulator = _mm_add_pd(accumulator, temp2);
    accumulator = _mm_add_pd(accumulator, temp3);
    _mm_store_pd(&m_matrix[2][2], accumulator);

    // Fourth row.
    otherMatrixFirstParam = _mm_set1_pd(mat.m_matrix[3][0]);
    otherMatrixSecondParam = _mm_set1_pd(mat.m_matrix[3][1]);
    otherMatrixThirdParam = _mm_set1_pd(mat.m_matrix[3][2]);
    otherMatrixFourthParam = _mm_set1_pd(mat.m_matrix[3][3]);

    // output30 and output31.
    accumulator = _mm_mul_pd(matrixBlockA, otherMatrixFirstParam);
    temp1 = _mm_mul_pd(matrixBlockC, otherMatrixSecondParam);
    temp2 = _mm_mul_pd(matrixBlockE, otherMatrixThirdParam);
    temp3 = _mm_mul_pd(matrixBlockG, otherMatrixFourthParam);

    accumulator = _mm_add_pd(accumulator, temp1);
    accumulator = _mm_add_pd(accumulator, temp2);
    accumulator = _mm_add_pd(accumulator, temp3);
    _mm_store_pd(&m_matrix[3][0], accumulator);

    // output32 and output33.
    accumulator = _mm_mul_pd(matrixBlockB, otherMatrixFirstParam);
    temp1 = _mm_mul_pd(matrixBlockD, otherMatrixSecondParam);
    temp2 = _mm_mul_pd(matrixBlockF, otherMatrixThirdParam);
    temp3 = _mm_mul_pd(matrixBlockH, otherMatrixFourthParam);

    accumulator = _mm_add_pd(accumulator, temp1);
    accumulator = _mm_add_pd(accumulator, temp2);
    accumulator = _mm_add_pd(accumulator, temp3);
    _mm_store_pd(&m_matrix[3][2], accumulator);
#else
    Matrix4 tmp;

    tmp[0][0] = (mat.m_matrix[0][0] * m_matrix[0][0] + mat.m_matrix[0][1] * m_matrix[1][0]
               + mat.m_matrix[0][2] * m_matrix[2][0] + mat.m_matrix[0][3] * m_matrix[3][0]);
    tmp[0][1] = (mat.m_matrix[0][0] * m_matrix[0][1] + mat.m_matrix[0][1] * m_matrix[1][1]
               + mat.m_matrix[0][2] * m_matrix[2][1] + mat.m_matrix[0][3] * m_matrix[3][1]);
    tmp[0][2] = (mat.m_matrix[0][0] * m_matrix[0][2] + mat.m_matrix[0][1] * m_matrix[1][2]
               + mat.m_matrix[0][2] * m_matrix[2][2] + mat.m_matrix[0][3] * m_matrix[3][2]);
    tmp[0][3] = (mat.m_matrix[0][0] * m_matrix[0][3] + mat.m_matrix[0][1] * m_matrix[1][3]
               + mat.m_matrix[0][2] * m_matrix[2][3] + mat.m_matrix[0][3] * m_matrix[3][3]);

    tmp[1][0] = (mat.m_matrix[1][0] * m_matrix[0][0] + mat.m_matrix[1][1] * m_matrix[1][0]
               + mat.m_matrix[1][2] * m_matrix[2][0] + mat.m_matrix[1][3] * m_matrix[3][0]);
    tmp[1][1] = (mat.m_matrix[1][0] * m_matrix[0][1] + mat.m_matrix[1][1] * m_matrix[1][1]
               + mat.m_matrix[1][2] * m_matrix[2][1] + mat.m_matrix[1][3] * m_matrix[3][1]);
    tmp[1][2] = (mat.m_matrix[1][0] * m_matrix[0][2] + mat.m_matrix[1][1] * m_matrix[1][2]
               + mat.m_matrix[1][2] * m_matrix[2][2] + mat.m_matrix[1][3] * m_matrix[3][2]);
    tmp[1][3] = (mat.m_matrix[1][0] * m_matrix[0][3] + mat.m_matrix[1][1] * m_matrix[1][3]
               + mat.m_matrix[1][2] * m_matrix[2][3] + mat.m_matrix[1][3] * m_matrix[3][3]);

    tmp[2][0] = (mat.m_matrix[2][0] * m_matrix[0][0] + mat.m_matrix[2][1] * m_matrix[1][0]
               + mat.m_matrix[2][2] * m_matrix[2][0] + mat.m_matrix[2][3] * m_matrix[3][0]);
    tmp[2][1] = (mat.m_matrix[2][0] * m_matrix[0][1] + mat.m_matrix[2][1] * m_matrix[1][1]
               + mat.m_matrix[2][2] * m_matrix[2][1] + mat.m_matrix[2][3] * m_matrix[3][1]);
    tmp[2][2] = (mat.m_matrix[2][0] * m_matrix[0][2] + mat.m_matrix[2][1] * m_matrix[1][2]
               + mat.m_matrix[2][2] * m_matrix[2][2] + mat.m_matrix[2][3] * m_matrix[3][2]);
    tmp[2][3] = (mat.m_matrix[2][0] * m_matrix[0][3] + mat.m_matrix[2][1] * m_matrix[1][3]
               + mat.m_matrix[2][2] * m_matrix[2][3] + mat.m_matrix[2][3] * m_matrix[3][3]);

    tmp[3][0] = (mat.m_matrix[3][0] * m_matrix[0][0] + mat.m_matrix[3][1] * m_matrix[1][0]
               + mat.m_matrix[3][2] * m_matrix[2][0] + mat.m_matrix[3][3] * m_matrix[3][0]);
    tmp[3][1] = (mat.m_matrix[3][0] * m_matrix[0][1] + mat.m_matrix[3][1] * m_matrix[1][1]
               + mat.m_matrix[3][2] * m_matrix[2][1] + mat.m_matrix[3][3] * m_matrix[3][1]);
    tmp[3][2] = (mat.m_matrix[3][0] * m_matrix[0][2] + mat.m_matrix[3][1] * m_matrix[1][2]
               + mat.m_matrix[3][2] * m_matrix[2][2] + mat.m_matrix[3][3] * m_matrix[3][2]);
    tmp[3][3] = (mat.m_matrix[3][0] * m_matrix[0][3] + mat.m_matrix[3][1] * m_matrix[1][3]
               + mat.m_matrix[3][2] * m_matrix[2][3] + mat.m_matrix[3][3] * m_matrix[3][3]);

    setMatrix(tmp);
#endif
    return *this;
}

void TransformationMatrix::multVecMatrix(double x, double y, double& resultX, double& resultY) const
{
    resultX = m_matrix[3][0] + x * m_matrix[0][0] + y * m_matrix[1][0];
    resultY = m_matrix[3][1] + x * m_matrix[0][1] + y * m_matrix[1][1];
    double w = m_matrix[3][3] + x * m_matrix[0][3] + y * m_matrix[1][3];
    if (w != 1 && w != 0) {
        resultX /= w;
        resultY /= w;
    }
}

void TransformationMatrix::multVecMatrix(double x, double y, double z, double& resultX, double& resultY, double& resultZ) const
{
    resultX = m_matrix[3][0] + x * m_matrix[0][0] + y * m_matrix[1][0] + z * m_matrix[2][0];
    resultY = m_matrix[3][1] + x * m_matrix[0][1] + y * m_matrix[1][1] + z * m_matrix[2][1];
    resultZ = m_matrix[3][2] + x * m_matrix[0][2] + y * m_matrix[1][2] + z * m_matrix[2][2];
    double w = m_matrix[3][3] + x * m_matrix[0][3] + y * m_matrix[1][3] + z * m_matrix[2][3];
    if (w != 1 && w != 0) {
        resultX /= w;
        resultY /= w;
        resultZ /= w;
    }
}

bool TransformationMatrix::isInvertible() const
{
    if (isIdentityOrTranslation())
        return true;

    double det = WebCore::determinant4x4(m_matrix);

    if (fabs(det) < SMALL_NUMBER)
        return false;

    return true;
}

Optional<TransformationMatrix> TransformationMatrix::inverse() const
{
    if (isIdentityOrTranslation()) {
        // identity matrix
        if (m_matrix[3][0] == 0 && m_matrix[3][1] == 0 && m_matrix[3][2] == 0)
            return TransformationMatrix();

        // translation
        return TransformationMatrix(1, 0, 0, 0,
                                    0, 1, 0, 0,
                                    0, 0, 1, 0,
                                    -m_matrix[3][0], -m_matrix[3][1], -m_matrix[3][2], 1);
    }

    TransformationMatrix invMat;
    // FIXME: Use LU decomposition to apply the inverse instead of calculating the inverse explicitly.
    // Calculating the inverse of a 4x4 matrix using cofactors is numerically unstable and unnecessary to apply the inverse transformation to a point.
    if (!WebCore::inverse(m_matrix, invMat.m_matrix))
        return Nullopt;

    return invMat;
}

void TransformationMatrix::makeAffine()
{
    m_matrix[0][2] = 0;
    m_matrix[0][3] = 0;

    m_matrix[1][2] = 0;
    m_matrix[1][3] = 0;

    m_matrix[2][0] = 0;
    m_matrix[2][1] = 0;
    m_matrix[2][2] = 1;
    m_matrix[2][3] = 0;

    m_matrix[3][2] = 0;
    m_matrix[3][3] = 1;
}

AffineTransform TransformationMatrix::toAffineTransform() const
{
    return AffineTransform(m_matrix[0][0], m_matrix[0][1], m_matrix[1][0],
                           m_matrix[1][1], m_matrix[3][0], m_matrix[3][1]);
}

static inline void blendFloat(double& from, double to, double progress)
{
    if (from != to)
        from = from + (to - from) * progress;
}

void TransformationMatrix::blend2(const TransformationMatrix& from, double progress)
{
    Decomposed2Type fromDecomp;
    Decomposed2Type toDecomp;
    from.decompose2(fromDecomp);
    decompose2(toDecomp);

    // If x-axis of one is flipped, and y-axis of the other, convert to an unflipped rotation.
    if ((fromDecomp.scaleX < 0 && toDecomp.scaleY < 0) || (fromDecomp.scaleY < 0 && toDecomp.scaleX < 0)) {
        fromDecomp.scaleX = -fromDecomp.scaleX;
        fromDecomp.scaleY = -fromDecomp.scaleY;
        fromDecomp.angle += fromDecomp.angle < 0 ? 180 : -180;
    }

    // Don't rotate the long way around.
    if (!fromDecomp.angle)
        fromDecomp.angle = 360;
    if (!toDecomp.angle)
        toDecomp.angle = 360;

    if (fabs(fromDecomp.angle - toDecomp.angle) > 180) {
        if (fromDecomp.angle > toDecomp.angle)
            fromDecomp.angle -= 360;
        else
            toDecomp.angle -= 360;
    }

    blendFloat(fromDecomp.m11, toDecomp.m11, progress);
    blendFloat(fromDecomp.m12, toDecomp.m12, progress);
    blendFloat(fromDecomp.m21, toDecomp.m21, progress);
    blendFloat(fromDecomp.m22, toDecomp.m22, progress);
    blendFloat(fromDecomp.translateX, toDecomp.translateX, progress);
    blendFloat(fromDecomp.translateY, toDecomp.translateY, progress);
    blendFloat(fromDecomp.scaleX, toDecomp.scaleX, progress);
    blendFloat(fromDecomp.scaleY, toDecomp.scaleY, progress);
    blendFloat(fromDecomp.angle, toDecomp.angle, progress);

    recompose2(fromDecomp);
}

void TransformationMatrix::blend4(const TransformationMatrix& from, double progress)
{
    Decomposed4Type fromDecomp;
    Decomposed4Type toDecomp;
    from.decompose4(fromDecomp);
    decompose4(toDecomp);

    blendFloat(fromDecomp.scaleX, toDecomp.scaleX, progress);
    blendFloat(fromDecomp.scaleY, toDecomp.scaleY, progress);
    blendFloat(fromDecomp.scaleZ, toDecomp.scaleZ, progress);
    blendFloat(fromDecomp.skewXY, toDecomp.skewXY, progress);
    blendFloat(fromDecomp.skewXZ, toDecomp.skewXZ, progress);
    blendFloat(fromDecomp.skewYZ, toDecomp.skewYZ, progress);
    blendFloat(fromDecomp.translateX, toDecomp.translateX, progress);
    blendFloat(fromDecomp.translateY, toDecomp.translateY, progress);
    blendFloat(fromDecomp.translateZ, toDecomp.translateZ, progress);
    blendFloat(fromDecomp.perspectiveX, toDecomp.perspectiveX, progress);
    blendFloat(fromDecomp.perspectiveY, toDecomp.perspectiveY, progress);
    blendFloat(fromDecomp.perspectiveZ, toDecomp.perspectiveZ, progress);
    blendFloat(fromDecomp.perspectiveW, toDecomp.perspectiveW, progress);

    slerp(&fromDecomp.quaternionX, &toDecomp.quaternionX, progress);

    recompose4(fromDecomp);
}

void TransformationMatrix::blend(const TransformationMatrix& from, double progress)
{
    if (from.isIdentity() && isIdentity())
        return;

    if (from.isAffine() && isAffine())
        blend2(from, progress);
    else
        blend4(from, progress);
}

bool TransformationMatrix::decompose2(Decomposed2Type& decomp) const
{
    if (isIdentity()) {
        memset(&decomp, 0, sizeof(decomp));
        decomp.scaleX = 1;
        decomp.scaleY = 1;
        decomp.m11 = 1;
        decomp.m22 = 1;
        return true;
    }

    return WebCore::decompose2(m_matrix, decomp);
}

bool TransformationMatrix::decompose4(Decomposed4Type& decomp) const
{
    if (isIdentity()) {
        memset(&decomp, 0, sizeof(decomp));
        decomp.perspectiveW = 1;
        decomp.scaleX = 1;
        decomp.scaleY = 1;
        decomp.scaleZ = 1;
        return true;
    }

    return WebCore::decompose4(m_matrix, decomp);
}

void TransformationMatrix::recompose2(const Decomposed2Type& decomp)
{
    makeIdentity();

    m_matrix[0][0] = decomp.m11;
    m_matrix[0][1] = decomp.m12;
    m_matrix[1][0] = decomp.m21;
    m_matrix[1][1] = decomp.m22;

    translate3d(decomp.translateX, decomp.translateY, 0);
    rotate(decomp.angle);
    scale3d(decomp.scaleX, decomp.scaleY, 1);
}

void TransformationMatrix::recompose4(const Decomposed4Type& decomp)
{
    makeIdentity();

    // First apply perspective.
    m_matrix[0][3] = decomp.perspectiveX;
    m_matrix[1][3] = decomp.perspectiveY;
    m_matrix[2][3] = decomp.perspectiveZ;
    m_matrix[3][3] = decomp.perspectiveW;

    // Next, translate.
    translate3d(decomp.translateX, decomp.translateY, decomp.translateZ);

    // Apply rotation.
    double xx = decomp.quaternionX * decomp.quaternionX;
    double xy = decomp.quaternionX * decomp.quaternionY;
    double xz = decomp.quaternionX * decomp.quaternionZ;
    double xw = decomp.quaternionX * decomp.quaternionW;
    double yy = decomp.quaternionY * decomp.quaternionY;
    double yz = decomp.quaternionY * decomp.quaternionZ;
    double yw = decomp.quaternionY * decomp.quaternionW;
    double zz = decomp.quaternionZ * decomp.quaternionZ;
    double zw = decomp.quaternionZ * decomp.quaternionW;

    // Construct a composite rotation matrix from the quaternion values.
    TransformationMatrix rotationMatrix(1 - 2 * (yy + zz), 2 * (xy - zw), 2 * (xz + yw), 0,
                           2 * (xy + zw), 1 - 2 * (xx + zz), 2 * (yz - xw), 0,
                           2 * (xz - yw), 2 * (yz + xw), 1 - 2 * (xx + yy), 0,
                           0, 0, 0, 1);

    multiply(rotationMatrix);

    // Apply skew.
    if (decomp.skewYZ) {
        TransformationMatrix tmp;
        tmp.setM32(decomp.skewYZ);
        multiply(tmp);
    }

    if (decomp.skewXZ) {
        TransformationMatrix tmp;
        tmp.setM31(decomp.skewXZ);
        multiply(tmp);
    }

    if (decomp.skewXY) {
        TransformationMatrix tmp;
        tmp.setM21(decomp.skewXY);
        multiply(tmp);
    }

    // Finally, apply scale.
    scale3d(decomp.scaleX, decomp.scaleY, decomp.scaleZ);
}

bool TransformationMatrix::isIntegerTranslation() const
{
    if (!isIdentityOrTranslation())
        return false;

    // Check for translate Z.
    if (m_matrix[3][2])
        return false;

    // Check for non-integer translate X/Y.
    if (static_cast<int>(m_matrix[3][0]) != m_matrix[3][0] || static_cast<int>(m_matrix[3][1]) != m_matrix[3][1])
        return false;

    return true;
}

TransformationMatrix TransformationMatrix::to2dTransform() const
{
    return TransformationMatrix(m_matrix[0][0], m_matrix[0][1], 0, m_matrix[0][3],
                                m_matrix[1][0], m_matrix[1][1], 0, m_matrix[1][3],
                                0, 0, 1, 0,
                                m_matrix[3][0], m_matrix[3][1], 0, m_matrix[3][3]);
}

void TransformationMatrix::toColumnMajorFloatArray(FloatMatrix4& result) const
{
    result[0] = m11();
    result[1] = m12();
    result[2] = m13();
    result[3] = m14();
    result[4] = m21();
    result[5] = m22();
    result[6] = m23();
    result[7] = m24();
    result[8] = m31();
    result[9] = m32();
    result[10] = m33();
    result[11] = m34();
    result[12] = m41();
    result[13] = m42();
    result[14] = m43();
    result[15] = m44();
}

bool TransformationMatrix::isBackFaceVisible() const
{
    // Back-face visibility is determined by transforming the normal vector (0, 0, 1) and
    // checking the sign of the resulting z component. However, normals cannot be
    // transformed by the original matrix, they require being transformed by the
    // inverse-transpose.
    //
    // Since we know we will be using (0, 0, 1), and we only care about the z-component of
    // the transformed normal, then we only need the m33() element of the
    // inverse-transpose. Therefore we do not need the transpose.
    //
    // Additionally, if we only need the m33() element, we do not need to compute a full
    // inverse. Instead, knowing the inverse of a matrix is adjoint(matrix) / determinant,
    // we can simply compute the m33() of the adjoint (adjugate) matrix, without computing
    // the full adjoint.

    double determinant = WebCore::determinant4x4(m_matrix);

    // If the matrix is not invertible, then we assume its backface is not visible.
    if (fabs(determinant) < SMALL_NUMBER)
        return false;

    double cofactor33 = determinant3x3(m11(), m12(), m14(), m21(), m22(), m24(), m41(), m42(), m44());
    double zComponentOfTransformedNormal = cofactor33 / determinant;

    return zComponentOfTransformedNormal < 0;
}

TextStream& operator<<(TextStream& ts, const TransformationMatrix& transform)
{
    ts << "\n";
    ts.increaseIndent();
    ts.writeIndent();
    ts << "[" << transform.m11() << " " << transform.m12() << " " << transform.m13() << " " << transform.m14() << "]\n";
    ts.writeIndent();
    ts << "[" << transform.m21() << " " << transform.m22() << " " << transform.m23() << " " << transform.m24() << "]\n";
    ts.writeIndent();
    ts << "[" << transform.m31() << " " << transform.m32() << " " << transform.m33() << " " << transform.m34() << "]\n";
    ts.writeIndent();
    ts << "[" << transform.m41() << " " << transform.m42() << " " << transform.m43() << " " << transform.m44() << "]";
    ts.decreaseIndent();
    return ts;
}

}