TransformationMatrix.cpp   [plain text]

/*
 * Copyright (C) 2005, 2006 Apple Computer, Inc.  All rights reserved.
 * Copyright (C) 2009 Torch Mobile, Inc.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY APPLE COMPUTER, INC. ``AS IS'' AND ANY
 * EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
 * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL APPLE COMPUTER, INC. OR
 * CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
 * EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
 * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
 * PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
 * OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
 * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 
 */

#include "config.h"
#include "TransformationMatrix.h"

#include "AffineTransform.h"
#include "FractionalLayoutRect.h"
#include "FloatPoint3D.h"
#include "FloatRect.h"
#include "FloatQuad.h"
#include "IntRect.h"

#include <wtf/Assertions.h>
#include <wtf/MathExtras.h>

using namespace std;

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 decompose(const TransformationMatrix::Matrix4& mat, TransformationMatrix::DecomposedType& 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::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(-1.0f, 1.0f);
}

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

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 kLargeNumber = 100000000;
        outX = copysign(kLargeNumber, outX);
        outY = copysign(kLargeNumber, 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) 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 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(!isnan(f));
    return min<float>(max<float>(f, -MAX_LAYOUT_UNIT / 2), MAX_LAYOUT_UNIT / 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 (isinf(mappedQuadBounds.x()) && isinf(mappedQuadBounds.width()))
        right = MAX_LAYOUT_UNIT / 2;
    else
        right = clampEdgeValue(ceilf(mappedQuadBounds.maxX()));

    float bottom;
    if (isinf(mappedQuadBounds.y()) && isinf(mappedQuadBounds.height()))
        bottom = MAX_LAYOUT_UNIT / 2;
    else
        bottom = clampEdgeValue(ceilf(mappedQuadBounds.maxY()));
    
    return LayoutRect(clampToLayoutUnit(left), clampToLayoutUnit(top), 
                      clampToLayoutUnit(right - left), clampToLayoutUnit(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]));

    double x, y;
    multVecMatrix(p.x(), p.y(), x, y);
    return FloatPoint(static_cast<float>(x), static_cast<float>(y));
}

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]));

    double x, y, z;
    multVecMatrix(p.x(), p.y(), p.z(), x, y, z);
    return FloatPoint3D(static_cast<float>(x), static_cast<float>(y), static_cast<float>(z));
}

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

FractionalLayoutRect TransformationMatrix::mapRect(const FractionalLayoutRect& r) const
{
    return enclosingFractionalLayoutRect(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 resultQuad = mapQuad(FloatQuad(r));
    return resultQuad.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(mapPoint(q.p1()));
    result.setP2(mapPoint(q.p2()));
    result.setP3(mapPoint(q.p3()));
    result.setP4(mapPoint(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)
{
    // angles are in degrees. Switch to radians
    angle = deg2rad(angle);
    
    angle /= 2.0f;
    double sinA = sin(angle);
    double cosA = cos(angle);
    double sinA2 = sinA * sinA;
    
    // normalize
    double length = sqrt(x * x + y * y + z * z);
    if (length == 0) {
        // bad vector, just use something reasonable
        x = 0;
        y = 0;
        z = 1;
    } else if (length != 1) {
        x /= length;
        y /= length;
        z /= length;
    }
    
    TransformationMatrix mat;

    // optimize case where axis is along major axis
    if (x == 1.0f && y == 0.0f && z == 0.0f) {
        mat.m_matrix[0][0] = 1.0f;
        mat.m_matrix[0][1] = 0.0f;
        mat.m_matrix[0][2] = 0.0f;
        mat.m_matrix[1][0] = 0.0f;
        mat.m_matrix[1][1] = 1.0f - 2.0f * sinA2;
        mat.m_matrix[1][2] = 2.0f * sinA * cosA;
        mat.m_matrix[2][0] = 0.0f;
        mat.m_matrix[2][1] = -2.0f * sinA * cosA;
        mat.m_matrix[2][2] = 1.0f - 2.0f * sinA2;
        mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0f;
        mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0f;
        mat.m_matrix[3][3] = 1.0f;
    } else if (x == 0.0f && y == 1.0f && z == 0.0f) {
        mat.m_matrix[0][0] = 1.0f - 2.0f * sinA2;
        mat.m_matrix[0][1] = 0.0f;
        mat.m_matrix[0][2] = -2.0f * sinA * cosA;
        mat.m_matrix[1][0] = 0.0f;
        mat.m_matrix[1][1] = 1.0f;
        mat.m_matrix[1][2] = 0.0f;
        mat.m_matrix[2][0] = 2.0f * sinA * cosA;
        mat.m_matrix[2][1] = 0.0f;
        mat.m_matrix[2][2] = 1.0f - 2.0f * sinA2;
        mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0f;
        mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0f;
        mat.m_matrix[3][3] = 1.0f;
    } else if (x == 0.0f && y == 0.0f && z == 1.0f) {
        mat.m_matrix[0][0] = 1.0f - 2.0f * sinA2;
        mat.m_matrix[0][1] = 2.0f * sinA * cosA;
        mat.m_matrix[0][2] = 0.0f;
        mat.m_matrix[1][0] = -2.0f * sinA * cosA;
        mat.m_matrix[1][1] = 1.0f - 2.0f * sinA2;
        mat.m_matrix[1][2] = 0.0f;
        mat.m_matrix[2][0] = 0.0f;
        mat.m_matrix[2][1] = 0.0f;
        mat.m_matrix[2][2] = 1.0f;
        mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0f;
        mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0f;
        mat.m_matrix[3][3] = 1.0f;
    } else {
        double x2 = x*x;
        double y2 = y*y;
        double z2 = z*z;
    
        mat.m_matrix[0][0] = 1.0f - 2.0f * (y2 + z2) * sinA2;
        mat.m_matrix[0][1] = 2.0f * (x * y * sinA2 + z * sinA * cosA);
        mat.m_matrix[0][2] = 2.0f * (x * z * sinA2 - y * sinA * cosA);
        mat.m_matrix[1][0] = 2.0f * (y * x * sinA2 - z * sinA * cosA);
        mat.m_matrix[1][1] = 1.0f - 2.0f * (z2 + x2) * sinA2;
        mat.m_matrix[1][2] = 2.0f * (y * z * sinA2 + x * sinA * cosA);
        mat.m_matrix[2][0] = 2.0f * (z * x * sinA2 + y * sinA * cosA);
        mat.m_matrix[2][1] = 2.0f * (z * y * sinA2 - x * sinA * cosA);
        mat.m_matrix[2][2] = 1.0f - 2.0f * (x2 + y2) * sinA2;
        mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0f;
        mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0f;
        mat.m_matrix[3][3] = 1.0f;
    }
    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;
    
    rz /= 2.0f;
    double sinA = sin(rz);
    double cosA = cos(rz);
    double sinA2 = sinA * sinA;
    
    mat.m_matrix[0][0] = 1.0f - 2.0f * sinA2;
    mat.m_matrix[0][1] = 2.0f * sinA * cosA;
    mat.m_matrix[0][2] = 0.0f;
    mat.m_matrix[1][0] = -2.0f * sinA * cosA;
    mat.m_matrix[1][1] = 1.0f - 2.0f * sinA2;
    mat.m_matrix[1][2] = 0.0f;
    mat.m_matrix[2][0] = 0.0f;
    mat.m_matrix[2][1] = 0.0f;
    mat.m_matrix[2][2] = 1.0f;
    mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0f;
    mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0f;
    mat.m_matrix[3][3] = 1.0f;
    
    TransformationMatrix rmat(mat);
    
    ry /= 2.0f;
    sinA = sin(ry);
    cosA = cos(ry);
    sinA2 = sinA * sinA;
    
    mat.m_matrix[0][0] = 1.0f - 2.0f * sinA2;
    mat.m_matrix[0][1] = 0.0f;
    mat.m_matrix[0][2] = -2.0f * sinA * cosA;
    mat.m_matrix[1][0] = 0.0f;
    mat.m_matrix[1][1] = 1.0f;
    mat.m_matrix[1][2] = 0.0f;
    mat.m_matrix[2][0] = 2.0f * sinA * cosA;
    mat.m_matrix[2][1] = 0.0f;
    mat.m_matrix[2][2] = 1.0f - 2.0f * sinA2;
    mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0f;
    mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0f;
    mat.m_matrix[3][3] = 1.0f;
    
    rmat.multiply(mat);

    rx /= 2.0f;
    sinA = sin(rx);
    cosA = cos(rx);
    sinA2 = sinA * sinA;
    
    mat.m_matrix[0][0] = 1.0f;
    mat.m_matrix[0][1] = 0.0f;
    mat.m_matrix[0][2] = 0.0f;
    mat.m_matrix[1][0] = 0.0f;
    mat.m_matrix[1][1] = 1.0f - 2.0f * sinA2;
    mat.m_matrix[1][2] = 2.0f * sinA * cosA;
    mat.m_matrix[2][0] = 0.0f;
    mat.m_matrix[2][1] = -2.0f * sinA * cosA;
    mat.m_matrix[2][2] = 1.0f - 2.0f * sinA2;
    mat.m_matrix[0][3] = mat.m_matrix[1][3] = mat.m_matrix[2][3] = 0.0f;
    mat.m_matrix[3][0] = mat.m_matrix[3][1] = mat.m_matrix[3][2] = 0.0f;
    mat.m_matrix[3][3] = 1.0f;
    
    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)
{
    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);
    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;
}

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;
    bool inverted = WebCore::inverse(m_matrix, invMat.m_matrix);
    if (!inverted)
        return TransformationMatrix();
    
    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::blend(const TransformationMatrix& from, double progress)
{
    if (from.isIdentity() && isIdentity())
        return;
        
    // decompose
    DecomposedType fromDecomp;
    DecomposedType toDecomp;
    from.decompose(fromDecomp);
    decompose(toDecomp);

    // interpolate
    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);
        
    // recompose
    recompose(fromDecomp);
}

bool TransformationMatrix::decompose(DecomposedType& decomp) const
{
    if (isIdentity()) {
        memset(&decomp, 0, sizeof(decomp));
        decomp.perspectiveW = 1;
        decomp.scaleX = 1;
        decomp.scaleY = 1;
        decomp.scaleZ = 1;
    }
    
    if (!WebCore::decompose(m_matrix, decomp))
        return false;
    return true;
}

void TransformationMatrix::recompose(const DecomposedType& decomp)
{
    makeIdentity();
    
    // first apply perspective
    m_matrix[0][3] = (float) decomp.perspectiveX;
    m_matrix[1][3] = (float) decomp.perspectiveY;
    m_matrix[2][3] = (float) decomp.perspectiveZ;
    m_matrix[3][3] = (float) decomp.perspectiveW;
    
    // now translate
    translate3d((float) decomp.translateX, (float) decomp.translateY, (float) 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);
    
    // now apply skew
    if (decomp.skewYZ) {
        TransformationMatrix tmp;
        tmp.setM32((float) decomp.skewYZ);
        multiply(tmp);
    }
    
    if (decomp.skewXZ) {
        TransformationMatrix tmp;
        tmp.setM31((float) decomp.skewXZ);
        multiply(tmp);
    }
    
    if (decomp.skewXY) {
        TransformationMatrix tmp;
        tmp.setM21((float) decomp.skewXY);
        multiply(tmp);
    }
    
    // finally, apply scale
    scale3d((float) decomp.scaleX, (float) decomp.scaleY, (float) 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;
}

}