path: root/cc/math_util.cc

// Copyright 2012 The Chromium Authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.

#include "cc/math_util.h"

#include <cmath>
#include <limits>

#include "ui/gfx/quad_f.h"
#include "ui/gfx/rect.h"
#include "ui/gfx/rect_conversions.h"
#include "ui/gfx/rect_f.h"
#include "ui/gfx/transform.h"
#include "ui/gfx/vector2d_f.h"

namespace cc {

const double MathUtil::PI_DOUBLE = 3.14159265358979323846;
const float MathUtil::PI_FLOAT = 3.14159265358979323846f;
const double MathUtil::EPSILON = 1e-9;

static HomogeneousCoordinate projectHomogeneousPoint(const gfx::Transform& transform, const gfx::PointF& p)
{
    // In this case, the layer we are trying to project onto is perpendicular to ray
    // (point p and z-axis direction) that we are trying to project. This happens when the
    // layer is rotated so that it is infinitesimally thin, or when it is co-planar with
    // the camera origin -- i.e. when the layer is invisible anyway.
    if (!transform.matrix().getDouble(2, 2))
        return HomogeneousCoordinate(0, 0, 0, 1);

    double x = p.x();
    double y = p.y();
    double z = -(transform.matrix().getDouble(2, 0) * x + transform.matrix().getDouble(2, 1) * y + transform.matrix().getDouble(2, 3)) / transform.matrix().getDouble(2, 2);
    // implicit definition of w = 1;

    double outX = x * transform.matrix().getDouble(0, 0) + y * transform.matrix().getDouble(0, 1) + z * transform.matrix().getDouble(0, 2) + transform.matrix().getDouble(0, 3);
    double outY = x * transform.matrix().getDouble(1, 0) + y * transform.matrix().getDouble(1, 1) + z * transform.matrix().getDouble(1, 2) + transform.matrix().getDouble(1, 3);
    double outZ = x * transform.matrix().getDouble(2, 0) + y * transform.matrix().getDouble(2, 1) + z * transform.matrix().getDouble(2, 2) + transform.matrix().getDouble(2, 3);
    double outW = x * transform.matrix().getDouble(3, 0) + y * transform.matrix().getDouble(3, 1) + z * transform.matrix().getDouble(3, 2) + transform.matrix().getDouble(3, 3);

    return HomogeneousCoordinate(outX, outY, outZ, outW);
}

static HomogeneousCoordinate mapHomogeneousPoint(const gfx::Transform& transform, const gfx::Point3F& p)
{
    double x = p.x();
    double y = p.y();
    double z = p.z();
    // implicit definition of w = 1;

    double outX = x * transform.matrix().getDouble(0, 0) + y * transform.matrix().getDouble(0, 1) + z * transform.matrix().getDouble(0, 2) + transform.matrix().getDouble(0, 3);
    double outY = x * transform.matrix().getDouble(1, 0) + y * transform.matrix().getDouble(1, 1) + z * transform.matrix().getDouble(1, 2) + transform.matrix().getDouble(1, 3);
    double outZ = x * transform.matrix().getDouble(2, 0) + y * transform.matrix().getDouble(2, 1) + z * transform.matrix().getDouble(2, 2) + transform.matrix().getDouble(2, 3);
    double outW = x * transform.matrix().getDouble(3, 0) + y * transform.matrix().getDouble(3, 1) + z * transform.matrix().getDouble(3, 2) + transform.matrix().getDouble(3, 3);

    return HomogeneousCoordinate(outX, outY, outZ, outW);
}

static HomogeneousCoordinate computeClippedPointForEdge(const HomogeneousCoordinate& h1, const HomogeneousCoordinate& h2)
{
    // Points h1 and h2 form a line in 4d, and any point on that line can be represented
    // as an interpolation between h1 and h2:
    //    p = (1-t) h1 + (t) h2
    //
    // We want to compute point p such that p.w == epsilon, where epsilon is a small
    // non-zero number. (but the smaller the number is, the higher the risk of overflow)
    // To do this, we solve for t in the following equation:
    //    p.w = epsilon = (1-t) * h1.w + (t) * h2.w
    //
    // Once paramter t is known, the rest of p can be computed via p = (1-t) h1 + (t) h2.

    // Technically this is a special case of the following assertion, but its a good idea to keep it an explicit sanity check here.
    DCHECK(h2.w != h1.w);
    // Exactly one of h1 or h2 (but not both) must be on the negative side of the w plane when this is called.
    DCHECK(h1.shouldBeClipped() ^ h2.shouldBeClipped());

    double w = 0.00001; // or any positive non-zero small epsilon

    double t = (w - h1.w) / (h2.w - h1.w);

    double x = (1-t) * h1.x + t * h2.x;
    double y = (1-t) * h1.y + t * h2.y;
    double z = (1-t) * h1.z + t * h2.z;

    return HomogeneousCoordinate(x, y, z, w);
}

static inline void expandBoundsToIncludePoint(float& xmin, float& xmax, float& ymin, float& ymax, const gfx::PointF& p)
{
    xmin = std::min(p.x(), xmin);
    xmax = std::max(p.x(), xmax);
    ymin = std::min(p.y(), ymin);
    ymax = std::max(p.y(), ymax);
}

static inline void addVertexToClippedQuad(const gfx::PointF& newVertex, gfx::PointF clippedQuad[8], int& numVerticesInClippedQuad)
{
    clippedQuad[numVerticesInClippedQuad] = newVertex;
    numVerticesInClippedQuad++;
}

gfx::Rect MathUtil::mapClippedRect(const gfx::Transform& transform, const gfx::Rect& srcRect)
{
    return gfx::ToEnclosingRect(mapClippedRect(transform, gfx::RectF(srcRect)));
}

gfx::RectF MathUtil::mapClippedRect(const gfx::Transform& transform, const gfx::RectF& srcRect)
{
    if (transform.IsIdentityOrTranslation())
        return srcRect + gfx::Vector2dF(static_cast<float>(transform.matrix().getDouble(0, 3)), static_cast<float>(transform.matrix().getDouble(1, 3)));
    
    // Apply the transform, but retain the result in homogeneous coordinates.

    double quad[4 * 2]; // input: 4 x 2D points
    quad[0] = srcRect.x();
    quad[1] = srcRect.y();
    quad[2] = srcRect.right();
    quad[3] = srcRect.y();
    quad[4] = srcRect.right();
    quad[5] = srcRect.bottom();
    quad[6] = srcRect.x();
    quad[7] = srcRect.bottom();

    double result[4 * 4];   // output: 4 x 4D homogeneous points
    transform.matrix().map2(quad, 4, result);

    HomogeneousCoordinate hc0(result[0], result[1], result[2], result[3]);
    HomogeneousCoordinate hc1(result[4], result[5], result[6], result[7]);
    HomogeneousCoordinate hc2(result[8], result[9], result[10], result[11]);
    HomogeneousCoordinate hc3(result[12], result[13], result[14], result[15]);
    return computeEnclosingClippedRect(hc0, hc1, hc2, hc3);
}

gfx::RectF MathUtil::projectClippedRect(const gfx::Transform& transform, const gfx::RectF& srcRect)
{
    if (transform.IsIdentityOrTranslation())
        return srcRect + gfx::Vector2dF(static_cast<float>(transform.matrix().getDouble(0, 3)), static_cast<float>(transform.matrix().getDouble(1, 3)));

    // Perform the projection, but retain the result in homogeneous coordinates.
    gfx::QuadF q = gfx::QuadF(srcRect);
    HomogeneousCoordinate h1 = projectHomogeneousPoint(transform, q.p1());
    HomogeneousCoordinate h2 = projectHomogeneousPoint(transform, q.p2());
    HomogeneousCoordinate h3 = projectHomogeneousPoint(transform, q.p3());
    HomogeneousCoordinate h4 = projectHomogeneousPoint(transform, q.p4());

    return computeEnclosingClippedRect(h1, h2, h3, h4);
}

void MathUtil::mapClippedQuad(const gfx::Transform& transform, const gfx::QuadF& srcQuad, gfx::PointF clippedQuad[8], int& numVerticesInClippedQuad)
{
    HomogeneousCoordinate h1 = mapHomogeneousPoint(transform, gfx::Point3F(srcQuad.p1()));
    HomogeneousCoordinate h2 = mapHomogeneousPoint(transform, gfx::Point3F(srcQuad.p2()));
    HomogeneousCoordinate h3 = mapHomogeneousPoint(transform, gfx::Point3F(srcQuad.p3()));
    HomogeneousCoordinate h4 = mapHomogeneousPoint(transform, gfx::Point3F(srcQuad.p4()));

    // The order of adding the vertices to the array is chosen so that clockwise / counter-clockwise orientation is retained.

    numVerticesInClippedQuad = 0;

    if (!h1.shouldBeClipped())
        addVertexToClippedQuad(h1.cartesianPoint2d(), clippedQuad, numVerticesInClippedQuad);

    if (h1.shouldBeClipped() ^ h2.shouldBeClipped())
        addVertexToClippedQuad(computeClippedPointForEdge(h1, h2).cartesianPoint2d(), clippedQuad, numVerticesInClippedQuad);

    if (!h2.shouldBeClipped())
        addVertexToClippedQuad(h2.cartesianPoint2d(), clippedQuad, numVerticesInClippedQuad);

    if (h2.shouldBeClipped() ^ h3.shouldBeClipped())
        addVertexToClippedQuad(computeClippedPointForEdge(h2, h3).cartesianPoint2d(), clippedQuad, numVerticesInClippedQuad);

    if (!h3.shouldBeClipped())
        addVertexToClippedQuad(h3.cartesianPoint2d(), clippedQuad, numVerticesInClippedQuad);

    if (h3.shouldBeClipped() ^ h4.shouldBeClipped())
        addVertexToClippedQuad(computeClippedPointForEdge(h3, h4).cartesianPoint2d(), clippedQuad, numVerticesInClippedQuad);

    if (!h4.shouldBeClipped())
        addVertexToClippedQuad(h4.cartesianPoint2d(), clippedQuad, numVerticesInClippedQuad);

    if (h4.shouldBeClipped() ^ h1.shouldBeClipped())
        addVertexToClippedQuad(computeClippedPointForEdge(h4, h1).cartesianPoint2d(), clippedQuad, numVerticesInClippedQuad);

    DCHECK(numVerticesInClippedQuad <= 8);
}

gfx::RectF MathUtil::computeEnclosingRectOfVertices(gfx::PointF vertices[], int numVertices)
{
    if (numVertices < 2)
        return gfx::RectF();

    float xmin = std::numeric_limits<float>::max();
    float xmax = -std::numeric_limits<float>::max();
    float ymin = std::numeric_limits<float>::max();
    float ymax = -std::numeric_limits<float>::max();

    for (int i = 0; i < numVertices; ++i)
        expandBoundsToIncludePoint(xmin, xmax, ymin, ymax, vertices[i]);

    return gfx::RectF(gfx::PointF(xmin, ymin), gfx::SizeF(xmax - xmin, ymax - ymin));
}

gfx::RectF MathUtil::computeEnclosingClippedRect(const HomogeneousCoordinate& h1, const HomogeneousCoordinate& h2, const HomogeneousCoordinate& h3, const HomogeneousCoordinate& h4)
{
    // This function performs clipping as necessary and computes the enclosing 2d
    // gfx::RectF of the vertices. Doing these two steps simultaneously allows us to avoid
    // the overhead of storing an unknown number of clipped vertices.

    // If no vertices on the quad are clipped, then we can simply return the enclosing rect directly.
    bool somethingClipped = h1.shouldBeClipped() || h2.shouldBeClipped() || h3.shouldBeClipped() || h4.shouldBeClipped();
    if (!somethingClipped) {
        gfx::QuadF mappedQuad = gfx::QuadF(h1.cartesianPoint2d(), h2.cartesianPoint2d(), h3.cartesianPoint2d(), h4.cartesianPoint2d());
        return mappedQuad.BoundingBox();
    }

    bool everythingClipped = h1.shouldBeClipped() && h2.shouldBeClipped() && h3.shouldBeClipped() && h4.shouldBeClipped();
    if (everythingClipped)
        return gfx::RectF();


    float xmin = std::numeric_limits<float>::max();
    float xmax = -std::numeric_limits<float>::max();
    float ymin = std::numeric_limits<float>::max();
    float ymax = -std::numeric_limits<float>::max();

    if (!h1.shouldBeClipped())
        expandBoundsToIncludePoint(xmin, xmax, ymin, ymax, h1.cartesianPoint2d());

    if (h1.shouldBeClipped() ^ h2.shouldBeClipped())
        expandBoundsToIncludePoint(xmin, xmax, ymin, ymax, computeClippedPointForEdge(h1, h2).cartesianPoint2d());

    if (!h2.shouldBeClipped())
        expandBoundsToIncludePoint(xmin, xmax, ymin, ymax, h2.cartesianPoint2d());

    if (h2.shouldBeClipped() ^ h3.shouldBeClipped())
        expandBoundsToIncludePoint(xmin, xmax, ymin, ymax, computeClippedPointForEdge(h2, h3).cartesianPoint2d());

    if (!h3.shouldBeClipped())
        expandBoundsToIncludePoint(xmin, xmax, ymin, ymax, h3.cartesianPoint2d());

    if (h3.shouldBeClipped() ^ h4.shouldBeClipped())
        expandBoundsToIncludePoint(xmin, xmax, ymin, ymax, computeClippedPointForEdge(h3, h4).cartesianPoint2d());

    if (!h4.shouldBeClipped())
        expandBoundsToIncludePoint(xmin, xmax, ymin, ymax, h4.cartesianPoint2d());

    if (h4.shouldBeClipped() ^ h1.shouldBeClipped())
        expandBoundsToIncludePoint(xmin, xmax, ymin, ymax, computeClippedPointForEdge(h4, h1).cartesianPoint2d());

    return gfx::RectF(gfx::PointF(xmin, ymin), gfx::SizeF(xmax - xmin, ymax - ymin));
}

gfx::QuadF MathUtil::mapQuad(const gfx::Transform& transform, const gfx::QuadF& q, bool& clipped)
{
    if (transform.IsIdentityOrTranslation()) {
        gfx::QuadF mappedQuad(q);
        mappedQuad += gfx::Vector2dF(static_cast<float>(transform.matrix().getDouble(0, 3)), static_cast<float>(transform.matrix().getDouble(1, 3)));
        clipped = false;
        return mappedQuad;
    }

    HomogeneousCoordinate h1 = mapHomogeneousPoint(transform, gfx::Point3F(q.p1()));
    HomogeneousCoordinate h2 = mapHomogeneousPoint(transform, gfx::Point3F(q.p2()));
    HomogeneousCoordinate h3 = mapHomogeneousPoint(transform, gfx::Point3F(q.p3()));
    HomogeneousCoordinate h4 = mapHomogeneousPoint(transform, gfx::Point3F(q.p4()));

    clipped = h1.shouldBeClipped() || h2.shouldBeClipped() || h3.shouldBeClipped() || h4.shouldBeClipped();

    // Result will be invalid if clipped == true. But, compute it anyway just in case, to emulate existing behavior.
    return gfx::QuadF(h1.cartesianPoint2d(), h2.cartesianPoint2d(), h3.cartesianPoint2d(), h4.cartesianPoint2d());
}

gfx::PointF MathUtil::mapPoint(const gfx::Transform& transform, const gfx::PointF& p, bool& clipped)
{
    HomogeneousCoordinate h = mapHomogeneousPoint(transform, gfx::Point3F(p));

    if (h.w > 0) {
        clipped = false;
        return h.cartesianPoint2d();
    }

    // The cartesian coordinates will be invalid after dividing by w.
    clipped = true;

    // Avoid dividing by w if w == 0.
    if (!h.w)
        return gfx::PointF();

    // This return value will be invalid because clipped == true, but (1) users of this
    // code should be ignoring the return value when clipped == true anyway, and (2) this
    // behavior is more consistent with existing behavior of WebKit transforms if the user
    // really does not ignore the return value.
    return h.cartesianPoint2d();
}

gfx::Point3F MathUtil::mapPoint(const gfx::Transform& transform, const gfx::Point3F& p, bool& clipped)
{
    HomogeneousCoordinate h = mapHomogeneousPoint(transform, p);

    if (h.w > 0) {
        clipped = false;
        return h.cartesianPoint3d();
    }

    // The cartesian coordinates will be invalid after dividing by w.
    clipped = true;

    // Avoid dividing by w if w == 0.
    if (!h.w)
        return gfx::Point3F();

    // This return value will be invalid because clipped == true, but (1) users of this
    // code should be ignoring the return value when clipped == true anyway, and (2) this
    // behavior is more consistent with existing behavior of WebKit transforms if the user
    // really does not ignore the return value.
    return h.cartesianPoint3d();
}

gfx::QuadF MathUtil::projectQuad(const gfx::Transform& transform, const gfx::QuadF& q, bool& clipped)
{
    gfx::QuadF projectedQuad;
    bool clippedPoint;
    projectedQuad.set_p1(projectPoint(transform, q.p1(), clippedPoint));
    clipped = clippedPoint;
    projectedQuad.set_p2(projectPoint(transform, q.p2(), clippedPoint));
    clipped |= clippedPoint;
    projectedQuad.set_p3(projectPoint(transform, q.p3(), clippedPoint));
    clipped |= clippedPoint;
    projectedQuad.set_p4(projectPoint(transform, q.p4(), clippedPoint));
    clipped |= clippedPoint;

    return projectedQuad;
}

gfx::PointF MathUtil::projectPoint(const gfx::Transform& transform, const gfx::PointF& p, bool& clipped)
{
    HomogeneousCoordinate h = projectHomogeneousPoint(transform, p);

    if (h.w > 0) {
        // The cartesian coordinates will be valid in this case.
        clipped = false;
        return h.cartesianPoint2d();
    }

    // The cartesian coordinates will be invalid after dividing by w.
    clipped = true;

    // Avoid dividing by w if w == 0.
    if (!h.w)
        return gfx::PointF();

    // This return value will be invalid because clipped == true, but (1) users of this
    // code should be ignoring the return value when clipped == true anyway, and (2) this
    // behavior is more consistent with existing behavior of WebKit transforms if the user
    // really does not ignore the return value.
    return h.cartesianPoint2d();
}

static inline float scaleOnAxis(double a, double b, double c)
{
    return std::sqrt(a * a + b * b + c * c);
}

gfx::Vector2dF MathUtil::computeTransform2dScaleComponents(const gfx::Transform& transform, float fallbackValue)
{
    if (transform.HasPerspective())
        return gfx::Vector2dF(fallbackValue, fallbackValue);
    float xScale = scaleOnAxis(transform.matrix().getDouble(0, 0), transform.matrix().getDouble(1, 0), transform.matrix().getDouble(2, 0));
    float yScale = scaleOnAxis(transform.matrix().getDouble(0, 1), transform.matrix().getDouble(1, 1), transform.matrix().getDouble(2, 1));
    return gfx::Vector2dF(xScale, yScale);
}

float MathUtil::smallestAngleBetweenVectors(gfx::Vector2dF v1, gfx::Vector2dF v2)
{
    double dotProduct = gfx::DotProduct(v1, v2) / v1.Length() / v2.Length();
    // Clamp to compensate for rounding errors.
    dotProduct = std::max(-1.0, std::min(1.0, dotProduct));
    return static_cast<float>(Rad2Deg(std::acos(dotProduct)));
}

gfx::Vector2dF MathUtil::projectVector(gfx::Vector2dF source, gfx::Vector2dF destination)
{
    float projectedLength = gfx::DotProduct(source, destination) / destination.LengthSquared();
    return gfx::Vector2dF(projectedLength * destination.x(), projectedLength * destination.y());
}

}  // namespace cc