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 // Copyright (c) 2013 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 "cobalt/math/matrix3_f.h" #include #include #include namespace cobalt { namespace math { namespace { // This is only to make accessing indices self-explanatory. enum MatrixCoordinates { M00, M01, M02, M10, M11, M12, M20, M21, M22, M_END }; template double Determinant3x3(T data[M_END]) { // This routine is separated from the Matrix3F::Determinant because in // computing inverse we do want higher precision afforded by the explicit // use of 'double'. return static_cast(data[M00]) * (static_cast(data[M11]) * data[M22] - static_cast(data[M12]) * data[M21]) + static_cast(data[M01]) * (static_cast(data[M12]) * data[M20] - static_cast(data[M10]) * data[M22]) + static_cast(data[M02]) * (static_cast(data[M10]) * data[M21] - static_cast(data[M11]) * data[M20]); } } // namespace Matrix3F::Matrix3F() {} Matrix3F::~Matrix3F() {} // static Matrix3F Matrix3F::Zeros() { Matrix3F matrix; matrix.SetMatrix(0.0f, 0.0f, 0.0f, 0.0f, 0.0f, 0.0f, 0.0f, 0.0f, 0.0f); return matrix; } // static Matrix3F Matrix3F::Ones() { Matrix3F matrix; matrix.SetMatrix(1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f); return matrix; } // static Matrix3F Matrix3F::Identity() { Matrix3F matrix; matrix.SetMatrix(1.0f, 0.0f, 0.0f, 0.0f, 1.0f, 0.0f, 0.0f, 0.0f, 1.0f); return matrix; } // static Matrix3F Matrix3F::FromOuterProduct(const Vector3dF& a, const Vector3dF& bt) { Matrix3F matrix; matrix.SetMatrix(a.x() * bt.x(), a.x() * bt.y(), a.x() * bt.z(), a.y() * bt.x(), a.y() * bt.y(), a.y() * bt.z(), a.z() * bt.x(), a.z() * bt.y(), a.z() * bt.z()); return matrix; } // static Matrix3F Matrix3F::FromArray(const float data[9]) { Matrix3F matrix; memcpy(matrix.data_, data, sizeof(float) * 9); return matrix; } // static Matrix3F Matrix3F::FromValues(float m00, float m01, float m02, float m10, float m11, float m12, float m20, float m21, float m22) { Matrix3F matrix; matrix.SetMatrix(m00, m01, m02, m10, m11, m12, m20, m21, m22); return matrix; } bool Matrix3F::IsZeros() const { return data_[M00] == 0.0f && data_[M01] == 0.0f && data_[M02] == 0.0f && data_[M10] == 0.0f && data_[M11] == 0.0f && data_[M12] == 0.0f && data_[M20] == 0.0f && data_[M21] == 0.0f && data_[M22] == 0.0f; } bool Matrix3F::IsIdentity() const { return data_[M00] == 1.0f && data_[M01] == 0.0f && data_[M02] == 0.0f && data_[M10] == 0.0f && data_[M11] == 1.0f && data_[M12] == 0.0f && data_[M20] == 0.0f && data_[M21] == 0.0f && data_[M22] == 1.0f; } bool Matrix3F::IsEqual(const Matrix3F& rhs) const { return 0 == memcmp(data_, rhs.data_, sizeof(data_)); } bool Matrix3F::IsNear(const Matrix3F& rhs, float precision) const { DCHECK_GE(precision, 0); for (int i = 0; i < M_END; ++i) { if (std::abs(data_[i] - rhs.data_[i]) > precision) return false; } return true; } Matrix3F Matrix3F::Inverse() const { Matrix3F inverse = Matrix3F::Zeros(); double determinant = Determinant3x3(data_); if (std::numeric_limits::epsilon() > std::abs(determinant)) return inverse; // Singular matrix. Return Zeros(). inverse.SetMatrix( (data_[M11] * data_[M22] - data_[M12] * data_[M21]) / determinant, (data_[M02] * data_[M21] - data_[M01] * data_[M22]) / determinant, (data_[M01] * data_[M12] - data_[M02] * data_[M11]) / determinant, (data_[M12] * data_[M20] - data_[M10] * data_[M22]) / determinant, (data_[M00] * data_[M22] - data_[M02] * data_[M20]) / determinant, (data_[M02] * data_[M10] - data_[M00] * data_[M12]) / determinant, (data_[M10] * data_[M21] - data_[M11] * data_[M20]) / determinant, (data_[M01] * data_[M20] - data_[M00] * data_[M21]) / determinant, (data_[M00] * data_[M11] - data_[M01] * data_[M10]) / determinant); return inverse; } float Matrix3F::Determinant() const { return static_cast(Determinant3x3(data_)); } Vector3dF Matrix3F::SolveEigenproblem(Matrix3F* eigenvectors) const { // The matrix must be symmetric. const float epsilon = std::numeric_limits::epsilon(); if (std::abs(data_[M01] - data_[M10]) > epsilon || std::abs(data_[M02] - data_[M20]) > epsilon || std::abs(data_[M12] - data_[M21]) > epsilon) { NOTREACHED(); return Vector3dF(); } float eigenvalues[3]; float p = data_[M01] * data_[M01] + data_[M02] * data_[M02] + data_[M12] * data_[M12]; bool diagonal = std::abs(p) < epsilon; if (diagonal) { eigenvalues[0] = data_[M00]; eigenvalues[1] = data_[M11]; eigenvalues[2] = data_[M22]; } else { float q = Trace() / 3.0f; p = (data_[M00] - q) * (data_[M00] - q) + (data_[M11] - q) * (data_[M11] - q) + (data_[M22] - q) * (data_[M22] - q) + 2 * p; p = std::sqrt(p / 6); // The computation below puts B as (A - qI) / p, where A is *this. Matrix3F matrix_b(*this); matrix_b.data_[M00] -= q; matrix_b.data_[M11] -= q; matrix_b.data_[M22] -= q; for (int i = 0; i < M_END; ++i) matrix_b.data_[i] /= p; double half_det_b = Determinant3x3(matrix_b.data_) / 2.0; // half_det_b should be in <-1, 1>, but beware of rounding error. double phi = 0.0f; if (half_det_b <= -1.0) phi = M_PI / 3; else if (half_det_b < 1.0) phi = acos(half_det_b) / 3; eigenvalues[0] = q + 2 * p * static_cast(cos(phi)); eigenvalues[2] = q + 2 * p * static_cast(cos(phi + 2.0 * M_PI / 3.0)); eigenvalues[1] = 3 * q - eigenvalues[0] - eigenvalues[2]; } // Put eigenvalues in the descending order. int indices[3] = {0, 1, 2}; if (eigenvalues[2] > eigenvalues[1]) { std::swap(eigenvalues[2], eigenvalues[1]); std::swap(indices[2], indices[1]); } if (eigenvalues[1] > eigenvalues[0]) { std::swap(eigenvalues[1], eigenvalues[0]); std::swap(indices[1], indices[0]); } if (eigenvalues[2] > eigenvalues[1]) { std::swap(eigenvalues[2], eigenvalues[1]); std::swap(indices[2], indices[1]); } if (eigenvectors != NULL && diagonal) { // Eigenvectors are e-vectors, just need to be sorted accordingly. *eigenvectors = Zeros(); for (int i = 0; i < 3; ++i) eigenvectors->Set(indices[i], i, 1.0f); } else if (eigenvectors != NULL) { // Consult the following for a detailed discussion: // Joachim Kopp // Numerical diagonalization of hermitian 3x3 matrices // arXiv.org preprint: physics/0610206 // Int. J. Mod. Phys. C19 (2008) 523-548 // TODO(motek): expand to handle correctly negative and multiple // eigenvalues. for (int i = 0; i < 3; ++i) { float l = eigenvalues[i]; // B = A - l * I Matrix3F matrix_b(*this); matrix_b.data_[M00] -= l; matrix_b.data_[M11] -= l; matrix_b.data_[M22] -= l; Vector3dF e1 = CrossProduct(matrix_b.column(0), matrix_b.column(1)); Vector3dF e2 = CrossProduct(matrix_b.column(1), matrix_b.column(2)); Vector3dF e3 = CrossProduct(matrix_b.column(2), matrix_b.column(0)); // e1, e2 and e3 should point in the same direction. if (DotProduct(e1, e2) < 0) e2 = -e2; if (DotProduct(e1, e3) < 0) e3 = -e3; Vector3dF eigvec = e1 + e2 + e3; // Normalize. eigvec.Scale(1.0f / eigvec.Length()); eigenvectors->set_column(i, eigvec); } } return Vector3dF(eigenvalues[0], eigenvalues[1], eigenvalues[2]); } Matrix3F Matrix3F::operator*(const Matrix3F& other) const { Matrix3F ret; ret.data_[M00] = data_[M00] * other.data_[M00] + data_[M01] * other.data_[M10] + data_[M02] * other.data_[M20]; ret.data_[M01] = data_[M00] * other.data_[M01] + data_[M01] * other.data_[M11] + data_[M02] * other.data_[M21]; ret.data_[M02] = data_[M00] * other.data_[M02] + data_[M01] * other.data_[M12] + data_[M02] * other.data_[M22]; ret.data_[M10] = data_[M10] * other.data_[M00] + data_[M11] * other.data_[M10] + data_[M12] * other.data_[M20]; ret.data_[M11] = data_[M10] * other.data_[M01] + data_[M11] * other.data_[M11] + data_[M12] * other.data_[M21]; ret.data_[M12] = data_[M10] * other.data_[M02] + data_[M11] * other.data_[M12] + data_[M12] * other.data_[M22]; ret.data_[M20] = data_[M20] * other.data_[M00] + data_[M21] * other.data_[M10] + data_[M22] * other.data_[M20]; ret.data_[M21] = data_[M20] * other.data_[M01] + data_[M21] * other.data_[M11] + data_[M22] * other.data_[M21]; ret.data_[M22] = data_[M20] * other.data_[M02] + data_[M21] * other.data_[M12] + data_[M22] * other.data_[M22]; return ret; } PointF Matrix3F::operator*(const PointF& rhs) const { float x = rhs.x() * data_[M00] + rhs.y() * data_[M01] + data_[M02]; float y = rhs.x() * data_[M10] + rhs.y() * data_[M11] + data_[M12]; float z = rhs.x() * data_[M20] + rhs.y() * data_[M21] + data_[M22]; DCHECK_NE(z, 0.f); return PointF(x / z, y / z); } RectF Matrix3F::MapRect(const RectF& rect) const { PointF points[4]; points[0] = *this * rect.origin(); points[1] = *this * rect.top_right(); points[2] = *this * rect.bottom_left(); points[3] = *this * rect.bottom_right(); float min_x = std::numeric_limits::max(); float max_x = -std::numeric_limits::max(); float min_y = std::numeric_limits::max(); float max_y = -std::numeric_limits::max(); for (int i = 0; i < 4; ++i) { min_x = std::min(min_x, points[i].x()); max_x = std::max(max_x, points[i].x()); min_y = std::min(min_y, points[i].y()); max_y = std::max(max_y, points[i].y()); } return RectF(min_x, min_y, max_x - min_x, max_y - min_y); } } // namespace math } // namespace cobalt