| // Copyright 2011 The Closure Library Authors. All Rights Reserved. |
| // |
| // Licensed under the Apache License, Version 2.0 (the "License"); |
| // you may not use this file except in compliance with the License. |
| // You may obtain a copy of the License at |
| // |
| // http://www.apache.org/licenses/LICENSE-2.0 |
| // |
| // Unless required by applicable law or agreed to in writing, software |
| // distributed under the License is distributed on an "AS-IS" BASIS, |
| // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. |
| // See the License for the specific language governing permissions and |
| // limitations under the License. |
| |
| /** |
| * @fileoverview Implements 3x3 matrices and their related functions which are |
| * compatible with WebGL. The API is structured to avoid unnecessary memory |
| * allocations. The last parameter will typically be the output vector and |
| * an object can be both an input and output parameter to all methods except |
| * where noted. Matrix operations follow the mathematical form when multiplying |
| * vectors as follows: resultVec = matrix * vec. |
| * |
| * The matrices are stored in column-major order. |
| * |
| */ |
| goog.provide('goog.vec.Mat3'); |
| |
| goog.require('goog.vec'); |
| |
| |
| /** @typedef {goog.vec.Float32} */ goog.vec.Mat3.Float32; |
| /** @typedef {goog.vec.Float64} */ goog.vec.Mat3.Float64; |
| /** @typedef {goog.vec.Number} */ goog.vec.Mat3.Number; |
| /** @typedef {goog.vec.AnyType} */ goog.vec.Mat3.AnyType; |
| |
| // The following two types are deprecated - use the above types instead. |
| /** @typedef {Float32Array} */ goog.vec.Mat3.Type; |
| /** @typedef {goog.vec.ArrayType} */ goog.vec.Mat3.Mat3Like; |
| |
| |
| /** |
| * Creates the array representation of a 3x3 matrix of Float32. |
| * The use of the array directly instead of a class reduces overhead. |
| * The returned matrix is cleared to all zeros. |
| * |
| * @return {!goog.vec.Mat3.Float32} The new matrix. |
| */ |
| goog.vec.Mat3.createFloat32 = function() { |
| return new Float32Array(9); |
| }; |
| |
| |
| /** |
| * Creates the array representation of a 3x3 matrix of Float64. |
| * The returned matrix is cleared to all zeros. |
| * |
| * @return {!goog.vec.Mat3.Float64} The new matrix. |
| */ |
| goog.vec.Mat3.createFloat64 = function() { |
| return new Float64Array(9); |
| }; |
| |
| |
| /** |
| * Creates the array representation of a 3x3 matrix of Number. |
| * The returned matrix is cleared to all zeros. |
| * |
| * @return {!goog.vec.Mat3.Number} The new matrix. |
| */ |
| goog.vec.Mat3.createNumber = function() { |
| var a = new Array(9); |
| goog.vec.Mat3.setFromValues(a, |
| 0, 0, 0, |
| 0, 0, 0, |
| 0, 0, 0); |
| return a; |
| }; |
| |
| |
| /** |
| * Creates the array representation of a 3x3 matrix of Float32. |
| * The returned matrix is cleared to all zeros. |
| * |
| * @deprecated Use createFloat32. |
| * @return {!goog.vec.Mat3.Type} The new matrix. |
| */ |
| goog.vec.Mat3.create = function() { |
| return goog.vec.Mat3.createFloat32(); |
| }; |
| |
| |
| /** |
| * Creates a 3x3 identity matrix of Float32. |
| * |
| * @return {!goog.vec.Mat3.Float32} The new 9 element array. |
| */ |
| goog.vec.Mat3.createFloat32Identity = function() { |
| var mat = goog.vec.Mat3.createFloat32(); |
| mat[0] = mat[4] = mat[8] = 1; |
| return mat; |
| }; |
| |
| |
| /** |
| * Creates a 3x3 identity matrix of Float64. |
| * |
| * @return {!goog.vec.Mat3.Float64} The new 9 element array. |
| */ |
| goog.vec.Mat3.createFloat64Identity = function() { |
| var mat = goog.vec.Mat3.createFloat64(); |
| mat[0] = mat[4] = mat[8] = 1; |
| return mat; |
| }; |
| |
| |
| /** |
| * Creates a 3x3 identity matrix of Number. |
| * The returned matrix is cleared to all zeros. |
| * |
| * @return {!goog.vec.Mat3.Number} The new 9 element array. |
| */ |
| goog.vec.Mat3.createNumberIdentity = function() { |
| var a = new Array(9); |
| goog.vec.Mat3.setFromValues(a, |
| 1, 0, 0, |
| 0, 1, 0, |
| 0, 0, 1); |
| return a; |
| }; |
| |
| |
| /** |
| * Creates the array representation of a 3x3 matrix of Float32. |
| * The returned matrix is cleared to all zeros. |
| * |
| * @deprecated Use createFloat32Identity. |
| * @return {!goog.vec.Mat3.Type} The new 9 element array. |
| */ |
| goog.vec.Mat3.createIdentity = function() { |
| return goog.vec.Mat3.createFloat32Identity(); |
| }; |
| |
| |
| /** |
| * Creates a 3x3 matrix of Float32 initialized from the given array. |
| * |
| * @param {goog.vec.Mat3.AnyType} matrix The array containing the |
| * matrix values in column major order. |
| * @return {!goog.vec.Mat3.Float32} The new, nine element array. |
| */ |
| goog.vec.Mat3.createFloat32FromArray = function(matrix) { |
| var newMatrix = goog.vec.Mat3.createFloat32(); |
| goog.vec.Mat3.setFromArray(newMatrix, matrix); |
| return newMatrix; |
| }; |
| |
| |
| /** |
| * Creates a 3x3 matrix of Float32 initialized from the given values. |
| * |
| * @param {number} v00 The values at (0, 0). |
| * @param {number} v10 The values at (1, 0). |
| * @param {number} v20 The values at (2, 0). |
| * @param {number} v01 The values at (0, 1). |
| * @param {number} v11 The values at (1, 1). |
| * @param {number} v21 The values at (2, 1). |
| * @param {number} v02 The values at (0, 2). |
| * @param {number} v12 The values at (1, 2). |
| * @param {number} v22 The values at (2, 2). |
| * @return {!goog.vec.Mat3.Float32} The new, nine element array. |
| */ |
| goog.vec.Mat3.createFloat32FromValues = function( |
| v00, v10, v20, v01, v11, v21, v02, v12, v22) { |
| var newMatrix = goog.vec.Mat3.createFloat32(); |
| goog.vec.Mat3.setFromValues( |
| newMatrix, v00, v10, v20, v01, v11, v21, v02, v12, v22); |
| return newMatrix; |
| }; |
| |
| |
| /** |
| * Creates a clone of a 3x3 matrix of Float32. |
| * |
| * @param {goog.vec.Mat3.Float32} matrix The source 3x3 matrix. |
| * @return {!goog.vec.Mat3.Float32} The new 3x3 element matrix. |
| */ |
| goog.vec.Mat3.cloneFloat32 = goog.vec.Mat3.createFloat32FromArray; |
| |
| |
| /** |
| * Creates a 3x3 matrix of Float64 initialized from the given array. |
| * |
| * @param {goog.vec.Mat3.AnyType} matrix The array containing the |
| * matrix values in column major order. |
| * @return {!goog.vec.Mat3.Float64} The new, nine element array. |
| */ |
| goog.vec.Mat3.createFloat64FromArray = function(matrix) { |
| var newMatrix = goog.vec.Mat3.createFloat64(); |
| goog.vec.Mat3.setFromArray(newMatrix, matrix); |
| return newMatrix; |
| }; |
| |
| |
| /** |
| * Creates a 3x3 matrix of Float64 initialized from the given values. |
| * |
| * @param {number} v00 The values at (0, 0). |
| * @param {number} v10 The values at (1, 0). |
| * @param {number} v20 The values at (2, 0). |
| * @param {number} v01 The values at (0, 1). |
| * @param {number} v11 The values at (1, 1). |
| * @param {number} v21 The values at (2, 1). |
| * @param {number} v02 The values at (0, 2). |
| * @param {number} v12 The values at (1, 2). |
| * @param {number} v22 The values at (2, 2). |
| * @return {!goog.vec.Mat3.Float64} The new, nine element array. |
| */ |
| goog.vec.Mat3.createFloat64FromValues = function( |
| v00, v10, v20, v01, v11, v21, v02, v12, v22) { |
| var newMatrix = goog.vec.Mat3.createFloat64(); |
| goog.vec.Mat3.setFromValues( |
| newMatrix, v00, v10, v20, v01, v11, v21, v02, v12, v22); |
| return newMatrix; |
| }; |
| |
| |
| /** |
| * Creates a clone of a 3x3 matrix of Float64. |
| * |
| * @param {goog.vec.Mat3.Float64} matrix The source 3x3 matrix. |
| * @return {!goog.vec.Mat3.Float64} The new 3x3 element matrix. |
| */ |
| goog.vec.Mat3.cloneFloat64 = goog.vec.Mat3.createFloat64FromArray; |
| |
| |
| /** |
| * Creates a 3x3 matrix of Float32 initialized from the given array. |
| * |
| * @deprecated Use createFloat32FromArray. |
| * @param {goog.vec.Mat3.Mat3Like} matrix The array containing the |
| * matrix values in column major order. |
| * @return {!goog.vec.Mat3.Type} The new, nine element array. |
| */ |
| goog.vec.Mat3.createFromArray = function(matrix) { |
| var newMatrix = goog.vec.Mat3.createFloat32(); |
| goog.vec.Mat3.setFromArray(newMatrix, matrix); |
| return newMatrix; |
| }; |
| |
| |
| /** |
| * Creates a 3x3 matrix of Float32 initialized from the given values. |
| * |
| * @deprecated Use createFloat32FromValues. |
| * @param {number} v00 The values at (0, 0). |
| * @param {number} v10 The values at (1, 0). |
| * @param {number} v20 The values at (2, 0). |
| * @param {number} v01 The values at (0, 1). |
| * @param {number} v11 The values at (1, 1). |
| * @param {number} v21 The values at (2, 1). |
| * @param {number} v02 The values at (0, 2). |
| * @param {number} v12 The values at (1, 2). |
| * @param {number} v22 The values at (2, 2). |
| * @return {!goog.vec.Mat3.Type} The new, nine element array. |
| */ |
| goog.vec.Mat3.createFromValues = function( |
| v00, v10, v20, v01, v11, v21, v02, v12, v22) { |
| var newMatrix = goog.vec.Mat3.create(); |
| goog.vec.Mat3.setFromValues( |
| newMatrix, v00, v10, v20, v01, v11, v21, v02, v12, v22); |
| return newMatrix; |
| }; |
| |
| |
| /** |
| * Creates a clone of a 3x3 matrix of Float32. |
| * |
| * @deprecated Use cloneFloat32. |
| * @param {goog.vec.Mat3.Mat3Like} matrix The source 3x3 matrix. |
| * @return {!goog.vec.Mat3.Type} The new 3x3 element matrix. |
| */ |
| goog.vec.Mat3.clone = goog.vec.Mat3.createFromArray; |
| |
| |
| /** |
| * Retrieves the element at the requested row and column. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix containing the value to |
| * retrieve. |
| * @param {number} row The row index. |
| * @param {number} column The column index. |
| * @return {number} The element value at the requested row, column indices. |
| */ |
| goog.vec.Mat3.getElement = function(mat, row, column) { |
| return mat[row + column * 3]; |
| }; |
| |
| |
| /** |
| * Sets the element at the requested row and column. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix containing the value to |
| * retrieve. |
| * @param {number} row The row index. |
| * @param {number} column The column index. |
| * @param {number} value The value to set at the requested row, column. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.setElement = function(mat, row, column, value) { |
| mat[row + column * 3] = value; |
| return mat; |
| }; |
| |
| |
| /** |
| * Initializes the matrix from the set of values. Note the values supplied are |
| * in column major order. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix to receive the |
| * values. |
| * @param {number} v00 The values at (0, 0). |
| * @param {number} v10 The values at (1, 0). |
| * @param {number} v20 The values at (2, 0). |
| * @param {number} v01 The values at (0, 1). |
| * @param {number} v11 The values at (1, 1). |
| * @param {number} v21 The values at (2, 1). |
| * @param {number} v02 The values at (0, 2). |
| * @param {number} v12 The values at (1, 2). |
| * @param {number} v22 The values at (2, 2). |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.setFromValues = function( |
| mat, v00, v10, v20, v01, v11, v21, v02, v12, v22) { |
| mat[0] = v00; |
| mat[1] = v10; |
| mat[2] = v20; |
| mat[3] = v01; |
| mat[4] = v11; |
| mat[5] = v21; |
| mat[6] = v02; |
| mat[7] = v12; |
| mat[8] = v22; |
| return mat; |
| }; |
| |
| |
| /** |
| * Sets the matrix from the array of values stored in column major order. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix to receive the values. |
| * @param {goog.vec.Mat3.AnyType} values The column major ordered |
| * array of values to store in the matrix. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.setFromArray = function(mat, values) { |
| mat[0] = values[0]; |
| mat[1] = values[1]; |
| mat[2] = values[2]; |
| mat[3] = values[3]; |
| mat[4] = values[4]; |
| mat[5] = values[5]; |
| mat[6] = values[6]; |
| mat[7] = values[7]; |
| mat[8] = values[8]; |
| return mat; |
| }; |
| |
| |
| /** |
| * Sets the matrix from the array of values stored in row major order. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix to receive the values. |
| * @param {goog.vec.Mat3.AnyType} values The row major ordered array |
| * of values to store in the matrix. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.setFromRowMajorArray = function(mat, values) { |
| mat[0] = values[0]; |
| mat[1] = values[3]; |
| mat[2] = values[6]; |
| mat[3] = values[1]; |
| mat[4] = values[4]; |
| mat[5] = values[7]; |
| mat[6] = values[2]; |
| mat[7] = values[5]; |
| mat[8] = values[8]; |
| return mat; |
| }; |
| |
| |
| /** |
| * Sets the diagonal values of the matrix from the given values. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix to receive the values. |
| * @param {number} v00 The values for (0, 0). |
| * @param {number} v11 The values for (1, 1). |
| * @param {number} v22 The values for (2, 2). |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.setDiagonalValues = function(mat, v00, v11, v22) { |
| mat[0] = v00; |
| mat[4] = v11; |
| mat[8] = v22; |
| return mat; |
| }; |
| |
| |
| /** |
| * Sets the diagonal values of the matrix from the given vector. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix to receive the values. |
| * @param {goog.vec.Vec3.AnyType} vec The vector containing the values. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.setDiagonal = function(mat, vec) { |
| mat[0] = vec[0]; |
| mat[4] = vec[1]; |
| mat[8] = vec[2]; |
| return mat; |
| }; |
| |
| |
| /** |
| * Sets the specified column with the supplied values. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix to recieve the values. |
| * @param {number} column The column index to set the values on. |
| * @param {number} v0 The value for row 0. |
| * @param {number} v1 The value for row 1. |
| * @param {number} v2 The value for row 2. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.setColumnValues = function(mat, column, v0, v1, v2) { |
| var i = column * 3; |
| mat[i] = v0; |
| mat[i + 1] = v1; |
| mat[i + 2] = v2; |
| return mat; |
| }; |
| |
| |
| /** |
| * Sets the specified column with the value from the supplied array. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix to receive the values. |
| * @param {number} column The column index to set the values on. |
| * @param {goog.vec.Vec3.AnyType} vec The vector elements for the column. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.setColumn = function(mat, column, vec) { |
| var i = column * 3; |
| mat[i] = vec[0]; |
| mat[i + 1] = vec[1]; |
| mat[i + 2] = vec[2]; |
| return mat; |
| }; |
| |
| |
| /** |
| * Retrieves the specified column from the matrix into the given vector |
| * array. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix supplying the values. |
| * @param {number} column The column to get the values from. |
| * @param {goog.vec.Vec3.AnyType} vec The vector elements to receive the |
| * column. |
| * @return {goog.vec.Vec3.AnyType} return vec so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.getColumn = function(mat, column, vec) { |
| var i = column * 3; |
| vec[0] = mat[i]; |
| vec[1] = mat[i + 1]; |
| vec[2] = mat[i + 2]; |
| return vec; |
| }; |
| |
| |
| /** |
| * Sets the columns of the matrix from the set of vector elements. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix to receive the values. |
| * @param {goog.vec.Vec3.AnyType} vec0 The values for column 0. |
| * @param {goog.vec.Vec3.AnyType} vec1 The values for column 1. |
| * @param {goog.vec.Vec3.AnyType} vec2 The values for column 2. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.setColumns = function(mat, vec0, vec1, vec2) { |
| goog.vec.Mat3.setColumn(mat, 0, vec0); |
| goog.vec.Mat3.setColumn(mat, 1, vec1); |
| goog.vec.Mat3.setColumn(mat, 2, vec2); |
| return mat; |
| }; |
| |
| |
| /** |
| * Retrieves the column values from the given matrix into the given vector |
| * elements. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix supplying the columns. |
| * @param {goog.vec.Vec3.AnyType} vec0 The vector to receive column 0. |
| * @param {goog.vec.Vec3.AnyType} vec1 The vector to receive column 1. |
| * @param {goog.vec.Vec3.AnyType} vec2 The vector to receive column 2. |
| */ |
| goog.vec.Mat3.getColumns = function(mat, vec0, vec1, vec2) { |
| goog.vec.Mat3.getColumn(mat, 0, vec0); |
| goog.vec.Mat3.getColumn(mat, 1, vec1); |
| goog.vec.Mat3.getColumn(mat, 2, vec2); |
| }; |
| |
| |
| /** |
| * Sets the row values from the supplied values. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix to receive the values. |
| * @param {number} row The index of the row to receive the values. |
| * @param {number} v0 The value for column 0. |
| * @param {number} v1 The value for column 1. |
| * @param {number} v2 The value for column 2. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.setRowValues = function(mat, row, v0, v1, v2) { |
| mat[row] = v0; |
| mat[row + 3] = v1; |
| mat[row + 6] = v2; |
| return mat; |
| }; |
| |
| |
| /** |
| * Sets the row values from the supplied vector. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix to receive the row values. |
| * @param {number} row The index of the row. |
| * @param {goog.vec.Vec3.AnyType} vec The vector containing the values. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.setRow = function(mat, row, vec) { |
| mat[row] = vec[0]; |
| mat[row + 3] = vec[1]; |
| mat[row + 6] = vec[2]; |
| return mat; |
| }; |
| |
| |
| /** |
| * Retrieves the row values into the given vector. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix supplying the values. |
| * @param {number} row The index of the row supplying the values. |
| * @param {goog.vec.Vec3.AnyType} vec The vector to receive the row. |
| * @return {goog.vec.Vec3.AnyType} return vec so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.getRow = function(mat, row, vec) { |
| vec[0] = mat[row]; |
| vec[1] = mat[row + 3]; |
| vec[2] = mat[row + 6]; |
| return vec; |
| }; |
| |
| |
| /** |
| * Sets the rows of the matrix from the supplied vectors. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix to receive the values. |
| * @param {goog.vec.Vec3.AnyType} vec0 The values for row 0. |
| * @param {goog.vec.Vec3.AnyType} vec1 The values for row 1. |
| * @param {goog.vec.Vec3.AnyType} vec2 The values for row 2. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.setRows = function(mat, vec0, vec1, vec2) { |
| goog.vec.Mat3.setRow(mat, 0, vec0); |
| goog.vec.Mat3.setRow(mat, 1, vec1); |
| goog.vec.Mat3.setRow(mat, 2, vec2); |
| return mat; |
| }; |
| |
| |
| /** |
| * Retrieves the rows of the matrix into the supplied vectors. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix to supplying the values. |
| * @param {goog.vec.Vec3.AnyType} vec0 The vector to receive row 0. |
| * @param {goog.vec.Vec3.AnyType} vec1 The vector to receive row 1. |
| * @param {goog.vec.Vec3.AnyType} vec2 The vector to receive row 2. |
| */ |
| goog.vec.Mat3.getRows = function(mat, vec0, vec1, vec2) { |
| goog.vec.Mat3.getRow(mat, 0, vec0); |
| goog.vec.Mat3.getRow(mat, 1, vec1); |
| goog.vec.Mat3.getRow(mat, 2, vec2); |
| }; |
| |
| |
| /** |
| * Makes the given 3x3 matrix the zero matrix. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @return {goog.vec.Mat3.AnyType} return mat so operations can be chained. |
| */ |
| goog.vec.Mat3.makeZero = function(mat) { |
| mat[0] = 0; |
| mat[1] = 0; |
| mat[2] = 0; |
| mat[3] = 0; |
| mat[4] = 0; |
| mat[5] = 0; |
| mat[6] = 0; |
| mat[7] = 0; |
| mat[8] = 0; |
| return mat; |
| }; |
| |
| |
| /** |
| * Makes the given 3x3 matrix the identity matrix. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @return {goog.vec.Mat3.AnyType} return mat so operations can be chained. |
| */ |
| goog.vec.Mat3.makeIdentity = function(mat) { |
| mat[0] = 1; |
| mat[1] = 0; |
| mat[2] = 0; |
| mat[3] = 0; |
| mat[4] = 1; |
| mat[5] = 0; |
| mat[6] = 0; |
| mat[7] = 0; |
| mat[8] = 1; |
| return mat; |
| }; |
| |
| |
| /** |
| * Performs a per-component addition of the matrices mat0 and mat1, storing |
| * the result into resultMat. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat0 The first addend. |
| * @param {goog.vec.Mat3.AnyType} mat1 The second addend. |
| * @param {goog.vec.Mat3.AnyType} resultMat The matrix to |
| * receive the results (may be either mat0 or mat1). |
| * @return {goog.vec.Mat3.AnyType} return resultMat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.addMat = function(mat0, mat1, resultMat) { |
| resultMat[0] = mat0[0] + mat1[0]; |
| resultMat[1] = mat0[1] + mat1[1]; |
| resultMat[2] = mat0[2] + mat1[2]; |
| resultMat[3] = mat0[3] + mat1[3]; |
| resultMat[4] = mat0[4] + mat1[4]; |
| resultMat[5] = mat0[5] + mat1[5]; |
| resultMat[6] = mat0[6] + mat1[6]; |
| resultMat[7] = mat0[7] + mat1[7]; |
| resultMat[8] = mat0[8] + mat1[8]; |
| return resultMat; |
| }; |
| |
| |
| /** |
| * Performs a per-component subtraction of the matrices mat0 and mat1, |
| * storing the result into resultMat. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat0 The minuend. |
| * @param {goog.vec.Mat3.AnyType} mat1 The subtrahend. |
| * @param {goog.vec.Mat3.AnyType} resultMat The matrix to receive |
| * the results (may be either mat0 or mat1). |
| * @return {goog.vec.Mat3.AnyType} return resultMat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.subMat = function(mat0, mat1, resultMat) { |
| resultMat[0] = mat0[0] - mat1[0]; |
| resultMat[1] = mat0[1] - mat1[1]; |
| resultMat[2] = mat0[2] - mat1[2]; |
| resultMat[3] = mat0[3] - mat1[3]; |
| resultMat[4] = mat0[4] - mat1[4]; |
| resultMat[5] = mat0[5] - mat1[5]; |
| resultMat[6] = mat0[6] - mat1[6]; |
| resultMat[7] = mat0[7] - mat1[7]; |
| resultMat[8] = mat0[8] - mat1[8]; |
| return resultMat; |
| }; |
| |
| |
| /** |
| * Multiplies matrix mat0 with the given scalar, storing the result |
| * into resultMat. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @param {number} scalar The scalar value to multiple to each element of mat. |
| * @param {goog.vec.Mat3.AnyType} resultMat The matrix to receive |
| * the results (may be mat). |
| * @return {goog.vec.Mat3.AnyType} return resultMat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.multScalar = function(mat, scalar, resultMat) { |
| resultMat[0] = mat[0] * scalar; |
| resultMat[1] = mat[1] * scalar; |
| resultMat[2] = mat[2] * scalar; |
| resultMat[3] = mat[3] * scalar; |
| resultMat[4] = mat[4] * scalar; |
| resultMat[5] = mat[5] * scalar; |
| resultMat[6] = mat[6] * scalar; |
| resultMat[7] = mat[7] * scalar; |
| resultMat[8] = mat[8] * scalar; |
| return resultMat; |
| }; |
| |
| |
| /** |
| * Multiplies the two matrices mat0 and mat1 using matrix multiplication, |
| * storing the result into resultMat. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat0 The first (left hand) matrix. |
| * @param {goog.vec.Mat3.AnyType} mat1 The second (right hand) matrix. |
| * @param {goog.vec.Mat3.AnyType} resultMat The matrix to receive |
| * the results (may be either mat0 or mat1). |
| * @return {goog.vec.Mat3.AnyType} return resultMat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.multMat = function(mat0, mat1, resultMat) { |
| var a00 = mat0[0], a10 = mat0[1], a20 = mat0[2]; |
| var a01 = mat0[3], a11 = mat0[4], a21 = mat0[5]; |
| var a02 = mat0[6], a12 = mat0[7], a22 = mat0[8]; |
| |
| var b00 = mat1[0], b10 = mat1[1], b20 = mat1[2]; |
| var b01 = mat1[3], b11 = mat1[4], b21 = mat1[5]; |
| var b02 = mat1[6], b12 = mat1[7], b22 = mat1[8]; |
| |
| resultMat[0] = a00 * b00 + a01 * b10 + a02 * b20; |
| resultMat[1] = a10 * b00 + a11 * b10 + a12 * b20; |
| resultMat[2] = a20 * b00 + a21 * b10 + a22 * b20; |
| resultMat[3] = a00 * b01 + a01 * b11 + a02 * b21; |
| resultMat[4] = a10 * b01 + a11 * b11 + a12 * b21; |
| resultMat[5] = a20 * b01 + a21 * b11 + a22 * b21; |
| resultMat[6] = a00 * b02 + a01 * b12 + a02 * b22; |
| resultMat[7] = a10 * b02 + a11 * b12 + a12 * b22; |
| resultMat[8] = a20 * b02 + a21 * b12 + a22 * b22; |
| return resultMat; |
| }; |
| |
| |
| /** |
| * Transposes the given matrix mat storing the result into resultMat. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix to transpose. |
| * @param {goog.vec.Mat3.AnyType} resultMat The matrix to receive |
| * the results (may be mat). |
| * @return {goog.vec.Mat3.AnyType} return resultMat so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.transpose = function(mat, resultMat) { |
| if (resultMat == mat) { |
| var a10 = mat[1], a20 = mat[2], a21 = mat[5]; |
| resultMat[1] = mat[3]; |
| resultMat[2] = mat[6]; |
| resultMat[3] = a10; |
| resultMat[5] = mat[7]; |
| resultMat[6] = a20; |
| resultMat[7] = a21; |
| } else { |
| resultMat[0] = mat[0]; |
| resultMat[1] = mat[3]; |
| resultMat[2] = mat[6]; |
| resultMat[3] = mat[1]; |
| resultMat[4] = mat[4]; |
| resultMat[5] = mat[7]; |
| resultMat[6] = mat[2]; |
| resultMat[7] = mat[5]; |
| resultMat[8] = mat[8]; |
| } |
| return resultMat; |
| }; |
| |
| |
| /** |
| * Computes the inverse of mat0 storing the result into resultMat. If the |
| * inverse is defined, this function returns true, false otherwise. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat0 The matrix to invert. |
| * @param {goog.vec.Mat3.AnyType} resultMat The matrix to receive |
| * the result (may be mat0). |
| * @return {boolean} True if the inverse is defined. If false is returned, |
| * resultMat is not modified. |
| */ |
| goog.vec.Mat3.invert = function(mat0, resultMat) { |
| var a00 = mat0[0], a10 = mat0[1], a20 = mat0[2]; |
| var a01 = mat0[3], a11 = mat0[4], a21 = mat0[5]; |
| var a02 = mat0[6], a12 = mat0[7], a22 = mat0[8]; |
| |
| var t00 = a11 * a22 - a12 * a21; |
| var t10 = a12 * a20 - a10 * a22; |
| var t20 = a10 * a21 - a11 * a20; |
| var det = a00 * t00 + a01 * t10 + a02 * t20; |
| if (det == 0) { |
| return false; |
| } |
| |
| var idet = 1 / det; |
| resultMat[0] = t00 * idet; |
| resultMat[3] = (a02 * a21 - a01 * a22) * idet; |
| resultMat[6] = (a01 * a12 - a02 * a11) * idet; |
| |
| resultMat[1] = t10 * idet; |
| resultMat[4] = (a00 * a22 - a02 * a20) * idet; |
| resultMat[7] = (a02 * a10 - a00 * a12) * idet; |
| |
| resultMat[2] = t20 * idet; |
| resultMat[5] = (a01 * a20 - a00 * a21) * idet; |
| resultMat[8] = (a00 * a11 - a01 * a10) * idet; |
| return true; |
| }; |
| |
| |
| /** |
| * Returns true if the components of mat0 are equal to the components of mat1. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat0 The first matrix. |
| * @param {goog.vec.Mat3.AnyType} mat1 The second matrix. |
| * @return {boolean} True if the the two matrices are equivalent. |
| */ |
| goog.vec.Mat3.equals = function(mat0, mat1) { |
| return mat0.length == mat1.length && |
| mat0[0] == mat1[0] && mat0[1] == mat1[1] && mat0[2] == mat1[2] && |
| mat0[3] == mat1[3] && mat0[4] == mat1[4] && mat0[5] == mat1[5] && |
| mat0[6] == mat1[6] && mat0[7] == mat1[7] && mat0[8] == mat1[8]; |
| }; |
| |
| |
| /** |
| * Transforms the given vector with the given matrix storing the resulting, |
| * transformed matrix into resultVec. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix supplying the transformation. |
| * @param {goog.vec.Vec3.AnyType} vec The vector to transform. |
| * @param {goog.vec.Vec3.AnyType} resultVec The vector to |
| * receive the results (may be vec). |
| * @return {goog.vec.Vec3.AnyType} return resultVec so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.multVec3 = function(mat, vec, resultVec) { |
| var x = vec[0], y = vec[1], z = vec[2]; |
| resultVec[0] = x * mat[0] + y * mat[3] + z * mat[6]; |
| resultVec[1] = x * mat[1] + y * mat[4] + z * mat[7]; |
| resultVec[2] = x * mat[2] + y * mat[5] + z * mat[8]; |
| return resultVec; |
| }; |
| |
| |
| /** |
| * Makes the given 3x3 matrix a translation matrix with x and y |
| * translation values. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @param {number} x The translation along the x axis. |
| * @param {number} y The translation along the y axis. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained. |
| */ |
| goog.vec.Mat3.makeTranslate = function(mat, x, y) { |
| goog.vec.Mat3.makeIdentity(mat); |
| return goog.vec.Mat3.setColumnValues(mat, 2, x, y, 1); |
| }; |
| |
| |
| /** |
| * Makes the given 3x3 matrix a scale matrix with x, y, and z scale factors. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The 3x3 (9-element) matrix |
| * array to receive the new scale matrix. |
| * @param {number} x The scale along the x axis. |
| * @param {number} y The scale along the y axis. |
| * @param {number} z The scale along the z axis. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained. |
| */ |
| goog.vec.Mat3.makeScale = function(mat, x, y, z) { |
| goog.vec.Mat3.makeIdentity(mat); |
| return goog.vec.Mat3.setDiagonalValues(mat, x, y, z); |
| }; |
| |
| |
| /** |
| * Makes the given 3x3 matrix a rotation matrix with the given rotation |
| * angle about the axis defined by the vector (ax, ay, az). |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @param {number} angle The rotation angle in radians. |
| * @param {number} ax The x component of the rotation axis. |
| * @param {number} ay The y component of the rotation axis. |
| * @param {number} az The z component of the rotation axis. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained. |
| */ |
| goog.vec.Mat3.makeRotate = function(mat, angle, ax, ay, az) { |
| var c = Math.cos(angle); |
| var d = 1 - c; |
| var s = Math.sin(angle); |
| |
| return goog.vec.Mat3.setFromValues(mat, |
| ax * ax * d + c, |
| ax * ay * d + az * s, |
| ax * az * d - ay * s, |
| |
| ax * ay * d - az * s, |
| ay * ay * d + c, |
| ay * az * d + ax * s, |
| |
| ax * az * d + ay * s, |
| ay * az * d - ax * s, |
| az * az * d + c); |
| }; |
| |
| |
| /** |
| * Makes the given 3x3 matrix a rotation matrix with the given rotation |
| * angle about the X axis. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @param {number} angle The rotation angle in radians. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained. |
| */ |
| goog.vec.Mat3.makeRotateX = function(mat, angle) { |
| var c = Math.cos(angle); |
| var s = Math.sin(angle); |
| return goog.vec.Mat3.setFromValues(mat, |
| 1, 0, 0, |
| 0, c, s, |
| 0, -s, c); |
| }; |
| |
| |
| /** |
| * Makes the given 3x3 matrix a rotation matrix with the given rotation |
| * angle about the Y axis. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @param {number} angle The rotation angle in radians. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained. |
| */ |
| goog.vec.Mat3.makeRotateY = function(mat, angle) { |
| var c = Math.cos(angle); |
| var s = Math.sin(angle); |
| return goog.vec.Mat3.setFromValues(mat, |
| c, 0, -s, |
| 0, 1, 0, |
| s, 0, c); |
| }; |
| |
| |
| /** |
| * Makes the given 3x3 matrix a rotation matrix with the given rotation |
| * angle about the Z axis. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @param {number} angle The rotation angle in radians. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained. |
| */ |
| goog.vec.Mat3.makeRotateZ = function(mat, angle) { |
| var c = Math.cos(angle); |
| var s = Math.sin(angle); |
| return goog.vec.Mat3.setFromValues(mat, |
| c, s, 0, |
| -s, c, 0, |
| 0, 0, 1); |
| }; |
| |
| |
| /** |
| * Rotate the given matrix by angle about the x,y,z axis. Equivalent to: |
| * goog.vec.Mat3.multMat( |
| * mat, |
| * goog.vec.Mat3.makeRotate(goog.vec.Mat3.create(), angle, x, y, z), |
| * mat); |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @param {number} angle The angle in radians. |
| * @param {number} x The x component of the rotation axis. |
| * @param {number} y The y component of the rotation axis. |
| * @param {number} z The z component of the rotation axis. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained. |
| */ |
| goog.vec.Mat3.rotate = function(mat, angle, x, y, z) { |
| var m00 = mat[0], m10 = mat[1], m20 = mat[2]; |
| var m01 = mat[3], m11 = mat[4], m21 = mat[5]; |
| var m02 = mat[6], m12 = mat[7], m22 = mat[8]; |
| |
| var cosAngle = Math.cos(angle); |
| var sinAngle = Math.sin(angle); |
| var diffCosAngle = 1 - cosAngle; |
| var r00 = x * x * diffCosAngle + cosAngle; |
| var r10 = x * y * diffCosAngle + z * sinAngle; |
| var r20 = x * z * diffCosAngle - y * sinAngle; |
| |
| var r01 = x * y * diffCosAngle - z * sinAngle; |
| var r11 = y * y * diffCosAngle + cosAngle; |
| var r21 = y * z * diffCosAngle + x * sinAngle; |
| |
| var r02 = x * z * diffCosAngle + y * sinAngle; |
| var r12 = y * z * diffCosAngle - x * sinAngle; |
| var r22 = z * z * diffCosAngle + cosAngle; |
| |
| return goog.vec.Mat3.setFromValues( |
| mat, |
| m00 * r00 + m01 * r10 + m02 * r20, |
| m10 * r00 + m11 * r10 + m12 * r20, |
| m20 * r00 + m21 * r10 + m22 * r20, |
| |
| m00 * r01 + m01 * r11 + m02 * r21, |
| m10 * r01 + m11 * r11 + m12 * r21, |
| m20 * r01 + m21 * r11 + m22 * r21, |
| |
| m00 * r02 + m01 * r12 + m02 * r22, |
| m10 * r02 + m11 * r12 + m12 * r22, |
| m20 * r02 + m21 * r12 + m22 * r22); |
| }; |
| |
| |
| /** |
| * Rotate the given matrix by angle about the x axis. Equivalent to: |
| * goog.vec.Mat3.multMat( |
| * mat, |
| * goog.vec.Mat3.makeRotateX(goog.vec.Mat3.create(), angle), |
| * mat); |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @param {number} angle The angle in radians. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained. |
| */ |
| goog.vec.Mat3.rotateX = function(mat, angle) { |
| var m01 = mat[3], m11 = mat[4], m21 = mat[5]; |
| var m02 = mat[6], m12 = mat[7], m22 = mat[8]; |
| |
| var c = Math.cos(angle); |
| var s = Math.sin(angle); |
| |
| mat[3] = m01 * c + m02 * s; |
| mat[4] = m11 * c + m12 * s; |
| mat[5] = m21 * c + m22 * s; |
| mat[6] = m01 * -s + m02 * c; |
| mat[7] = m11 * -s + m12 * c; |
| mat[8] = m21 * -s + m22 * c; |
| |
| return mat; |
| }; |
| |
| |
| /** |
| * Rotate the given matrix by angle about the y axis. Equivalent to: |
| * goog.vec.Mat3.multMat( |
| * mat, |
| * goog.vec.Mat3.makeRotateY(goog.vec.Mat3.create(), angle), |
| * mat); |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @param {number} angle The angle in radians. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained. |
| */ |
| goog.vec.Mat3.rotateY = function(mat, angle) { |
| var m00 = mat[0], m10 = mat[1], m20 = mat[2]; |
| var m02 = mat[6], m12 = mat[7], m22 = mat[8]; |
| |
| var c = Math.cos(angle); |
| var s = Math.sin(angle); |
| |
| mat[0] = m00 * c + m02 * -s; |
| mat[1] = m10 * c + m12 * -s; |
| mat[2] = m20 * c + m22 * -s; |
| mat[6] = m00 * s + m02 * c; |
| mat[7] = m10 * s + m12 * c; |
| mat[8] = m20 * s + m22 * c; |
| |
| return mat; |
| }; |
| |
| |
| /** |
| * Rotate the given matrix by angle about the z axis. Equivalent to: |
| * goog.vec.Mat3.multMat( |
| * mat, |
| * goog.vec.Mat3.makeRotateZ(goog.vec.Mat3.create(), angle), |
| * mat); |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @param {number} angle The angle in radians. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained. |
| */ |
| goog.vec.Mat3.rotateZ = function(mat, angle) { |
| var m00 = mat[0], m10 = mat[1], m20 = mat[2]; |
| var m01 = mat[3], m11 = mat[4], m21 = mat[5]; |
| |
| var c = Math.cos(angle); |
| var s = Math.sin(angle); |
| |
| mat[0] = m00 * c + m01 * s; |
| mat[1] = m10 * c + m11 * s; |
| mat[2] = m20 * c + m21 * s; |
| mat[3] = m00 * -s + m01 * c; |
| mat[4] = m10 * -s + m11 * c; |
| mat[5] = m20 * -s + m21 * c; |
| |
| return mat; |
| }; |
| |
| |
| /** |
| * Makes the given 3x3 matrix a rotation matrix given Euler angles using |
| * the ZXZ convention. |
| * Given the euler angles [theta1, theta2, theta3], the rotation is defined as |
| * rotation = rotation_z(theta1) * rotation_x(theta2) * rotation_z(theta3), |
| * with theta1 in [0, 2 * pi], theta2 in [0, pi] and theta3 in [0, 2 * pi]. |
| * rotation_x(theta) means rotation around the X axis of theta radians. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @param {number} theta1 The angle of rotation around the Z axis in radians. |
| * @param {number} theta2 The angle of rotation around the X axis in radians. |
| * @param {number} theta3 The angle of rotation around the Z axis in radians. |
| * @return {goog.vec.Mat3.AnyType} return mat so that operations can be |
| * chained. |
| */ |
| goog.vec.Mat3.makeEulerZXZ = function(mat, theta1, theta2, theta3) { |
| var c1 = Math.cos(theta1); |
| var s1 = Math.sin(theta1); |
| |
| var c2 = Math.cos(theta2); |
| var s2 = Math.sin(theta2); |
| |
| var c3 = Math.cos(theta3); |
| var s3 = Math.sin(theta3); |
| |
| mat[0] = c1 * c3 - c2 * s1 * s3; |
| mat[1] = c2 * c1 * s3 + c3 * s1; |
| mat[2] = s3 * s2; |
| |
| mat[3] = -c1 * s3 - c3 * c2 * s1; |
| mat[4] = c1 * c2 * c3 - s1 * s3; |
| mat[5] = c3 * s2; |
| |
| mat[6] = s2 * s1; |
| mat[7] = -c1 * s2; |
| mat[8] = c2; |
| |
| return mat; |
| }; |
| |
| |
| /** |
| * Decomposes a rotation matrix into Euler angles using the ZXZ convention so |
| * that rotation = rotation_z(theta1) * rotation_x(theta2) * rotation_z(theta3), |
| * with theta1 in [0, 2 * pi], theta2 in [0, pi] and theta3 in [0, 2 * pi]. |
| * rotation_x(theta) means rotation around the X axis of theta radians. |
| * |
| * @param {goog.vec.Mat3.AnyType} mat The matrix. |
| * @param {goog.vec.Vec3.AnyType} euler The ZXZ Euler angles in |
| * radians as [theta1, theta2, theta3]. |
| * @param {boolean=} opt_theta2IsNegative Whether theta2 is in [-pi, 0] instead |
| * of the default [0, pi]. |
| * @return {goog.vec.Vec3.AnyType} return euler so that operations can be |
| * chained together. |
| */ |
| goog.vec.Mat3.toEulerZXZ = function(mat, euler, opt_theta2IsNegative) { |
| // There is an ambiguity in the sign of sinTheta2 because of the sqrt. |
| var sinTheta2 = Math.sqrt(mat[2] * mat[2] + mat[5] * mat[5]); |
| |
| // By default we explicitely constrain theta2 to be in [0, pi], |
| // so sinTheta2 is always positive. We can change the behavior and specify |
| // theta2 to be negative in [-pi, 0] with opt_Theta2IsNegative. |
| var signTheta2 = opt_theta2IsNegative ? -1 : 1; |
| |
| if (sinTheta2 > goog.vec.EPSILON) { |
| euler[2] = Math.atan2(mat[2] * signTheta2, mat[5] * signTheta2); |
| euler[1] = Math.atan2(sinTheta2 * signTheta2, mat[8]); |
| euler[0] = Math.atan2(mat[6] * signTheta2, -mat[7] * signTheta2); |
| } else { |
| // There is also an arbitrary choice for theta1 = 0 or theta2 = 0 here. |
| // We assume theta1 = 0 as some applications do not allow the camera to roll |
| // (i.e. have theta1 != 0). |
| euler[0] = 0; |
| euler[1] = Math.atan2(sinTheta2 * signTheta2, mat[8]); |
| euler[2] = Math.atan2(mat[1], mat[0]); |
| } |
| |
| // Atan2 outputs angles in [-pi, pi] so we bring them back to [0, 2 * pi]. |
| euler[0] = (euler[0] + Math.PI * 2) % (Math.PI * 2); |
| euler[2] = (euler[2] + Math.PI * 2) % (Math.PI * 2); |
| // For theta2 we want the angle to be in [0, pi] or [-pi, 0] depending on |
| // signTheta2. |
| euler[1] = ((euler[1] * signTheta2 + Math.PI * 2) % (Math.PI * 2)) * |
| signTheta2; |
| |
| return euler; |
| }; |