矩陣的逆矩陣 和 轉置矩陣

這幾天用到了逆矩陣，就在這裏總結一下逆矩陣和轉置矩陣。

逆矩陣

逆矩陣就是一個矩陣的逆向。比如一個點乘以一個矩陣後得到了一個新的點的位置，如果想通過這個點再獲得矩陣轉換前的位置，那我們就需要乘以這個矩陣的逆矩陣。
在Three.js裏面，我們可以通過new THREE.Matrix4().getInverse(matrix4)方法來獲得一個矩陣的逆矩陣。

具有的性質：
可逆矩陣一定是方陣。
如果矩陣是可逆的，那它的逆矩陣具有唯一性。
矩陣A的逆矩陣的逆矩陣，等於它自身。

Three.js獲得一個矩陣的逆矩陣：

var m = `new THREE.Matrix4().getInverse(matrix4);

Three.js求逆矩陣源碼：

		getInverse: function ( m, throwOnDegenerate ) {

			// based on http://www.euclideanspace.com/maths/algebra/matrix/functions/inverse/fourD/index.htm
			var te = this.elements,
				me = m.elements,

				n11 = me[ 0 ], n21 = me[ 1 ], n31 = me[ 2 ], n41 = me[ 3 ],
				n12 = me[ 4 ], n22 = me[ 5 ], n32 = me[ 6 ], n42 = me[ 7 ],
				n13 = me[ 8 ], n23 = me[ 9 ], n33 = me[ 10 ], n43 = me[ 11 ],
				n14 = me[ 12 ], n24 = me[ 13 ], n34 = me[ 14 ], n44 = me[ 15 ],

				t11 = n23 * n34 * n42 - n24 * n33 * n42 + n24 * n32 * n43 - n22 * n34 * n43 - n23 * n32 * n44 + n22 * n33 * n44,
				t12 = n14 * n33 * n42 - n13 * n34 * n42 - n14 * n32 * n43 + n12 * n34 * n43 + n13 * n32 * n44 - n12 * n33 * n44,
				t13 = n13 * n24 * n42 - n14 * n23 * n42 + n14 * n22 * n43 - n12 * n24 * n43 - n13 * n22 * n44 + n12 * n23 * n44,
				t14 = n14 * n23 * n32 - n13 * n24 * n32 - n14 * n22 * n33 + n12 * n24 * n33 + n13 * n22 * n34 - n12 * n23 * n34;

			var det = n11 * t11 + n21 * t12 + n31 * t13 + n41 * t14;

			if ( det === 0 ) {

				var msg = "THREE.Matrix4: .getInverse() can't invert matrix, determinant is 0";

				if ( throwOnDegenerate === true ) {

					throw new Error( msg );

				} else {

					console.warn( msg );

				}

				return this.identity();

			}

			var detInv = 1 / det;

			te[ 0 ] = t11 * detInv;
			te[ 1 ] = ( n24 * n33 * n41 - n23 * n34 * n41 - n24 * n31 * n43 + n21 * n34 * n43 + n23 * n31 * n44 - n21 * n33 * n44 ) * detInv;
			te[ 2 ] = ( n22 * n34 * n41 - n24 * n32 * n41 + n24 * n31 * n42 - n21 * n34 * n42 - n22 * n31 * n44 + n21 * n32 * n44 ) * detInv;
			te[ 3 ] = ( n23 * n32 * n41 - n22 * n33 * n41 - n23 * n31 * n42 + n21 * n33 * n42 + n22 * n31 * n43 - n21 * n32 * n43 ) * detInv;

			te[ 4 ] = t12 * detInv;
			te[ 5 ] = ( n13 * n34 * n41 - n14 * n33 * n41 + n14 * n31 * n43 - n11 * n34 * n43 - n13 * n31 * n44 + n11 * n33 * n44 ) * detInv;
			te[ 6 ] = ( n14 * n32 * n41 - n12 * n34 * n41 - n14 * n31 * n42 + n11 * n34 * n42 + n12 * n31 * n44 - n11 * n32 * n44 ) * detInv;
			te[ 7 ] = ( n12 * n33 * n41 - n13 * n32 * n41 + n13 * n31 * n42 - n11 * n33 * n42 - n12 * n31 * n43 + n11 * n32 * n43 ) * detInv;

			te[ 8 ] = t13 * detInv;
			te[ 9 ] = ( n14 * n23 * n41 - n13 * n24 * n41 - n14 * n21 * n43 + n11 * n24 * n43 + n13 * n21 * n44 - n11 * n23 * n44 ) * detInv;
			te[ 10 ] = ( n12 * n24 * n41 - n14 * n22 * n41 + n14 * n21 * n42 - n11 * n24 * n42 - n12 * n21 * n44 + n11 * n22 * n44 ) * detInv;
			te[ 11 ] = ( n13 * n22 * n41 - n12 * n23 * n41 - n13 * n21 * n42 + n11 * n23 * n42 + n12 * n21 * n43 - n11 * n22 * n43 ) * detInv;

			te[ 12 ] = t14 * detInv;
			te[ 13 ] = ( n13 * n24 * n31 - n14 * n23 * n31 + n14 * n21 * n33 - n11 * n24 * n33 - n13 * n21 * n34 + n11 * n23 * n34 ) * detInv;
			te[ 14 ] = ( n14 * n22 * n31 - n12 * n24 * n31 - n14 * n21 * n32 + n11 * n24 * n32 + n12 * n21 * n34 - n11 * n22 * n34 ) * detInv;
			te[ 15 ] = ( n12 * n23 * n31 - n13 * n22 * n31 + n13 * n21 * n32 - n11 * n23 * n32 - n12 * n21 * n33 + n11 * n22 * n33 ) * detInv;

			return this;

		}

轉置矩陣

說到轉置矩陣，這裏就要說一下矩陣的排列。矩陣的排列有兩種方式：行優先和列優先。兩種方式在數學上沒有什麼不同，大多數人都習慣上使用行優先。在Three.js中，我們需要通過行優先設置，而在存儲中，則使用列優先存儲數據到elements。
設置矩陣時使用行優先：

var m = new Matrix4();

m.set( 11, 12, 13, 14,
       21, 22, 23, 24,
       31, 32, 33, 34,
       41, 42, 43, 44 );

元素數組elements將存儲爲:

m.elements = [ 11, 21, 31, 41,
               12, 22, 32, 42,
               13, 23, 33, 43,
               14, 24, 34, 44 ];

在Three.js中，我們可以同過transpose()方法，將矩陣轉置：

matrix.transpose();

Three.js的轉置源碼：

	transpose: function () {

		var te = this.elements;
		var tmp;

		tmp = te[ 1 ]; te[ 1 ] = te[ 4 ]; te[ 4 ] = tmp;
		tmp = te[ 2 ]; te[ 2 ] = te[ 8 ]; te[ 8 ] = tmp;
		tmp = te[ 6 ]; te[ 6 ] = te[ 9 ]; te[ 9 ] = tmp;

		tmp = te[ 3 ]; te[ 3 ] = te[ 12 ]; te[ 12 ] = tmp;
		tmp = te[ 7 ]; te[ 7 ] = te[ 13 ]; te[ 13 ] = tmp;
		tmp = te[ 11 ]; te[ 11 ] = te[ 14 ]; te[ 14 ] = tmp;

		return this;

	},