矢量

1. 概念

矢量表示的是方向与大小，并不表示位置

1）矢量的分量--components

2）矢量--vector

3）矢量的模--magnitude

The magnitude or length of a vector r is written

4）位置矢量--Position Vectors

Given any point P(x, y, z), a position vector p is created by assuming that P is the vector’s head and the origin is its tail. As the tail coordinates are (0, 0, 0) the vector’s components are x, y, z

5）单位矢量--Unit Vectors

2. 操作

1）缩放--Scaling a Vector

2）加减--Vector Addition and Subtraction

 

3）单位化

4）笛卡尔矢量--Cartesian Vectors

A Cartesian vector is constructed from three unit vectors: i, j and k, aligned with the x-, y- and z-axis, respectively:

By employing the rules of vector addition and subtraction, we can compose a vector r by summing three scaled Cartesian unit vectors as follows：

 

相等于

5）标量积--Scalar Product/Dot Product

在计算机图形学中的应用点：

(1) 计算灯光与表面的法线之间的夹角

(2) 判断三角面的朝向是正面，还是背面

(3) 三维空间中的三角形面积计算

6）叉积--Vector Product/Cross Product

注意：叉积不满足交换律

           叉积在三维世界中需要遵守右手规则

在计算机图形学中的应用点：

(1) 三角形表面的法线计算

(2) 二维空间中的三角形面积计算