Interpolation, Lines(插值、线性插值)

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Interpolation is a term used to describe the process of finding values that lie between a set of known points(插值的意思是从一群已知的点的集合中找出某一个值的过程). Consider the equation of the line passing through points A and B(想象一下一个连接A点到B点的直线):

where P is any point on the line and is the vector from A to B(这里P表示的时候AB之间任意一点):

We can therefore write this equation as(因此我们可以把这个等式写成：)

It is easy to see that when t is 0, P is equal to A; and when t is 1, P is equal to A + B − A, which is simply B. Such a line is shown in Figure 4.13(当t是0的时候，p就相当于A点，当t的值是1的时候，p就相当于B点，如图4.13所示)

If t lies between 0.0 and 1.0, then P will end up somewhere between A and B(如果t的值在0到1之间，那么p就落在A与B之间的某处). Values of t outside this range will push P off the ends of the line. You should be able to see that by smoothly varying t, we can move point P from A to B and back. This is known as linear interpolation(如果t的值超出了这个范围，那么p则也会超出AB的范围。可以看到，当让t从0到1慢慢变化时，p会从A移动到B，这叫线性插值). The values of A and B (and therefore P) can have any number of dimensions(A和B的值可以有任意多个维度). For example, they could be scalar values; two-dimensional values such as points on a graph; three-dimensional values such as coordinates in 3D space, colors, and so on(比如这里A和B的值可以是一个标量，可以是二维的点，可以是三位的点、颜色等等); or even higher-dimension quantities such as matrices, arrays, or even whole images(甚至是更高维度的的矩阵、数组亦或是整个图像). In many cases, linear interpolation doesn’t make much sense (for example, linearly interpolating between two matrices generally doesn’t produce a meaningful result), but angles, positions, and other coordinates can normally be interpolated safely(在很多情况下，线性插值不怎么好使，但是在角度、位置以及其他坐标方面，一般来说这就够了). Linear interpolation is such a common operation in graphics that GLSL includes a builtin function specifically for this purpose, mix:(在GLSL里，能用来进行线性插值的内置函数是mix)

vec4 mix(vec4 A, vec4 B, float t);
The mix function comes in several versions taking different dimensionalities of vectors or scalars as the A and B inputs and taking scalars or matching vectors for t(该函数接收A、B两个数据，然后变量t，来对数据进行线性插值).

本日的翻译就到这里，明天见，拜拜~~

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