Interpolation, Lines(插值、線性插值)

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Interpolation is a term used to describe the process of finding values that lie between a set of known points(插值的意思是從一群已知的點的集合中找出某一個值的過程). Consider the equation of the line passing through points A and B(想象一下一個連接A點到B點的直線):

where P is any point on the line and is the vector from A to B(這里P表示的時候AB之間任意一點):

We can therefore write this equation as(因此我們可以把這個等式寫成：)

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(當t是0的時候，p就相當于A點，當t的值是1的時候，p就相當于B點，如圖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之間，那么p就落在A與B之間的某處). 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的值超出了這個范圍，那么p則也會超出AB的范圍。可以看到，當讓t從0到1慢慢變化時，p會從A移動到B，這叫線性插值). The values of A and B (and therefore P) can have any number of dimensions(A和B的值可以有任意多個維度). 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(比如這里A和B的值可以是一個標量，可以是二維的點，可以是三位的點、顏色等等); or even higher-dimension quantities such as matrices, arrays, or even whole images(甚至是更高維度的的矩陣、數組亦或是整個圖像). 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(在很多情況下，線性插值不怎么好使，但是在角度、位置以及其他坐標方面，一般來說這就夠了). Linear interpolation is such a common operation in graphics that GLSL includes a builtin function specifically for this purpose, mix:(在GLSL里，能用來進行線性插值的內置函數是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(該函數接收A、B兩個數據，然后變量t，來對數據進行線性插值).

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