The gist of the function is that is 0 everywhere except from p-1 to p+1, where it linearly rises, peaks at a height of 1 and then falls back to 0.

It's a fairly big function but it is easy to calculate and it is made up of only elementary operations, and it's behavior is useful for a lot of applications that I can think of. Here's one example:

Suppose you have these 6 [x,y] points:

[1,1], [6,3.5], [0,3.5],[5,1],[3,5],[1,1]

You can make a function that connects each of these with a straight line between them over time. You make a parametric function:

Now I realize if each T(p) were written out instead of using this notation this function would be hideous looking but it is still an elementary parametric equation there are just a lot of elementary operations.

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