I’m thinking I might want to devote Monday posts to math topics. Maybe these posts should be called Math Mondays. Today seems like a find day to start.
I remember in my early twenties wondering why things always blur in a Gaussian function e-x2. Then one day it occurred to me that there is no other possibility.
What I realized is that if you blur something horizontally, and also blur it vertically, then the blur needs to be circular. If you got any other result, it would not really be blurring.
And the Gaussian function is the one function that does exactly that, because when you multiply two functions ab and ac, you are really just adding their exponents ab+c.
Which is why Gaussians (and only Gaussians) make a circular blur shape when you blur in x (horizontally) and also blur in y (vertically):
e-x2 * e-y2 = e-(x2+y2)
If you look at the resulting exponent, you realize that the blur is circularly symmetric. It drops off as the square of distance. And the Gaussian function is the only “drop off with distance” function for which this is true.
As Pascal might have said, et voila!