mask expansion and rounded corners

Someone just asked on a forum why expanding a rectangular mask resulted in a mask with rounded corners. I made a little visual aid to answer the question, so I thought that I should post it here.

Increasing mask expansion is not the same thing scaling a mask. Scaling preserves shape. Expansion extends the influence of the mask by growing outward from each point along the mask path by a certain number of pixels.

For each point on the original mask path, imagine a circle radiating outward by the number of pixels by which you're expanding.

In the image above, the white rectangle is the original mask path. The red circles indicate the expansion radius. Note how these circles define a sharp inner rectangle, which is what you get with a negative expansion; and they define a rounded outer rectangle, which is what you get with a positive expansion.