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Gru

I am unable to get an intuitive idea of what a distance function is. What do d1(x) and d2(x) above mean?

keenan

Something confusing here may be that "x" refers to a vector in space, i.e., x = (x1,x2). For any point x in the plane, d1(x) is the distance to the closest point on the first shape; d2(x) is the distance to the closest point on the second shape.

A concrete example is the (signed) distance to a circle with center c and radius r, given by

dC(x) = |x-c|-r,

where |x-c| is the Euclidean norm of the vector x-c. Notice that this function will be zero exactly when x is a point on the circle, i.e., when the distance from the center to x is equal to the radius. It will be positive when we are outside the circle (and increasing as we get further away), and negative when we are inside.

Here for instance is the signed distance to a butterfly: