Sequence of smooth functions approximating a 2d cylinder step function

Question

Let $f(x,y)=1$ if $(x,y)$ is in the unit disk and $f=0$ otherwise. I would like to approximate $f$ by a sequence of smooth functions. The functions need to be evaluated quickly so the results I'm getting from mollification are to awkward to deal with...

Thanks in advance..

Answer 1

You might want to try any of formulae 17 to 27 in the MathWorld page for the unit step function, since the function you're interested in is expressible as

$$1-H(x^2+y^2-1)$$

Answer 2

You can convolve your function with a bivariate Gaussian density kernel, $\Phi(x,y; \sigma_x, \sigma_y)$, and obtain a smooth representation. Then you can let the variance parameters $\sigma_x$ and $\sigma_y$ go to zero to obtain arbitrarily accurate approximation to $f(x,y)$.