The problem I want to solve, is to quickly decide, whether a point $p=(x^*,y^*)$ is inside or outside of a polygon $P := (p_1, p_2,..., p_n=p_1), p_i := (x_i,y_i)$, with $n$ potentially very large.

My idea was to try to find the series-expansion of a bivariate function $f(x,y)$ with the following properties:

• $f(x,y) := \sum_{i=0}^{\infty}\sum_{j=0}^{\infty}a_{ij}x^iy^j$
• $\sum_{i=0}^{n}\sum_{j=0}^{n}a_{ij}x^iy^j=0$ resembles a jordan curve, that does not intersect the polygon $P$
• if the jordan curve encloses the polygon, then the enclosed area should be minimal and maximal if the jordan curve is inside the polygon.
• the jordan curves related to different partial sums should be nested

Questions:

• are there better functions than the one's resembling the Schwarz-Christoffel mapping http://en.wikipedia.org/wiki/Schwarz%E2%80%93Christoffel_mapping of a circle to the polygon $P$?
• are there practical methods of determining such functions?

Note: I formulated the series expansion as a Taylor series but other expansions would also be appreciated.