Topology preserving inside-outside functions for simple closed curves

We examine a simple closed curve and to develop an “inside/outside function” for it. This function should be able to determine whether a given point belongs to the interior or to the exterior of our curve. In other words, the above mentioned function should provide information about the mutual state of a given point relative to the given curve. The creation of this “inside/outside function” is based on the following fundamentals: a continuous and single-valued function f is found. If f operates on a simple closed curve α, a simple closed curve β will result, in a way that the interior of α will be mapped on the interior of β, and the exterior of α will be mapped on the exterior of β. Therefore, if a transformation f* can be constructed in such a way that it will map the given curve on the unit circle, it can be used in the following way: for a given point ξ (ξ represent the pair (x,y) in the ℜ² plane) we calculate f*(ξ), and examine the mutual state of f*(ξ) relative to the unit circle: if f* (ξ) is within the interior of the unit circle then we can conclude that the point ξ is located inside the original curve. On the other hand, if f (ξ) is not within the boundaries of the unit circle then it can be concluded that the point ξ is located outside the original curve. This approach enables us to deal with the problem of the mutual state of a given point relative to the unit circle, instead of dealing with the original problem that is highly complex