# Topology preserving inside-outside functions for simple closed curves

Lori Ben-Bassat, Daniel Keren

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

## Abstract

The purpose of this article is to 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 abovementioned function should provide us the information about the mutual state of a given point relative to the given curve. The creation of this " inside/outside function " will be based on the following fundamentals: A continuous and single-valued function f will be found. If f operates on a simple closed curve a , a simple closed curve β will result, in a way that the interior of a will be mapped on the interior of p , and the exterior of a will be mapped on the exterior of p . 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 plane) we will 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 5 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 6 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.

Original language English 21st IEEE Convention of the Electrical and Electronic Engineers in Israel, Proceedings Institute of Electrical and Electronics Engineers Inc. 110-113 4 0780358422, 9780780358423 https://doi.org/10.1109/EEEI.2000.924337 Published - 2000 21st IEEE Convention of the Electrical and Electronic Engineers in Israel, IEEEI 2000 - Tel-Aviv, IsraelDuration: 11 Apr 2000 → 12 Apr 2000

### Publication series

Name 21st IEEE Convention of the Electrical and Electronic Engineers in Israel, Proceedings

### Conference

Conference 21st IEEE Convention of the Electrical and Electronic Engineers in Israel, IEEEI 2000 Israel Tel-Aviv 11/04/00 → 12/04/00