# Analytical Solution for the Problem of Point Location in Arbitrary Planar Domains
## 9 Discussion and Conclusions
The dependency on scale is particularly relevant near the boundaries: if very high accuracy is required, the samples over the curved boundaries must be more dense; hence, a larger number of triangles is generated in the mesh. This dependency on scale (with consequences on the accuracy in areas closer to the borders) does not occur in the proposed approach because it is analytical. The next list of topics explains the advantages of the proposed approach in this paper, the reasons for better performance than existing solutions, and the reason for that to happen. * It involves fewer operations: besides the common sums and subtractions, at most only one floating point division is required per segment (or two multiplications instead) and also fewer comparisons, for the case of polygonal shaped regions. This makes it faster than state of the art solutions (about 5 times as fast was demonstrated for polygons with many sides and a large number of data points for the algorithm implemented in Matlab [8]). * It has no restrictions on which points define the polygons (or the generically shaped region) nor on the testing points that can be over the border, or being themselves the very vertices, and no particular post processing or operations are required. * It accepts the geometry of boundary segments to be other than straight lines and is applicable virtually to any type of curve that can be expressed parametrically. * The parametric formulation of the boundary makes it straightforward to define the curve segments and their extremes (anchor points). * The parametric definition is intrinsically associated with the concept of sense or circulation, which allows the detection of the sense of axis intersection, and not only the fact that there are intersections; and the sense of intersection is crucial to obtaining the global winding number. * The parametric description allows a separation of coordinates, making algebraic operations independent, and allows us to overcome limitations in degenerate situations that would occur in two coordinate point representations but do not occur in separate coordinate representations. Although the formal and analytic procedures for point location in generic planar domains have been fully established and demonstrated in this paper, the future may hold improvements in the computational component of the algorithms for specific curves. This can be true, namely for some curves of very high complexity, as well as for the detection of specific situations of point location relative to those more complex boundaries, in order to accelerate the calculation operations. An early part of this work was developed under the Ph.D. Grant BD-1657/91_IA from JNICT, Programa CIENCIA, Portugal. **Data Availability Statement:** Algorithms and data to support the results of this paper are available at a GitHub repository: [[https://github.com/vitoruapt/PointInclusion](https://github.com/vitoruapt/PointInclusion)]([https://github.com/vitoruapt/PointInclusion](https://github.com/vitoruapt/PointInclusion)). **Acknowledgments:** The author acknowledges the EC Joint Research Center of Ispra, Italy, for hosting him as a Ph.D. student in the early 1990s' where the basic idea that lead to this research started. **Conflicts of Interest:** The author declares no conflicts of interest.