A new hierarchical triangle-based point-in-polygon data structure

Abstract

The point-in-polygon or containment test is fundamental in computational geometry and is applied intensively in geographical information systems. When this test is repeated several times with the same polygon a data structure is necessary in order to reduce the linear time needed to obtain an inclusion result. In the literature different approaches, like grids or quadtrees, have been proposed for reducing the complexity of these algorithms. We propose a new data structure based on hierarchical subdivisions by means of tri-cones, which reduces the time necessary for this inclusion test. This data structure, named tri-tree, decomposes the whole space by means of tri-cones and classifies the edges of the polygon. For the inclusion test only the edges classified in one tri-cone are considered. The tri-tree has a set of properties which makes it specially suited to this aim, with a similar complexity to other data structures, but well suited to polygons which represent regions. We include a theoretical and practical study in which the new data structure is compared with classical ones, showing a significant improvement.

Introduction

The containment test is widely used in areas like computational geometry, computer graphics and geographical information systems (GIS). In some situations we must consider the inclusion not only of a point in a polygon, but also of many points and polygons, and a common task is to assign points to polygons. The point-in-polygon operation is popular in GIS analysis and makes sense from both the discrete object and the field perspective (Longley et al., 2001).

The point-in-polygon test could be used in a very large set of applications, such as the superposition of polygons, the classification of entities or queries about properties of regions. Imagine an area in which it is necessary to determine spatial coincidence between hydrologic unit codes and regions. A point-in-polygon routine is needed for assigning regions to hydrologic units. These hydrologic units are useful for the prediction of zones of erosion and deposition in a catchment, and can be expanded when torrential rains occur. This special situation is considered in the “Andalucía” region of Spain.

The inclusion test consists of obtaining a boolean result regarding the containment of a point in the interior or on the boundary of a polygon represented by a set of ordered vertices.

The definition of polygon changes for different applications, and special algorithms have been developed for different types of polygons (like convex or non-convex) and for different definitions of polygons (like simple or non-simple). We consider a polygon as an ordered set of vertices (in counter-clockwise order) which form edges making a closed loop without edge intersection (simple polygon). We allow convex or non-convex polygons, manifold or non-manifold, and polygons with or without holes (Fig. 1). We employ this type of polygons which represent spatial areas in a GIS perspective.

When an inclusion test is needed repetitively for the same polygon and there is enough memory in the system for extra information, some data structures are introduced as pre-processing in order to reduce the time complexity of the algorithms. In these cases, some features of the polygon are classified or arranged and the inclusion time is reduced by means of a determined search in these structures.

In this paper we present a new data structure specially suited to the point-in-polygon test. This triangle-based structure subdivides recursively the space occupied by the polygon into tri-cones¹ and classifies the edges of the polygon in pre-processing time. Two bounding triangles delimit the area occupied by the classified edges. In query time the points are classified in this data structure, and a point-in-polygon test is performed only with the edges classified in the corresponding tri-cone (Fig. 2). For the containment test several algorithms could be used, but the ray-crossing method (Haines, 1994) and the triangle-based algorithm (Feito et al., 1995) are well suited.

The new data structure is suitable for polygons which represent regions because it fits properly to the contour, which means a quick inclusion test. The expected complexity is $O(\log n)$ for query, being n the number of vertices of the polygon, with $O(n)$ space and $O(n\log n)$ construction time. This new data structure is constructed in less time than other data structures like quadtrees, obtaining an improvement in the inclusion test and being suitable for some inclusion methods like the ray-crossings or the triangle-based ones.

In the next section we will revise data structures traditionally used for the point-in-polygon test, as well as their relevant properties. A review of point-in-polygon methods is also shown. Then we will propose a new data structure and its fundamental features. We will follow with the point-in-polygon algorithms adapted to the use of this data structure. Finally we will carry out a time study in which we measure the times obtained for diverse inclusion methods and data structures.