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In computational geometry, the point-in-polygon (PIP) problem asks whether a given point in the plane lies inside, outside, or on the boundary of a polygon. It is a special case of point location problems and finds applications in areas that deal with processing geometrical data, such as computer graphics , computer vision , geographic ...
The point location problem is a fundamental topic of computational geometry. It finds applications in areas that deal with processing geometrical data: computer graphics , geographic information systems (GIS), motion planning , and computer aided design (CAD).
The closest pair of points problem or closest pair problem is a problem of computational geometry: given points in metric space, find a pair of points with the smallest distance between them. The closest pair problem for points in the Euclidean plane [ 1 ] was among the first geometric problems that were treated at the origins of the systematic ...
Point in polygon: Decide whether a point is inside or outside a given polygon. In many applications this problem is treated as a single-shot one, i.e., belonging to the first class. For example, in many applications of computer graphics a common problem is to find which area on the screen is clicked by a pointer. However, in some applications ...
An important special case, in which the points are given in the order of traversal of a simple polygon's boundary, is described later in a separate subsection. If not all points are on the same line, then their convex hull is a convex polygon whose vertices are some of the points in the input set. Its most common representation is the list of ...
That is, an input polygon or the convex hull of an input point set must be subdivided into triangles that meet edge-to-edge and vertex-to-vertex, in such a way as to minimize the sum of the perimeters of the triangles. The problem is NP-hard for point set inputs, but may be approximated to any desired degree of accuracy. For polygon inputs, it ...
The article Optimal Reliable Point-in-Polygon Test and Differential Coding Boolean Operations on Polygons describes a point-in-polygon algorithm that is more accurate than the winding number algorithm featured in this Wikipedia article. The winding number algorithm fails for some points if there are parallel or vertical edges in a polygon.
If the point is closer than ε to the line segment, then any points not currently marked to be kept can be discarded without the simplified curve being worse than ε. If the point farthest from the line segment is greater than ε from the approximation then that point must be kept. The algorithm recursively calls itself with the first point and ...