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In mathematics, projections onto convex sets (POCS), sometimes known as the alternating projection method, is a method to find a point in the intersection of two closed convex sets. It is a very simple algorithm and has been rediscovered many times. [1] The simplest case, when the sets are affine spaces, was analyzed by John von Neumann.
intersection of two polygons: window test. If one wants to determine the intersection points of two polygons, one can check the intersection of any pair of line segments of the polygons (see above). For polygons with many segments this method is rather time-consuming. In practice one accelerates the intersection algorithm by using window tests ...
Therefore, two line segments that share an endpoint, or a line segment that contains an endpoint of another segment, both count as an intersection of two line segments. When multiple line segments intersect at the same point, create and process a single event point for that intersection.
The algorithm is based on the definition of the "inside" of a polygon based on the winding number. It considers regions with odd winding number to be inside the polygon; this is known as the even–odd rule. It takes two lists of polygons as input. In its original form, the algorithm is divided into three phases:
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 ...
Convex polygons will only have one intersecting polygon. The same algorithm can be used for merging two polygons by starting at the outbound intersections rather than the inbound ones. However this can produce counter-clockwise holes. Some polygon combinations may be difficult to resolve, especially when holes are allowed.
The x and y coordinates of the point of intersection of two non-vertical lines can easily be found using the following substitutions and rearrangements. Suppose that two lines have the equations y = ax + c and y = bx + d where a and b are the slopes (gradients) of the lines and where c and d are the y-intercepts of the lines.
First, find out if the line produced by the ray intersects with the plane that the triangle is on, and if it does, find the coordinates of that intersection. The only way that the line will not intersect the plane is if the ray's direction vector is parallel to the plane. [3]