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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 ...
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines , which either is one point (sometimes called a vertex ) or does not exist (if the lines are parallel ).
Additional vertices are inserted into both polygons at the points of intersection; an intersection vertex holds a pointer to its counterpart in the other polygon. In the second phase, each intersection is marked as either an entry intersection or an exit intersection. This is accomplished by evaluating the even–odd rule at the first vertex ...
No two line segment endpoints or crossings have the same x-coordinate; No line segment endpoint lies upon another line segment; No three line segments intersect at a single point. In such a case, L will always intersect the input line segments in a set of points whose vertical ordering changes only at a finite set of discrete events ...
Note that if the subject polygon was concave at vertices outside the clipping polygon, the new polygon may have coincident (i.e., overlapping) edges – this is acceptable for rendering, but not for other applications such as computing shadows. All steps for clipping concave polygon 'W' with a 5-sided convex polygon
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 Möller–Trumbore ray-triangle intersection algorithm, named after its inventors Tomas Möller and Ben Trumbore, is a fast method for calculating the intersection of a ray and a triangle in three dimensions without needing precomputation of the plane equation of the plane containing the triangle. [1]
Rendering of constructive solid geometry is particularly simple when ray tracing.Ray tracers intersect a ray with both primitives that are being operated on, apply the operator to the intersection intervals along the 1D ray, and then take the point closest to the camera along the ray as being the result.