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Assume that we want to find intersection of two infinite lines in 2-dimensional space, defined as a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0. We can represent these two lines in line coordinates as U 1 = (a 1, b 1, c 1) and U 2 = (a 2, b 2, c 2). The intersection P′ of two lines is then simply given by [4]
The Shamos–Hoey algorithm [1] applies this principle to solve the line segment intersection detection problem, as stated above, of determining whether or not a set of line segments has an intersection; the Bentley–Ottmann algorithm works by the same principle to list all intersections in logarithmic time per intersection.
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 ...
Shortest path (A, C, E, D, F) between vertices A and F in the weighted directed graph. In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
The rotating calipers technique for designing geometric algorithms may also be interpreted as a form of the plane sweep, in the projective dual of the input plane: a form of projective duality transforms the slope of a line in one plane into the x-coordinate of a point in the dual plane, so the progression through lines in sorted order by their ...
PLL problems generally have 2 solutions. As shown above, if a circle is tangent to two given lines, its center must lie on one of the two lines that bisect the angle between the two given lines. By symmetry, if such a circle passes through a given point P, it must also pass through a point Q that is the "mirror image" of P about the angle bisector.
An example of coplanar points. Two lines in three-dimensional space are coplanar if there is a plane that includes them both. This occurs if the lines are parallel, or if they intersect each other. Two lines that are not coplanar are called skew lines.
It can be viewed as a three-dimensional version of the line clipping problem. [ 1 ] If the polyhedron is given as the intersection of a finite number of halfspaces , then one may partition the halfspaces into three subsets: the ones that include only one infinite end of the line, the ones that include the other end, and the ones that include ...