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Let X be a Riemann surface.Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function :), we can associate a differential form of compact support, the Poincaré dual of c, with the property that integrals along c can be calculated by integrals over X:
The Bentley–Ottmann algorithm processes a sequence of + events, where denotes the number of input line segments and denotes the number of crossings. Each event is processed by a constant number of operations in the binary search tree and the event queue, and (because it contains only segment endpoints and crossings between adjacent segments ...
In order to find the intersection point of a set of lines, we calculate the point with minimum distance to them. Each line is defined by an origin a i and a unit direction vector n̂ i . The square of the distance from a point p to one of the lines is given from Pythagoras:
If the winding number is non-zero, the point lies inside the polygon. This algorithm is sometimes also known as the nonzero-rule algorithm. To check if a given point lies inside or outside a polygon: Draw a horizontal line to the right of each point and extend it to infinity. Count the number of times the line intersects with polygon edges.
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]
Python sets are very much like mathematical sets, and support operations like set intersection and union. Python also features a frozenset class for immutable sets, see Collection types. Dictionaries (class dict) are mutable mappings tying keys and corresponding values. Python has special syntax to create dictionaries ({key: value})
In these trees, each node contains one of the input points. Since the division of the plane is decided by the order of point-insertion, the tree's height is sensitive to and dependent on insertion order. Inserting in a "bad" order can lead to a tree of height linear in the number of input points (at which point it becomes a linked-list).
The intersection number can also be found in polynomial time for graphs whose maximum degree is five, but is NP-hard for graphs of maximum degree six. [38] [39] On planar graphs, computing the intersection number exactly remains NP-hard, but it has a polynomial-time approximation scheme based on Baker's technique. [21]