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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:
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 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]
The intersection points are: (−0.8587, 0.7374, −0.6332), (0.8587, 0.7374, 0.6332). A line–sphere intersection is a simple special case. Like the case of a line and a plane, the intersection of a curve and a surface in general position consists of discrete points, but a curve may be partly or totally contained in a surface.
The numbers ,, are called the intersection numbers of the graph. They satisfy the equation + + =, where = is the valency, i.e., the number of ...
In terms of intersection forms, we say the plane has one of type x 2 (there is only one class of lines, and they all intersect with each other). Note that on the affine plane, one might push off L to a parallel line, so (thinking geometrically) the number of intersection points depends on the choice of push-off. One says that “the affine ...
Numbers were the symbols of the divine order of the universe. “However, the Greek mathematician and mystic Pythagoras studied these great teachings and formed a more cohesive philosophy of numbers.
The intersection of a ray of light with each plane is used to produce an image of the surface. In vision-based 3D reconstruction, a subfield of computer vision, depth values are commonly measured by so-called triangulation method, which finds the intersection between light plane and ray reflected toward camera.