Twoplane

Twoplane refers to a pair of planes in three-dimensional Euclidean space. In geometry, the relationship between two planes is determined by their relative orientation and position: they may intersect along a line, be parallel without meeting, or be coincident if they are the same plane.

A plane in three-space can be described by a linear equation using a normal vector. Let P1 be defined by n1 · x = d1 and P2 by n2 · x = d2, where n1 and n2 are the normals and x is the position vector (x, y, z). If n1 and n2 are not parallel, the planes intersect in a straight line. The direction of this line is given by the cross product l = n1 × n2. A particular point on the line can be found by solving the two equations for two coordinates in terms of a parameter, using standard linear algebra.

The angle between two planes is defined as the angle between their normal vectors. If theta is this angle, cos(theta) = (n1 · n2) / (|n1||n2|). The smaller angle is often taken as the dihedral angle between the planes.

Special cases include parallel planes, where n1 and n2 are parallel; if additionally d1 ≠ d2, the planes do not intersect. If n1 and n2 are parallel and d1 = d2, the planes are coincident (the same plane). If the planes are not parallel, they intersect along a line, and the two-plane configuration is fully determined by n1, n2, d1, and d2.

Applications of the concept appear in computer-aided design, computer graphics, architecture, robotics, and geometric reasoning, where two-plane relationships are used to model surfaces, clipping volumes, and intersection tests.