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Line–plane intersection. The three possible plane-line relationships in three dimensions. (Shown in each case is only a portion of the plane, which extends infinitely far.) In analytic geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or a line. It is the entire line if that line is ...
Vector projection. The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as or a∥b. The vector component or vector resolute of a perpendicular to b, sometimes ...
The real projective plane can be thought of as the Euclidean plane with additional points added, which are called points at infinity, and are considered to lie on a new line, the line at infinity. There is a point at infinity corresponding to each direction (numerically given by the slope of a line), informally defined as the limit of a point ...
Parallel planes are planes in the same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called skew lines.
In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or another line. Distinguishing these cases and finding the intersection have uses, for example, in computer graphics, motion planning, and collision detection. In three-dimensional Euclidean geometry, if two lines are not in the same plane, they have no ...
A vector tangent to the path is: ... Let two section planes be given: ... Assuming that the two planes are not parallel, the line of intersection is on both planes.
A bivector is an oriented plane element, in much the same way that a vector is an oriented line element. Given two vectors a and b, one can view the bivector a ∧ b as the oriented parallelogram spanned by a and b. The cross product is then obtained by taking the Hodge star of the bivector a ∧ b, mapping 2-vectors to vectors:
The following remarks apply only to finite planes. There are two main kinds of finite plane geometry: affine and projective. In an affine plane, the normal sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a unique point