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An arbitrary line and cylinder may have no intersection at all. Or there may be one or two points of intersection. [1] Or a line may lie along the surface of a cylinder, parallel to its axis, resulting in infinitely many points of intersection. The method described here distinguishes between these cases, and when intersections exist, computes ...
In the simplest case, the intersection of two non-parallel planes in Euclidean 3-space is a line. In general, an intersection curve consists of the common points of two transversally intersecting surfaces, meaning that at any common point the surface normals are not parallel. This restriction excludes cases where the surfaces are touching or ...
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines , which either is one point (sometimes called a vertex ) or does not exist (if the lines are parallel ).
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 embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single point.
In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective ...
This is found by noticing that the line must be perpendicular to both plane normals, and so parallel to their cross product (this cross product is zero if and only if the planes are parallel, and are therefore non-intersecting or entirely coincident).
Hyperbolas can degenerate to two intersecting lines (the asymptotes), as in =, or to two parallel lines: =, or to the double line =, as a goes to 0. Parabolas can degenerate to two parallel lines: x 2 − a y − 1 = 0 {\displaystyle x^{2}-ay-1=0} or the double line x 2 − a y = 0 , {\displaystyle x^{2}-ay=0,} as a goes to 0; but, because ...
Because parallel lines in a Euclidean plane are equidistant there is a unique distance between the two parallel lines. Given the equations of two non-vertical, non-horizontal parallel lines, = + = +, the distance between the two lines can be found by locating two points (one on each line) that lie on a common perpendicular to the parallel lines ...