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Two intersecting 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 particular it is important to assure that for two given line segments, a new line segment can be constructed, such that its length equals the product of lengths of the other two. Similarly one needs to be able to construct, for a line segment of length , a new line segment of length . The intercept theorem can be used to show that for both ...
In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal.
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 elliptic geometry, two lines perpendicular to a given line must intersect. In fact, all perpendiculars to a given line intersect at a single point called the absolute pole of that line. Every point corresponds to an absolute polar line of which it is the absolute pole. Any point on this polar line forms an absolute conjugate pair with the
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). Other types of geometric intersection include: Line–plane intersection; Line–sphere intersection; Intersection of a polyhedron with a line
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The analytic determination of the intersection curve of two surfaces is easy only in simple cases; for example: a) the intersection of two planes, b) plane section of a quadric (sphere, cylinder, cone, etc.), c) intersection of two quadrics in special cases. For the general case, literature provides algorithms, in order to calculate points of ...