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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.
Line–plane intersection. The intersection of a line and a plane in general position in three dimensions is a point. Commonly a line in space is represented parametrically ((), (), ()) and a plane by an equation + + =. Inserting the parameter representation into the equation yields the linear equation
We wish to find a point which is on both planes (i.e. on their intersection), so insert this equation into each of the equations of the planes to get two simultaneous equations which can be solved for and .
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 ...
Lines A, B and C are concurrent in Y. In geometry, lines in a plane or higher-dimensional space are concurrent if they intersect at a single point.. The set of all lines through a point is called a pencil, and their common intersection is called the vertex of the pencil.
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 embedded in the plane, and is ...
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. [1] [2] Such a drawing is called a plane graph, or a planar embedding of the graph.
Thus the axiom of projective geometry, requiring all pairs of lines in a plane to intersect, is confirmed. [2] Given P and Q in σ, the elliptic distance between them is the measure of the angle POQ, usually taken in radians. Arthur Cayley initiated the study of elliptic geometry when he wrote "On the definition of distance".
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