Search results
Results from the WOW.Com Content Network
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.
There are two types, points and lines, and one "incidence" relation between points and lines. The three axioms are: G1: Every line contains at least 3 points; G2: Every two distinct points, A and B, lie on a unique line, AB. G3: If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C).
The Napoleon points and generalizations of them are points of concurrency. For example, the first Napoleon point is the point of concurrency of the three lines each from a vertex to the centroid of the equilateral triangle drawn on the exterior of the opposite side from the vertex. A generalization of this notion is the Jacobi point.
The image represents the projective line as a circle where antipodal points are identified, and shows the two homeomorphisms of a real line to the projective line; as antipodal points are identified, the image of each line is represented as an open half circle, which can be identified with the projective line with a single point removed.
That is, a plane duality σ will map points to lines and lines to points (P σ = L and L σ = P) in such a way that if a point Q is on a line m (denoted by Q I m) then Q I m ⇔ m σ I ∗ Q σ. A plane duality which is an isomorphism is called a correlation. [6] The existence of a correlation means that the projective plane C is self-dual.
The intersection points are: (−0.8587, 0.7374, −0.6332), (0.8587, 0.7374, 0.6332). A line–sphere intersection is a simple special case. Like the case of a line and a plane, the intersection of a curve and a surface in general position consists of discrete points, but a curve may be partly or totally contained in a surface.
Let l 1 = [a 1, b 1, c 1] and l 2 = [a 2, b 2, c 2] be a pair of distinct lines. Then the intersection of lines l 1 and l 2 is point a P = (x 0, y 0, z 0) that is the simultaneous solution (up to a scalar factor) of the system of linear equations: a 1 x + b 1 y + c 1 z = 0 and a 2 x + b 2 y + c 2 z = 0. The solution of this system gives: x 0 ...
Adding four new points, each being added to all the lines of a single parallel class (so all of these lines now intersect), and one new line containing just these four new points produces the projective plane of order three, a (13 4) configuration. Conversely, starting with the projective plane of order three (it is unique) and removing any ...