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The linear maps (or linear functions) of vector spaces, viewed as geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations. In projective geometry these linear mappings are called homographies and are just one type of collineation.
Tangential – intersecting a curve at a point and parallel to the curve at that point. Collinear – in the same line; Parallel – in the same direction. Transverse – intersecting at any angle, i.e. not parallel. Orthogonal (or perpendicular) – at a right angle (at the point of intersection).
Simply, a collineation is a one-to-one map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. One may formalize this using various ways of presenting a projective space. Also, the case of the projective line is special, and hence generally treated ...
The number of vertices is smaller when some lines are parallel, or when some vertices are crossed by more than two lines. [4] An arrangement can be rotated, if necessary, to avoid axis-parallel lines. After this step, each ray that forms an edge of the arrangement extends either upward or downward from its endpoint; it cannot be horizontal.
The line AB is the interval AB and the two rays A/B and B/A. Points on the line AB are said to be collinear. An angle consists of a point O (the vertex) and two non-collinear rays out from O (the sides). A triangle is given by three non-collinear points (called vertices) and their three segments AB, BC, and CA.
One also says "two generic lines intersect in a point", which is formalized by the notion of a generic point. Similarly, three generic points in the plane are not collinear ; if three points are collinear (even stronger, if two coincide), this is a degenerate case .
The Simson line LN (red) of the triangle ABC with respect to point P on the circumcircle. In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. [1] The line through these points is the Simson line of P, named for Robert Simson. [2]
Affine planes satisfy the following axioms (Cameron 1991, chapter 2): (in which two lines are called parallel if they are equal or disjoint): Any two distinct points lie on a unique line. Given a point and line there is a unique line that contains the point and is parallel to the line; There exist three non-collinear points.