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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.
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 ...
In geometry, an incidence relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a point lies on a line" or "a line is contained in a plane" are used. The most basic incidence relation is that between a point, P , and a line, l , sometimes denoted P I l .
By Desargues' theorem, the points X, Y, Z are collinear. The line of collinearity is the axis of perspectivity of triangle ABC and triangle DEF. The line XYZ is the trilinear polar of the point P. [1] The points X, Y, Z can also be obtained as the harmonic conjugates of D, E, F with respect to the pairs of points (B, C), (C, A), (A, B ...
A permutation of the seven points that carries collinear points (points on the same line) to collinear points is called a collineation or symmetry of the plane. The collineations of a geometry form a group under composition, and for the Fano plane this group (PΓL(3, 2) = PGL(3, 2)) has 168 elements.
Monge's theorem states that the three such points given by the three pairs of circles always lie in a straight line. In the case of two of the circles being of equal size, the two external tangent lines are parallel. In this case Monge's theorem asserts that the other two intersection points must lie on a line parallel to those two external ...
In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following construction: Given three collinear points A, B, C , let L be a point not lying on their join and let any line through C meet LA, LB at M, N respectively.
For any point A and line l not incident with it (an anti-flag) there is exactly one line m incident with A (that is, A I m), that does not meet l (known as Playfair's axiom), and satisfying the non-degeneracy condition: There exists a triangle, i.e. three non-collinear points. The lines l and m in the statement of Playfair's axiom are said to ...