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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 1847, von Staudt demonstrated that the algebraic structure is implicit in projective geometry, by creating an algebra based on construction of the projective harmonic conjugate, which he called a throw (German: Wurf): given three points on a line, the harmonic conjugate is a fourth point that makes the cross ratio equal to −1.
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 .
There exist four points such that no three are collinear (points not on a single line). In an affine plane, two lines are called parallel if they are equal or disjoint. Using this definition, Playfair's axiom above can be replaced by: [2] Given a point and a line, there is a unique line which contains the point and is parallel to the line.
the points AB ∩ ab, AC ∩ ac and BC ∩ bc are collinear. The points A, B, a and b are coplanar (lie in the same plane) because of the assumed concurrency of Aa and Bb. Therefore, the lines AB and ab belong to the same plane and must intersect. Further, if the two triangles lie on different planes, then the point AB ∩ ab belongs to
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 ...
For instance, the Sylvester–Gallai theorem, stating that any non-collinear set of points in the plane has an ordinary line containing exactly two points, transforms under projective duality to the statement that any projective arrangement of finitely many lines with more than one vertex has an ordinary point, a vertex where only two lines cross.
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related to: collinear points means in geometry terms crossword pdf