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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 ...
Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affine , Euclidean , absolute , and hyperbolic geometry (but not for projective geometry).
In geometry, the Euler line, named after Leonhard Euler (/ ˈ ɔɪ l ər / OY-lər), is a line determined from any triangle that is not equilateral.It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.
While the projective linear group of the projective line is 3-transitive (any three distinct points can be mapped to any other three points), and indeed simply 3-transitive (there is a unique projective map taking any triple to another triple), with the cross ratio thus being the unique projective invariant of a set of four points, there are ...
A projective plane geometry is a nonempty set X (whose elements are called "points"), along with a nonempty collection L of subsets of X (whose elements are called "lines"), such that: For every two distinct points, there is exactly one line that contains both points. The intersection of any two distinct lines contains exactly one point.
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.
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.