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In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an isomorphism between projective spaces, or an automorphism from a
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations.This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (projective space) and a selective set of basic geometric concepts.
A mapping of a geometry to itself which sends lines to lines is called a collineation; it preserves the collinearity property. 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.
A central collineation (traditionally these were called perspectivities, [8] but this term may be confusing, having another meaning; see Perspectivity) is a bijection α from P to P, such that there exists a hyperplane H (called the axis of α), which is fixed pointwise by α (that is, α(X) = X for all points X in H) and a point O (called the ...
By the Fundamental theorem of projective geometry, the full collineation group (or automorphism group, or symmetry group) is the projective linear group PGL(3, 2), [a] Hirschfeld 1979, p. 131 [3] This is a well-known group of order 168 = 2 3 ·3·7, the next non-abelian simple group after A 5 of order 60 (ordered by size).
A matter collineation (sometimes matter symmetry and abbreviated to MC) is a vector field that satisfies the condition, = where are the energy–momentum tensor components. The intimate relation between geometry and physics may be highlighted here, as the vector field is regarded as preserving certain physical quantities along the flow lines of , this being true for any two observers.
A plane conic passing through the circular points at infinity. For real projective geometry this is much the same as a circle in the usual sense, but for complex projective geometry it is different: for example, circles have underlying topological spaces given by a 2-sphere rather than a 1-sphere. circuit A component of a real algebraic curve.
In synthetic geometry, point and line are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the ...