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If the geometric dimension of a pappian projective space is at least 2, then every collineation is the product of a homography (a projective linear transformation) and an automorphic collineation. More precisely, the collineation group is the projective semilinear group, which is the semidirect product of homographies by automorphic collineations.
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.
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).
This is a list of two-dimensional geometric shapes in Euclidean and other geometries. For mathematical objects in more dimensions, see list of mathematical shapes. For a broader scope, see list of shapes.
In geometry, collinearity of a set of points is the property of their lying on a single line. [1] A set of points with this property is said to be collinear (sometimes spelled as colinear [ 2 ] ). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".
The list of 53 types is given in Dembowski (1968, pp. 124–125) and a table of the then known existence results (for both collineation groups and planes having such a collineation group) in both the finite and infinite cases appears on page 126. As of 2007, "36 of them exist as finite groups.
For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak. Vertex figure: not itself an element of a polytope, but a diagram showing how the elements meet.
The geometric view of a central collineation is easiest to see in a projective plane. Given a central collineation α, consider a line ℓ that does not pass through the center O, and its image under α, ℓ ′ = α(ℓ). Setting R = ℓ ∩ ℓ ′, the axis of α is some line M through R.
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