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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".
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, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. [1] The line through these points is the Simson line of P, named for Robert Simson. [2] The concept was first published, however, by William Wallace in 1799, [3] and is sometimes called the Wallace line. [4]
A collineation, automorphism, or symmetry of the Fano plane is a permutation of the 7 points that preserves collinearity: that is, it carries collinear points (on the same line) to collinear points. By the Fundamental theorem of projective geometry , the full collineation group (or automorphism group , or symmetry group ) is the projective ...
The Nagel point is the isotomic conjugate of the Gergonne point.The Nagel point, the centroid, and the incenter are collinear on a line called the Nagel line.The incenter is the Nagel point of the medial triangle; [2] [3] equivalently, the Nagel point is the incenter of the anticomplementary triangle.
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.
Another type of non-Euclidean geometry is the hyperbolic plane, and arrangements of lines in this geometry have also been studied. [50] Any finite set of lines in the Euclidean plane has a combinatorially equivalent arrangement in the hyperbolic plane (e.g. by enclosing the vertices of the arrangement by a large circle and interpreting the ...
A net V of plane curves of some degree d such that the base locus of a generic pencil of V is the base locus of V together with d–1 collinear points (Dolgachev 2012, theorem 7.3.5) (Coolidge 1931, p. 423) lemniscate A lemniscate is a curve resembling a figure 8. See Salmon (1879, p.42) limaçon