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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".
Möbius' designation can be expressed by saying, collinear points are mapped by a permutation to collinear points, or in plain speech, straight lines stay straight. Contemporary mathematicians view geometry as an incidence structure with an automorphism group consisting of mappings of the underlying space that preserve incidence. Such a mapping ...
For example, right triangles, isosceles triangles and equilateral triangles are non-generic and non-degenerate. In fact, degenerate cases often correspond to singularities, either in the object or in some configuration space. For example, a conic section is degenerate if and only if it has singular points (e.g., point, line, intersecting lines ...
In particular, the situation may be as in the next diagram, which is the diagram for Lemma X. Just as before, we have (J, G; D, B) = (J, Z; H, E). Pappus does not explicitly prove this; but Lemma X is a converse, namely that if these two cross ratios are the same, and the straight lines BE and DH cross at A, then the points G, A, and Z must be ...
A semipartial geometry is a partial geometry if and only if = (+) . It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters ( 1 + s ( t + 1 ) + s ( t + 1 ) t ( s − α + 1 ) / μ , s ( t + 1 ) , s − 1 + t ( α − 1 ) , μ ) {\displaystyle (1+s(t+1)+s(t+1)t(s-\alpha +1)/\mu ,s(t+1 ...
In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear.
In a projective plane, every non-collinear set of n points determines at least n distinct lines. As the authors pointed out, since their proof was combinatorial, the result holds in a larger setting, in fact in any incidence geometry in which there is a unique line through every pair of distinct points.
In the complex plane (formed by viewing the real and imaginary parts of a complex number as the x and y Cartesian coordinates of the plane), concyclicity has a particularly simple formulation: four points in the complex plane are either concyclic or collinear if and only if their cross-ratio is a real number. [17]
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