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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 ...
It is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry. The cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property. It was extensively studied in the 19th century. [1]
Since one line of the cone lies in each plane, the apex of each cone must lie in both planes, and hence somewhere on the line of intersection of the two planes. Therefore, the three external homothetic centers are collinear. Monge's theorem can also be proved by using Desargues' theorem.
Let x, y, and z refer to a coordinate system with the x- and y-axis in the sensor plane. Denote the coordinates of the point P on the object by ,,, the coordinates of the image point of P on the sensor plane by x and y and the coordinates of the projection (optical) centre by ,,.
Thus (E, H; J, G) = (E, K; D, L), so by Lemma X, the points H, M, and K are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon ADEGBZ are collinear. Lemmas XV and XVII are that, if the point M is determined as the intersection of HK and BG, then the points A, M, and D are collinear.
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, an incidence relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a point lies on a line" or "a line is contained in a plane" are used. The most basic incidence relation is that between a point, P , and a line, l , sometimes denoted P I l .
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