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For instance, in spherical geometry, where lines are represented in the standard model by great circles of a sphere, sets of collinear points lie on the same great circle. Such points do not lie on a "straight line" in the Euclidean sense, and are not thought of as being in a row.
Let one side of an inscribed regular n-gon have length s n and touch the circle at points A and B. Let A′ be the point opposite A on the circle, so that A′A is a diameter, and A′AB is an inscribed triangle on a diameter. By Thales' theorem, this is a right triangle with right angle at B.
For example, three (non-collinear) points determine a circle: the generic circle is given by the equation () + = where the center is located at (a, b) and the radius is r. Equivalently, by expanding the squared terms, the generic equation is x 2 − 2 a x + y 2 − 2 b y = k , {\displaystyle x^{2}-2ax+y^{2}-2by=k,} where k = r 2 − a 2 − b 2 ...
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 ...
The equations originate from the central projection of a point of the object through the optical centre of the camera to the image on the sensor plane. [1] The three points P, Q and R are projected on the plane S through the projection centre C x- and z-axis of the projection of P through the projection centre C
The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a disc. The circle has been known since before the beginning of recorded history.
Explicitly, let the conic be the unit circle. For any two points P and Q, inside the unit circle . If the line connecting them intersects the circle in two points, X and Y and the points are, in order, X, P, Q, Y. Then the hyperbolic distance between P and Q in the Cayley–Klein model of the hyperbolic plane can be expressed as
Gauss's circle problem asks how many points there are inside this circle of the form (,) where and are both integers. Since the equation of this circle is given in Cartesian coordinates by x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} , the question is equivalently asking how many pairs of integers m and n there are such that