# Tetracyclic coordinates

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of a point in the plane

A kind of homogeneous coordinates for a point in the complex inversive plane. The numbers , not all zero, are connected by the relation

All points which satisfy a linear equation

are said to form a circle with "coordinates" . Two circles and are orthogonal if , tangent if

If two circles and intersect, the expression

measures the cosine of their angle (or the hyperbolic cosine of their inversive distance).

In three dimensions, with an extra coordinate , one obtains the analogous pentaspherical coordinates, which lead to spheres instead of circles.

According to an alternative definition, involving only real numbers, tetracyclic coordinates of points and circles in the plane can be introduced using stereographic projection. Here the tetracyclic coordinates of a point in the plane are the homogeneous coordinates of the point on the sphere corresponding to it under stereographic projection. The tetracyclic coordinates of a circle in the plane are the homogeneous coordinates of the point in space that is the pole of the plane of the circle on the sphere which corresponds to the circle in the plane under stereographic projection with respect to that sphere.