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The reciprocal of a real or complex number z!=0 is its multiplicative inverse 1/z=z^(-1), i.e., z to the power -1. The reciprocal of zero is undefined. A plot of the ...

The midsphere, also called the intersphere, reciprocating sphere, or inversion sphere, is a sphere with respect to which the polyhedron vertices of a polyhedron are the ...

Reciprocation is an incidence-preserving transformation in which points are transformed into their polars. A projective geometry-like duality principle holds for ...

Given collinear points W, X, Y, and Z, Y and Z are harmonic conjugates with respect to W and X if (|WY|)/(|YX|)=(|WZ|)/(|XZ|). (1) W and X are also harmonic conjugates with ...

Given an obtuse triangle, the polar circle has center at the orthocenter H. Call H_i the feet. Then the square of the radius r is given by r^2 = HA^_·HH_A^_ (1) = HB^_·HH_B^_ ...

The reciprocal curve of a given circle is the locus of a point which moves so that its distance from the center of reciprocation varies as its distance from the line which is ...

If a, b, c, and d are points in the extended complex plane C^*, their cross ratio, also called the cross-ratio (Courant and Robbins 1996, p. 172; Durell 1928, p. 73), ...

A type of plot invented by M. Trott that shows the behavior of a function near and far from the origin simultaneously. Inside a given radius, the plot shows the actual value ...

A cyclide formed by inversion of a standard torus when inversion sphere is tangent to the torus.

A parabolic cyclide formed by inversion of a horn torus when the inversion sphere is tangent to the torus.