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This page is a collection of various results related to circle inversions and Mobius transformations. Some of the results will be described in more detail my thesis, and others I plan to describe in more detail in the future.

Mobius frame

A Mobius frame is a curved coordinate frame that is defined by 3 points and that defines a Mobius transformation. Mobius transformations are conformal and preserve circles, where lines are circles with infinite radius.
 

Mobius frame inversion

A Mobius frame is an orientation-reversing similarity frame under circle inversion.

Steady Mobius pattern

Two Mobius frames define a steady Mobius pattern or motion. Steady Mobius patterns have a loxodrome (double spiral) shape.

Loxodrome through 4 points

A loxodrome interpolates 4 points. The curve is the path traced by a point under a steady Mobius motion.

"Mobius rotation" (elliptic Mobius transformation) from 4 points

Four points define an elliptic steady Mobius motion. This is a rotation under a circle inversion.

Steady Mobius 2-pattern

A 2-directional pattern of circles may be defined as a Mobius pattern of a Mobius pattern of circles.

Trans-Mobius Interpolant (TMI) map

A TMI map is a generalization of several existing maps, including Mobius transformations, Corner-Operated Tran-Similar (COTS) maps, and Four-Point Interpolant maps. A TMI map is a circle inversion of a COTS map. It may be defined by 5 points on the boundary of the map (4 of which are corners). Consecutive tiles along a particular direction in the map are related by the same Mobius transformation, and all tiles are related by a Mobius transformation. Angular distortion is evenly distributed throughout the map.

Steady Four-Point Interpolant (FPI) patterns and motions

A repeated application of the same FPI transformation yields a steady FPI pattern. A steady FPI pattern or motion is a steady affine pattern or motion under a circle inversion. A steady FPI pattern or motion may be defined by 7 points, 4 that define one FPI map and another 3 that define the next FPI map in the steady pattern.

Steady FPI spiral through 5 points

An interesting double spiral (a generalization of the loxodrome) interpolates 5 points. The curve is the path traced by a point under a steady FPI motion.

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