Miquelian inversive plane

This term is related to: incidence geometry
View other terms related to incidence geometry | View facts related to incidence geometry

This term is related to: inversive geometry
View other terms related to inversive geometry | View facts related to inversive geometry

Definition

An inversive plane is said to be 'Miquelian if it satisfies the following. Let $C_{1},C_{2},C_{3},C_{4}$ be four circles and let the intersection of $C_{i}$ with $C_{i+1}$ (read modulo 4) be the set $\{a_{i},b_{i}\}$ ($a_{i}$ may be equal to $b_{i}$). Then, $a_{i}$s are concircular if and only if the $b_{i}$s are concircular.

It turns out that an inversive plane is Miquelian if and only if it arises from a non-ruled quadric in a three-dimensional geometry over a commutative field.

Relation with other properties

Weaker properties

• Inversive plane satisfying bundle theorem

References

Finite Geometries by Peter Dembowski, Chapter 6. Inversive planes