Math 445 Week 6: Introduction to Inversive Geometry

Inversive geometry is a way of studying the geometry of circles by transformational geometry.  In addition to being interesting in its own right, it will provide the tools for our later study of non-Euclidean geometry.

Each of the 3 books* associated with the course has a lot of information about inversive geometry.  It is a great resource, but also a possible source of confusion, that the approach of each book is different.  The books differ in their ordering of topics and their choice of geometrical tools.  In the end, they all get to about the same place, but this embarrassment of riches may make it more challenging to organize the material in your mind at the initial stages.  Thus, in this part of the course, we will follow a quite structured path through the subject.

Reading Guide:  (Note: Inversion and radical axis are also defined in Birkhoff &Beatley.)


Book pages





Definition of inversion




Constructions of Inversion




Harmonic Ratio




Power of a Point




Orthogonal Circles




Radical Axis




Pencils of circles




Images under inversion




Math 445 Monday, 2/10

Topic: Power of a point

Learn definition.  Review circle theorems that prove power can be computed from secants.  Learn relationship with orthogonal circle radii.

Topic: Radical axis

Learn definition of radical axis of two circles.  Use the radical axis in constructing orthogonal circles.

Topic:  Orthogonal Circles, with connection to Power of a point

Power of a point gives radius of orthogonal circle.  Point on radical axis of two circles is the center of a circle orthogonal to both (if the point is exterior to the circles).

Math 445 Wednesday, 2/12

Topic: Radical axis

Prove the radical axis is a line using coordinate equations.

Topic: Inversion of a Point in a Circle

Learn definition of inversion.  Circles through P and P' = inversion of P are orthogonal to mirror circle (except when P is on c and so P = P').

Topic:  Orthogonal Circles, with connection to Power of a point and Inversion

Power of a point gives radius of orthogonal circle.  Circles through P and inversion of P are orthogonal to mirror circle.

Math 487, Wednesday, 2/12

Topic: Constructions of Inversion of a Point in a Circle

Learn constructions for inversion. 

Topic: Construction of orthogonal circles

Construct orthogonal circles using inversion.  Construct orthogonal circles using radical axes.

Math 445 Friday, 2/14

Topic: Images under inversion

Learn proofs that images of lines and circles are either circles or lines.  State invariance of angle under inversion.

Topic: Solving geometry problems by inversion

Problem: Given 3 circles through a point A with the other ponts of intersection of pairs of circles at points B, C, D. Then the arcs of the circles define a sort of triangle BCD or arcs. What is the angle sum?

Answer: Invert the figure in a circle with center A. The figure becomes an ordinary triangle. Since angles are preserved, the angle sum is 180 degrees.

Topic: The Inversive Plane

The inversive plane is an ordinary Euclidean plane with one point added at infinity. The "circles" or i-circles or inversive circles are of two kinds (a) a Euclidean circle or (b) a Euclidean line with the point at infinity added. Using this concept, the inversion of an i-circle in an i-circle is an i-circle.

Topic: Connection with Stereographic Projection

The ordinary Euclidean plane can be mapped to a sphere minus the "north pole" N by (inverse) stereographic projection. Also, the the circles on the plane are mapped to circles on the sphere (not through N) and the lines in the plane are mapped to circles through N. Thus if we define the stereographic projection of the point at infinity to be N, then the each i-circle is mapped to an ordinary circle on the sphere. Thus the inversive plane can be modeled by the points on the sphere with the -icircles being ordinary circles on the sphere.

Math 445 Wednesday, 2/19

Topic: Pencils of Circles

Define 3 kinds of pencils and their images under inversion.