1. viz 1 (available in VR)
  2. viz 2 (available in VR)
  3. viz 3 (available in VR)

Introduction to inversion

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Inversion is a simple geometric operation exchanging the inside and the outside of a circle or sphere.

Here is the definition for a sphere. Note that the same definition applies to circles. Given a circle or sphere of radius \(r\), inversion sends a point \(P\) to a point \(P'\) where \(OP \cdot OP' = r^2\) and the two points lie on the same line going through the center of the inverting sphere. Note that points on the boundary are fixed. In its most basic definition, the inversion map is not defined at the center of the sphere. The definition can be extended to exchange the center with a point at infinity.

Inversion can also be given a more geometric definition which is shown in the visualization above. If the given point is in the interior of the sphere, we take a perpendicular plane to the point and intersect it with the sphere. The inversion maps to the unique point on the axial line that intersects the tangent plane there. If the point is on the outside of the sphere, we just reverse the process by first finding a tangent and then a perpendicular plane at the tangent point to arrive at a point on the axial line in the interior of the sphere.

From the visualization, if you select the sphere checkbox, we see a neat property of inversion: it maps spheres to spheres. It also takes circles to circles.

Applying inversion to the image of a point gives you back the original point. So you can think of the red point as being the inverted image of the blue point or vice versa.

Inversion may seem like a classical curiosity from acquity, but it is actually relevant for doing modern mathematics. For instance, inversions relate different models of hyperbolic geometry to each other and also represent isometries of hyperbolic space in the disk model.

Relationship to stereographic projection

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Inversions also have a neat relationship with stereographic projection, which we looked at in another Vinequai post. Stereographic projection of a sphere from its north pole can be viewed as the restriction of an inversion of a containing sphere whose radius is twice as large. This is shown in the visualization above. You can view the red point as the inversion of the blue point with respect to the yellow sphere, or you can view it as the stereographic projection of the blue point with respect to the blue sphere.

What this means is that facts that we prove about inversion automatically apply to stereographic projection. For instance as we will show below, inversion maps circles to circles, which means that stereographic projection also maps circles to circles. We also saw this property of stereographic projection in the stereographic projection Vinequai post

Characterisation via orthogonal circles and spheres

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Inversion also has an equivalent definition with respect to orthogonal spheres. It is the unique map that exchanges the inside of the sphere with the outside of the sphere and maps spheres that intersect orthogonally to themselves. This is a bit of a mouthful, so the visualization above can help show what I mean.

To see how this definition is connected to the original one. All orthogonal spheres passing through a given point in the inverting sphere share a unique point in common at the exterior of this sphere. This point at the exterior is the image of the given point.

The fact that the two definitions are the same boils down to the tangent secant theorem

This characterisation by orthogonal spheres is a useful one, as we can use it to prove interesting properties about inversion. For instance inversion preserves angles. For any point and any two direction verctors, we can find two orthogonal spheres that meet at that point along each of the direction vectors. By symmetry across the plane containing the centers of all orthogonal spheres, we see that the two orthogonal spheres meet at the same angle at the image point.

To see the fact that we can achieve any desired direction through the given point with an orthogonal sphere, you can use the visualization above. For shallow angles, we place the center of the orthogonal sphere near the midpoint of the red and blue points. For a steeper angle, we place the center farther away.

The orthogonal sphere characterisation can also be used to show that inversion takes circles to circles and spheres to spheres. Roughly speaking, since inversion preserves orthgonal spheres and preserves angles, it must take tangent spheres to tangent spheres.

For spheres that are not tangent, we can gradually blow them up or shrink them down until they are tangent and then apply the same reasoning.

All spheres passing through a point in the interior and orthogonal to the inverting sphere pass through a point on the exterior. This fact can be seen in the visualization above. These points are the inversion pair. Each orthogonal sphere (purple) intersects the inverting sphere (gold) in a circle at 90 degrees. The intrsection is indicated in green.


Material for this math demonstration comes from Chapters 1.2 and 2.2 of Bill Thurston's Three-Dimensional Geometry and Topology. This book is a treasure trove of beautiful geometric ideas and will be a rich source of inspiration for future Vinequai posts.