Inversive Geometry


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Inversive geometry is a nice way to introduce the idea of geometry mappings, which can lead to the study of the Poincaré disk. With a little effort, the ideas are accessible, and there are an abundance of concepts to learn that can be uncovered with geometry or algebra.

Inverse geometry requires that we have a circle with known radius and center Geometry doesn't require coordinates for the location of the circle, but algebra will, so we will assume this is the unit circle with and centered at the origin.

Given any point in the plane at distance from the center of the circle, then is the point on ray at distance from the center of the circle. When does Which points map into the interior of the disk?

Inverse of a point through the unit circle.

You can play with this in desmos by clicking on the graph below. The orange disk is the unit circle. The red dot can be dragged. The green dashed line segment from the red dot towards the origin leads to the purple point, which is at a distance that is the reciprocal of the red dot's distance from origin.

InverseGeometryDesmos

What shape is traced out by the inverse point as we slide the red dot along the line? Is this always the case?

Now, consider infinite graph paper with integer coordinates. The lines consist of all vertical lines with integer values, and all horizontal lines with integer values. What do you get if you find the inverse of all those lines?

This mapping of the entire plane outside the disk into the unit disk (and vice versa) is a conformal mapping, which means it preserves angles. If we define distance between two points in the disk to be the same as the distance between the inverses of the points, then we are preserving distance. Angles are the same and distances are the same, so we could study the ideas in high school geometry, but staying entirely inside a disk. What project ideas does this give you?
inverseEuclideanPlan

inversiveGeometry
A rectangular grid, inverted, and laser cut on plywood.

Inverse of the plane with geometrically spaced lines from Pete Benson on Vimeo.

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