Stereographic Projection and Inversion
Hubert Shutrick communicated to me a delightful connection between the stereographic projection and the inversion transform. Inversion is a generalization of a reflection in line. Wonderfully, there is even a more direct connection between inversion and reflection.
The basic configuration consists of a plane and a circle
The diagram below shows a cross-section of the configuration by the xz-plane
The diagram shows several points:
The main result of this page is the observation that two points R and S are the inversive images of each other in
Proof
Assume R and S are inversive images of each other: rs = a², implying the proportion
In another direction, assume S' and T' are two antipodal points on the sphere (in xz-plane). Then since ∠S'NT' = 90°, so is ∠SNT, meaning NO² = OT×OS. This shows that S and T are the images of each other under inversion with negative power. If R' is the reflection of T' in NO, then it is also the reflection of S' in the equatorial plane. Its stereographic image R relates to S by the inversion in
There is also an algebraic derivation. Pick point
Since NO is the diameter of the sphere (and also of the shown circle) angles OS'N and SON are both right, making triangles OS'N and SS'O right and similar. From their similarity we obtain the proportion:
OS / OS' = NO / NS'
from which OS = NO · OS' / NS'. We easily find
NO | = a, | |
OS' | = √u² + v², | |
NS' | = √u² + (a - v)². |
This gives a formula for expressing stereographic projection in a coordinate form:
s = a √u² + v² / √u² + (a - v)². |
Two points S'(u, y, v) and R'(w, y, z) on the sphere are reflections in the equatorial plane if and only if
r = a √u² + (a - v)² / √u² + v².
Multiplying the two gives rs = a².
Both inversion and stereographic projection preserve angles and map (generalized, i.e., including straight lines) circles on (generalized) circles. The revealed connection between the two transformations sheds additional light on why this is so. It is also noteworthy that the circles or lines that are orthogonal to the circle of inversion invert to themselves and are images of the vertical circles under stereographic projection.
Finally, there are at least two popular configurations that fall under the rubric of stereographic projection. The one, as above, has a sphere standing on a plane. In the other, the plane cuts through the center of the sphere so that the 2D and 3D coordinate systems share the origin. In the latter, the asserted property holds even more obviously with the circle of inversion being the equatorial circle of the sphere. Among other things, it is immediately obvious that as the reflection in the equatorial plane exchanges the upper and lower semispheres so the inversion exchanges the interior of the circle of inversion with its exterior.
The algebraic derivation is even simpler. For
rs | = au / (a - v) × au / (a + v) | |
= a²u² / (a² - v²) | ||
= a²u² / u² | ||
= a². |
Note: Stereographic projection serves an enlightening model for another geometric construct, the radical axis of two circles.
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