Inversion in a sphere


In geometry, inversion in a sphere is a transformation of Euclidean space that fixes the points of a sphere while sending the points inside of the sphere to the outside of the sphere, and vice versa. Intuitively, it "swaps the inside and outside" of the sphere while leaving the points on the sphere unchanged. Inversion is a conformal transformation, and is the basic operation of inversive geometry; it is a generalization of inversion in a circle.

Definition

Inversion in a sphere is most easily described using polar coordinates. Choose a system of affine coordinates so that the centre of the sphere is at the origin and the radius of the sphere is 1. Then every point can be written in the form rv, where r is the distance from the point to the origin and v is a unit vector; moreover, for every point apart from the origin this representation is unique. Given such a representation of a point, its image under spherical inversion is defined to be the point r−1v. This defines a homeomorphism from to itself. As a map from Euclidean space to itself, the spherical inversion map is not defined at the origin, but we can extend it to, the one-point compactification of, by specifying that 0 should be sent to infinity and infinity should be sent to 0. Thus, spherical inversion can be thought of as a homeomorphism of.

Properties

Inversion is self-inverse, and fixes the points lying on the sphere. The inverse of a line is a circle through the centre of the reference sphere, and vice versa. The inverse of a plane is a sphere through the centre of the reference sphere, and vice versa. Otherwise the inverse of a circle is a circle; the inverse of a sphere is a sphere.
Inversion in a sphere is a powerful transformation. One simple example is in map projection.
The usual projection of the North or South Pole is an inversion from the Earth to a plane.
If instead of making a pole the centre, we chose a city, then Inversion could produce a map where all the shortest routes for flying from that city would appear as straight lines, which would simplify the flight path, for passengers at least.

Proofs

Let the reference sphere be Σ, with centre O and radius r denoted by. All inverses, in this paper, are in the sphere Σ.
The results in this article are dependent on three simple ideas:

Definition

draw tangents, in the plane, from P to Σ, meeting it at S, T.

Inversion of a pair of points

Inverse of a line



Note 4: Generally, the inverse of a line is a circle through the centre of reference.

Inverse of a plane

Note 5: Generally, the inverse of a plane is a sphere through the centre of reference.

Inverse of a Sphere







Note 6: Generally the inverse of a sphere is a sphere

Inverse of a circle

Note 7: Generally the inverse of a circle is a circle.

Results of inversion in a sphere

  1. A line through the centre of inversion is self-inverse.
  2. Generally, the inverse of a line is a circle through the centre of inversion.
  3. The inverse of a circle through the centre of inversion is a line.
  4. Generally the inverse of a circle is a circle.
  5. A plane through the centre of inversion is self-inverse.
  6. Generally, the inverse of a plane is a sphere through the centre of inversion.
  7. The inverse of a sphere through the centre of inversion is a plane.
  8. Generally the inverse of a sphere is a sphere.