Talk:Inversion in a sphere

It has been stated that "These results are well known", but they need to be proved. if they are to be cited.

The proofs of inversion in a sphere, of lines, and circles where the centre of reference is in the same plane, are almost identical to the proofs of their inversion in a circle, but the inversion of planes, spheres, and a general circle are three dimensional. In particular, the inversion of a general circle (one where the plane of the circle does not contain the centre of reference) needs further investigation.

The cone of projection of a general circle, with the centre of reference at the apex, is oblate, i.e. is right elliptical.

The inverse of the circle lies on this cone, and also on the inverse of the plane of the circle, which is a sphere containing the centre of reference.

In general the intersection of a sphere and a right- elliptical cone is a biquadratic curve (something like the connection on a tennis ball).

It is fortunate that in this instance, it can be shown to be a circle! 10:52, 4 April 2012 (UTC)

The basic geometry required has been listed, and explanations have been added in the text.

08:21, 11 April 2012 (UTC)

The illustration of the inversion of a cylinder,is beautiful.

With the radius of the cylinder and the sphere the same, it is even neater!

Could we see another illustration, where the cylinder passes through the centre of inversion?

18:20, 20 June 2012 (UTC)~ — Preceding unsigned comment added by Bparslow (talk • contribs)

Still waiting for the new illustration. Some hints (supposing that the cylinder is vertical): the inverse of the vertical 'line' (L1, say) of the cylinder that passes through the sphere centre is self inverse; the inverse of every other 'line' is a circle through the centre, passing though the points where that line cuts the sphere; the horizontal circular cut of the cylinder that passes through the centre is a horizontal line, tangential to the sphere and the cylinder; any other circular cut is a circle through the inverse of the point, where that circle cuts L1, and, if the circle cuts the sphere, through those two points too; all circles outside the sphere invert into circles inside the sphere. An interesting shape 13:58, 25 July 2012 (UTC)~ — Preceding unsigned comment added by Mastrud (talk • contribs)

This article should be deleted or redirected to inversive geometry

Since the contents of this article is contained in inversive geometry and Dupin cyclide the article should be deleted or redirected to inversive geometry.--Ag2gaeh (talk) 07:23, 10 September 2016 (UTC)