Oriented circles and relativistic geometry II | Geometric Linear Algebra 35 | NJ Wildberger

We continue our discussion of oriented, or signed, or directed circles in the plane, which are also called cycles, and the intimate connection with relativistic geometry in three dimensions. This correspondence makes it easier for us to apply linear algebraic ideas to the geometry of circles, but it also provides an interesting geometrical interpretation of the framework of Einstein's Special Relativity. This theory was developed in the 19th century and is called cyclography, but it is not as well-known these days as it ought to be, and benefits from a purely algebraic development.

In this lecture we talk about the homothetic centre of two circles (sometimes also called a center of similitude), how to find this, and then describe a lovely theorem of G. Monge on the homothetic centres of three circles. Tangency of signed circles has a natural interpretation of the relativistic quadrance being zero, and we show how the locus of a circle tangent to two given ones has a relativistic meaning as an intersection of two cones, yielding a hyperbola of centres in the original plane. The famous problem of Apollonius makes its appearance.

We describe how spheres in the relativistic space, which appear as hyperboloids of one and two sheets, can be viewed from the point of view of circle geometry, involving interesting pencils of circles.

Finally we give a perhaps new theorem that describes the geometrical meaning of the relativistic quadrance between two signed circles.

This lecture has a lot of material in it, so go slowly!

Video Chapters:
00:00 Introduction
6:22 Lines of circles and homothetic centers
14:05 Monge's theorem
18:21 (Relativistic) quadrance between signed circles
22:15 Quadrance of zero
27:07 Mutually tangent circles
31:42 Problem of Apollonius
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