Course MT3818 Topics in Geometry

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(The projective groups)
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More projective groups

The projective group PGL(2, C).

This is the group that acts on the complex projective line P(C2) = CP1.
As in the last example the elements of this group can be regarded as the set of rational analytic functions:
z goesto (az + b)/(cz + d) with a, b, c, d belongs C with ad - bc noteq0,
from C union {infinity} to itself.

Remarks

  1. These transformations are sometimes called Möbius functions (after the German mathematician August Möbius (1790 to 1868) best known for his results in Number Theory and for the so-called Möbius band).

  2. Complex variable theory shows that these mappings are conformal (angle preserving) transformations at most points.

  3. As in the last example, we may define the cross-ratio of four points (a complex number this time) and this is preserved by these transformations.

The projective group PGL(3, R).

This is the group that acts on the real projective plane P(R3) = RP2.
One may show that there is a unique element of this group taking any four points in RP2 (no three on a line) into any other four points in RP2.
As in the projective line case, one may take standard reference points to be: [1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1]. These are the points at infinity on the x and y axes, the origin and the point (1, 1). We may manipulate these as in the earlier case.
We may define the cross-ratio of four points on a line and these transformations preserve that.


Previous page
(The projective groups)
Contents Next page
(Some theorems in plane projective geometry)

JOC March 2003