Clifford parallel

Clifford parallel

A Clifford parallel is a line which lies at a constant distance from some "base" line but, unlike an ordinary parallel line, does not lie in the same plane. Such lines do not exist in ordinary Euclidean space, but only in certain others such as elliptic space.

Clifford parallels were first described in 1873 by the English mathematician William Kingdon Clifford.

The lines on 1 in elliptic space are described by versors with a fixed axis r:

\lbrace e^{ar} :\ 0 \le a < \pi \rbrace

For an arbitrary point u in elliptic space, the right Clifford parallel to this line is

\lbrace u e^{ar}:\ 0 \le a < \pi \rbrace, and the left Clifford parallel is
\lbrace e^{ar}u:\ 0 \le a < \pi \rbrace.

Clifford parallels are also called paratactic lines.

Clifford surfaces

Rotating a Clifford parallel around the base line creates a Clifford surface.

Given two square roots of minus one in the quaternions, written r and s, the Clifford surface through them is given by

\lbrace e^{ar}e^{br} :\ 0 \le a,b < \pi \rbrace.

A Clifford surface is a ruled surface: every point is on two lines, each contained in the surface.

See also

References

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