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Topic: [HM] Name needed for a point: Hirst?
Replies: 4   Last Post: Jan 10, 2000 3:57 AM

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Clark Kimberling

Posts: 20
Registered: 12/3/04
[HM] Name needed for a point: Hirst?
Posted: Jan 6, 2000 9:16 AM
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Suppose P=u:v:w and W=x:y:z are points in homogeneous coordinates
(e.g., barycentric or trilinear) in the plane of a triangle. Define

P*Q = vwxx-yzuu : wuyy-zxvv : uvzz-xyww.

If P is fixed and Q variable, then P* is an involution. Someone
mentioned that this is a HIRST TRANSFORMATION. Unfortunately the
mentioner gave no details and his name is not known to me. As for
Hirst, this is Thomas Archer Hirst (1830-1892), described in Historia
Mathematica 1 (1974) 181-184.

Hirst's involution yields a kind of conjugate. That is, for any
particular point Q, we have P*(P*Q) = Q, in the same vein as isogonal
conjugate, isotomic conjugate, and Ceva P-conjugate.

Can someone cite Hirst's introduction of this transformation? Is
there a better (short, please!) name for the point P*Q than this:
"Hirst P-conjugate of Q"?

Clark Kimberling

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