Date: Jan 7, 2000 1:01 PM
Author: Gunter Weiss
Subject: Re: [HM] Name needed for a point: Hirst?


Dear all,

Thank you for the great lot of interesting questions and informative replies!
Every morning I am really looking forward to read my new 'Newspaper', namely
your discussions on HM. So even I am neither strong in English nor in History
of Mathematics, I dare to send a reply to the following question:

> [HM] Name needed for a point: Hirst?

> From: Clark Kimberling <ck6@evansville.edu>
>
> 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"?
>
> Thanks!
> Clark Kimberling
>



At first some references:

T.A. HIRST: Rep. Brit. Ass. Adv. Sciences 34, London 1865 (Bath 1864), p.3
J.S. VANECEK: Paris Compte Rendues 94(1882), p.1042

in Encyklop"adie der mathematischen Wissenschaften III.2.2.B:
III C 11, L.BERZOLARI: Algebraische Transformatione und Korrespondenzen.

concerning CREMONA-transformations (where Hirst-Inversiones belong to)
see e.g.

H.P. HUDSON: Cremona Transformations in plane and Space. Cambridge U.P.
1927

From what I read and learned I got the impression, that Hirst simply
generalised the classical 'inversion': Given a center C and a (regular
or singular) polarity \pi in a projective space, then the (C,\pi)-inverse X'
to a point X (not= C) is defined by
(HI-1) C, X, X' are collinear,
(HI-2) X, X' are \pi-conjugate.
One will call X' the Hirst inverse to X with respect to the inversion
center C and the polarity \pi.
So two points X, X' are involutorically related, because of \pi. There is
a certain singularity set, containing at least C and depending on the
arithmetization field (finite, real, complex ....) to the projective space.
Thus your final question should possibly best be answered by:
Replace the word 'conjugate' by 'inverse' and say "X, X' are
(Hirst-)inverse", when it is clear that You deal with a certain C and
a certain \pi, but write "X, X' are (C,\pi)-inverse", when You want
to be exact.

Finally some explanations, (please omit them, when You think they are
trivial): For \pi ... polarsystem to a euclidean circle, C its center,
this mapping is the usual inversion.
For \pi singular, e.g. inducing an involutoric projectivity in a pencil
of lines with a pair of fixed lines a, u (the latter being the ideal
line of the projectively extended euclidean plane), and let C be an ideal
point, then the Hirst inversion is the (skew or orthogonal) reflection at
a. Hirst inversion is a very useful concept, when dealing with (euclidean
and non-euclidean) circle geometries and (hyper)sphere geometries,
(M"obius-, Minkowski-, Lie-geometries): M"obius transformations are
products of similarities and Hirst inversions. The M"obius extension of
the plane or space uses exactly the singularity set of a Hirst inversion
as the ideal point set! Example: Euclidean planar Lie-geometry 'is'
pseudo-euclidean M"obius-geometry in 3-space, whereby spheres are the
planes and one- and two-sheet hyperboloids with translatorically equal
asymptotic cones. Such a cone represents the singularity set of the
corresponding inversion and has to be added to the 3-space as the set
of ideal points. Reference to circlegeometries: W.BENZ: Geometrie der
Algebren. Springer Berlin 1983.

There are of course deeper generalizations of the classical inversion
then Hirst's. See e.g. Berzolari and Hudson as well as Rudolf STURM (Die
Lehre von den geometrischen Verwandtschaften, Vol.4, Teubner Leipzig 1908).

Final remark:
In (high) school classes, besides some elementary geometry one should
also present the principle of generalizing concepts of elementary geometry.
This would give pupils a feeling how mathematics works, that it still
grows and never will stop growing. And it would end the isolation (and
extinction) of elementary geometry within our curricula. Hirst-inversion
is a good example to 'explain' mathematics in this sense.

Best regards
Gunter Weiss (Dresden)
weiss@math.tu-dresden.de