Newsgroups: geometry.puzzles
From: haoyuep@aol.com (Dan Hoey)
Date: Wed, 5 Dec 2001 18:12:01 +0000 (UTC)
Subject: Re: Reply to tangent

Rouben Rostamian <rostamian@umbc.edu> wrote:
> "Javier Orman" <perlman@adinet.com.uy> wrote:
> > Does anyone know the geometric (not analytical) definition
> > of tangent to a curve?
[...]

> As I wrote somewhere else in this thread, a non-analytic
> definition will have a very hard time coping with a complicated
> curve [....]

I think you have to essentially duplicate the analytic version
in geometry.  For instance, let  C  be a curve in the plane
containing point  P.  For any positive number  N,  let us define
an  N-rectangle  of  C  at  P  to be a rectangle  QRST with
center  P,  for which  C  intersects triangles  PQR  and  PST
only at  P,  and for which segment  QR  is  N  times as long as
segment  RS.

Then a line  L  is said to be tangent to  C  at  P  when
for every  N  there is an  N-rectangle  QRST  of  C  at  P  for
which  L intersects segments  RS  and  QT.

It is amusing that the proposition that  P  is on  L  is now a
theorem rather than part of the definition.

Dan Hoey
haoyuep@aol.com