Newsgroups:     sci.math
Subject:        bifocals?
From:  (Noam Elkies)
Date:           20 Sep 92 15:12:26 EDT
Organization:   Harvard Math Department
Summary:        does a bifocal curve exist?

Say that a point P in the interior of a simple closed curve C in the
plane is a _focal point_ of C if C is starlike w.r.t. P and all the
chords of C passing through P have the same length.

Is it known whether a simple closed curve can have more than one
focal point?

[C is starlike w.r.t. P means every ray from P intersects C in exactly
one point; this is a condition weaker than convexity (though C is
convex iff it is starlike w.r.t. every interior point).  There are
plenty of curves with one focal point.]

--Noam D. Elkies (
  Dept. of Mathematics, Harvard University

From:  (Curtis T. McMullen)
Newsgroups:     sci.math
Subject:        Re: bifocals?
Date:           20 Sep 1992 20:21:39 GMT
Organization:   U.C. Berkeley Math. Department.

Let a *wheel* be a Jordan curve W in the plane.
A chord  for W is a line segment with endpoints on W.
A point p in the plane is an *axle* for W if there
exists an L>0 such that for every w in W,
there is a w' in W such that |w-w'| = L and the segment
[w,w'] contains p in its interior.
Then a less restrictive form of Elkies' question is:

	does there exist a wheel with two axles?

Here are some recollection of my thoughts
on this problem from 1990 or so.
It can be related to dynamical systems as follows.
Let p and q be two (candidate) axles; I think one can show
they both must have the same L, by considering the line
through p and q.  Given a point w, allegedly on the wheel,
one can construct two more points w_1 and w_2 on the wheel by drawing lines
of length L from w through p and w through q.  Repeating this process,
you can make a computer drawing of (lots of) points which must
lie on the wheel W to be consistent with the data (w,p,q,L).
By starting with points w very close to the line through p and q,
you can generate continua which must lie in W.  These continua
tend to be non-locally connected, due to incoherence between
stable and unstable manifolds in this dynamical system
(a well-known phenomena).  I suspect this can be made into a proof
that there is no wheel with two axes.  It should be considerably
simpler with the assumption that the wheel is star-shaped from
p and q.  But perhaps the discussion is "academic", since a solution
of a large portion of the problem appears in:

	author={R. Sch\"afke and H. Volkmer},
	title="Asymptotic analysis of the equichordal problem",
	journal="J. reine angew. Math.",

-Curt McMullen

From:  (Joseph O'Rourke)
Newsgroups:     sci.math
Subject:        Equichordal problem
Date:           9 Jan 91 17:36:34 GMT
Reply-To: (Joseph O'Rourke)
Organization:   Smith College, Northampton MA USA

A year ago I heard that the equichordal problem had at long last
been solved, but I have not seen the paper.  Does anyone have a
	The equichordal problem is to determine whether a compact
convex subset S of E^2 might have more than one equichordal point.
A point p is equichordal in S if every chord through p has the same

Date:           2 Oct 1997 21:55:41 -0400
From:           Marek Rychlik <>
Organization:   Epigone
Subject:        Re: Equichordal points

The best known outstanding problem regarding equichordal points 
has just recently been solved by me (solution published in
Inventiones Mathematicae, 129(1), pp. 141-212, 1997). This problem 
was posed in 1916 by Fujiwara and in 1917 by Blaschke et. al. :

A point P inside a closed convex curve C is called equichordal if    
every chord of C drawn through point P has the same
length. Is there a curve with two equichordal points?

It proves that a curve cannot have two equichordal points. The
assumption of convexity can be significantly weakened, to include all
Jordan curves. For further references my Web site can be consulted:

-Marek Rychlik