the ellipse and its evolute
and the distance problem:
updating Apollonius

by Frederick Hartmann and Robert Jantzen [2004, 2008]

• introduction and article [This is the link to the Top Level Web Page with animation]
   
• 2008 reformulation for MAA website Convergence:
  now available 2008-03:
  http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1709
  
  pdf; tex source zip (with original EPS figure files);
  small gif figure file zip; large jpg figure file zip
   
• 2004 PDF article
• 2004 maple 9.5 worksheet
  [in progress 2008: update to Maple 11 Standard, LinearAlgebra, VectorCalculus packages]
   
• html export of 2004 maple worksheet

For another example of the amazing geometrical ability of Apollonius see:

• Apollonius' Problem: A Study of Solutions and Their Relations
  by David Gisch and Jason M. Ribando
  American Journal of Undergraduate Research
  Vol 3 No 1 June 2003, 15-23.
• Kissing circles: A French romance in MetaPost
  by Denis Roegel     [Advanced -- MetaPost construction of the "Apollonian gasket" of tangent circles]
  TUGboat Vol 26 No 1, 2005, 10-17.

For a complete study of the evolutes of the graphs of polynomials, see

• Normal Lines and the Evolute Curve
  by David Sanchez and Kirby C. Smith [Texas A&M University, College Station, TX]
  The College Mathematics Journal (MAA), Vol 31 No 5 November 2000, 397-403

A similar study of the evolutes of the ellipsoid and its geometry was begun but left unfinished:

• ../ellipsoid_normals_2008.mw

A few miscellaneous links:

• http://en.wikipedia.org/wiki/Apollonius_of_Tyana
• http://www.gap-system.org/~history/Biographies/Apollonius.html