next to The spectral curve for for a Discrete Conformal Map

Discrete Conformal Maps as discrete integrable systems.

A discrete conformal map is a map tex2html_wrap_inline72 with the property that the cross-ratio of four adjacent points is constant. Specifically, for four points a,b,c,d on tex2html_wrap_inline76 define their cross-ratio to be


then the map is discrete conformal when


for some constant tex2html_wrap_inline78 for all k,m. More rigid definitions take q=-1. The motivation for this definition is that if tex2html_wrap_inline84 is smooth then it is (weakly) conformal precisely when


i.e. when tex2html_wrap_inline86 .

It is a remarkable fact that when the discrete map z has one period (i.e. tex2html_wrap_inline90 for all k,m and some fixed period n) then it can be treated as an algebraically integrable dynamical system and may be written down in terms of the Riemann theta functions of an associated Riemann surface, called its spectral curve. The pictures below are parts of discrete conformal maps with cross-ratio -1. For these pictures the spectral curve is a rational curve with two nodes. These are easy to compute with since the tex2html_wrap_inline98 -function is a quadratic function of two exponentials.

To see how this Riemann surface arises we consider the map as a discrete conformal flow of a discrete curve as follows.

Some pictures:
next to The spectral curve for for a Discrete Conformal Map

Ian McIntosh
Wed May 13 14:19:03 EDT 1998