
		Lim Version 2.2 

	Limit Sets of Kleinian Groups

Type "cd src; make" to install.
Then type "cd ../examples; ./Example.run" to test.
Other *.run files contain other examples.
(If you have trouble on a Mac, make sure you have the 
Xcode package installed.)

The program lim draws the orbits of circles
under the action of a group of Mobius transformations.

Usage:
lim [options]
    -a [agree] circles are regarded the same if center
       and radius differ by less than epsilon/agree.
    -b put box around image
    -c [c.x c.y r] clipping circle 
    -d [depth] - generations
    -e [epsilon] - radius of circles to ignore
    -h [huge-radius] huge cutoff radius
    -l [linewidth] - in 1/72nd of inch
    -m [maximum] number of circles to draw
    -q quiet
    -s plot on sphere
    -t [gray] make sphere transparent.  Circles
       on the back colored gray (0-1.0).
    -w [xmin ymin xmax ymax] window 
    < data.file
    > postscript.file

Entries in the data.file are of the following forms:

c center-x center-y radius
	for circle in the limit set
r center-x center-y radius
	for reflection in the group
m ax ay bx by cx cy dx dy
	for mobius transformation in the group
t ax ay bx by cx cy dx dy
	for mobius transformation to be applied
	to the coordinate system
u cx cy r
	for reflection in a circle to be applied
	to the coordinate system

LINES as CIRCLES
Any circle in data.file can also be a line (that is, a circle
through infinity).  A line L is specified by setting
	(center-x, center-y)  =  
	a point on L, usually the point closest to the origin; and

	radius = a number <= 0, usually in [-1,0], such
	that the angle -PI*radius is the slope of the line.

Examples: c 0 0 0 is the real axis; c 0 0 -0.5 is the imaginary
axis; c 0 0 -0.25 has slope 45 degrees and passes through (1,1).
See examples/lines.run for an example.

SUBDIRECTORIES:
examples/
	The .run files in this directory are shell 
	scripts giving many examples of running lim.

HISTORY:
Version 1.0 Written at MSRI, March 1995, to prepare slides
	for a conference at Harvard.

Version 1.4 June 1998:  Updated compare_circles() in lim.c
	to pass pointers to structures, for compatibility with 
	vanilla qsort().

Version 2.0 July 1999:  Allows lines for circles.
        2.1 June 2001:  Better windowing for lines.
        2.2 Aug  2001:  Corrected great circles.

Copyright (c) 2001

Curtis T McMullen
ctm@math.harvard.edu
