Very good idea. I used to draw such curves with basic too. So here's a perl one, just to disprove blazar's statement.

#!perl

use warnings; use strict;

sub pie() {
    2 * atan2(1, 0);
}
sub asin {
    atan2($_[0], sqrt(1 - $_[0]**2));
}

sub liss {
    my($p_m, $p_n, $p_d) = @_;
    my($ht, $wd) = (24, 60);
    for my $l (0 .. $ht - 1) {
        my $ou = " " x ($wd + 1);
        my($y0, $y1) = (2 * $l / $ht - 1, 2 * ($l + 1) / $ht - 1);
        my($su0, $su1) = (asin($y0), asin($y1));
        for my $br (0 .. $p_m - 1) {
            for my $sgn (0, 1) {
                my($u0, $u1) = 
                    ($br * 2 * pie + $sgn * pie + $su0 * ($sgn?-1:1), 
+$br * 2 * pie + $sgn * pie + $su1 * ($sgn?-1:1));
                my($v0, $v1) = ($u0 / $p_m * $p_n + $p_d, $u1 / $p_m *
+ $p_n + $p_d);
                my($x0, $x1) = (sin($v0), sin($v1));
                $x1 < $x0 and ($x0, $x1) = ($x1, $x0);
                int(($v0 - pie / 2) / 2 / pie) == int(($v1 - pie / 2) 
+/ 2 / pie) or
                    $x1 = 1;
                int(($v0 + pie / 2) / 2 / pie) == int(($v1 + pie / 2) 
+/ 2 / pie) or
                    $x1 = -1;
                my($p0, $p1) = (($x0 + 1) / 2 * $wd, ($x1 + 1) / 2 * $
+wd);
                my ($ps, $pd) = 
                    int($p0) < int($p1) ?
                        (int($p0), int($p1) - int($p0)) :
                        (int(($p0 + $p1 + 1) / 2), 1);
                substr($ou, $p0, $pd) =~ y/ /*/;
            }
        }
        print $ou, "\n";
    }
}

# -log(r)/p = x

for (0 .. 19) {
    print "\n" x 4;
    my $r = sub { int(1 - log(1-rand(0.999)) / $_[0]) };
    my($m, $n) = (&$r(0.7), &$r(0.9));
    liss $m, $n, rand(2*pie);
    sleep 3;
}

__END__
[download]

Update: I just want to mention that a classmate of mine has told me some years ago that you can render conic sections fast this way. This program uses basically the same algorithm except that it uses trigonometric functions instead of just square root. (Update: Yeah, except that this one is also cheating: where it says (p0+p1)/2, it should really use the p for (x0+x1)/2 really.)

Update: posted changed version at Lissajous curves

Nice effect. Reminds me of lissajous from my Basic days...

I didn't know that particular program, in fact I've managed to (almost) always stay away from Basic but of course I'm familiar with Lissajous figures, which I also considered, although those would have required either ANSI sequences (and then are nice to be seen while being drawn - I wanted something to paste into email instead) or mangling with a buffer, whereas my program is both much simpler and gives me what I wanted, that is a potentially very long output. Lissajous figures by contrast are always constrained in a rectangle (if you don't consider rescaling, a square).