[Angelinaaiiiiiiiii]

Offhand I'd say that you would probably benefit from using polar coordinates instead of cartesian coordinates, but I haven't used polar coordinates with enough regularity to come up with an algorithm off the top of my head.

[maks_holi]

There's no way to do this without trig

The best way is via polar coords.

consider a direction as an iterative series of actions (turn plus step) on the following triplex: d,a,h

h is heading, d is distance from origin, a is angle of current position from x axis.

I will write out form of iteration when not on phone. Two directions are equivalent if their final triplexes match in d and a (h doesn't matter). Iteration involves law of sines and law of cosines.

[paulaglober]

So the non-trig way to do it with pi/4 turns is a special case, the because the results of trig are rational at every step!?

[RaicickKida]

You're building a new bar pickup line?

I'm happily married, thank you..