Equivalence of directions (a programming question)

[onodera]

A direction is a sequence of turns separated by unit distances. A turn is expressed in multiples of pi/10, so a u-turn is +5 or -5. Positive values mean CCW turns.
We start at origin, heading towards (1,0)
If I use relative turns (think "turn left", not "go north"), how can I determine if two directions are equivalent (i.e. they lead to the same point)?
For example, [+1, -2] and [-1, +2], lead to the same point (phi, 0)
Or [+1, +2, +3] equals [+3] or [+5, -2, -3]
I know that if I use unit distances I can simply trig my way down to the destination, round the values down to a single decimal and compare them directly, but what if I don't want to or can't use trigonometric functions at all?
Will switching to absolute (i.e., always measure the turns from A+(1,0)) turns help?

P.S. Extra geek cred to the one who can guess what task I need this for.

[loinburger]

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.

[onodera]

Well, yes, a direction is simply a list of polar coordinates, except the distance is always 1.

[KrazyHorse]

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.

[KrazyHorse]

The iterative step will involve a sqrt, an inverse sine, a sine and a cosine. All other opperations are arithmetic (sums, products). The algo is linear in the number of turns.

[onodera]

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!?

[VetLegion]

P.S. Extra geek cred to the one who can guess what task I need this for.

You're building a new bar pickup line?

[onodera]

You're building a new bar pickup line?

I'm happily married, thank you..

[Fve Crathva]

You could use a relative control model and translate it to an absolute model behind the scenes by tracking the pose (x, y, θ). Then you only have to store a lookup table implementing the motion model as translations in x and y (pre-computed using trigonometry). If at step t the pose is (x_t, y_t, θ_t) and the desired control primitive is dθ, then the resulting pose at time t+1 is (x_t + lt[θ_t + dθ][x], y_t + lt[θ_t + dθ][y], θ_t + dθ), assuming the rotation precedes a unit translation and that this is a noise-free simulation. In this example lt[θ_t + dθ][x] corresponds to the lookup table entry for the x component of the translation vector resulting from the motion primitive θ_t + dθ.

SP