I have a cube of unit length. Each face of the cube is divided into 10 x 10 equal segments. Consider an object of size equal to that of a segment moving through the surface of the cube ( or any 3D object ).
I need to mathematically compute an equation through which I can get the next segment in the cube given the direction of the object.
EDIT:
I agree with the comments that this problem can be approached progammatically. I was looking for a way to do it in a generic way ( maybe like a parametric surface ) where you specify a domain (u, v) and a range of values (x,y,z) for a 3D Object.
An example would be like this - http://chimera.labs.oreilly.com/books/1234000001814/ch03.html#ch03_id36001767 . Can we define one such mathematical function for a cube?
To put in simple words, can a function be defined for the domain f(u, v) which will give a range of (x, y, z) values for a cube?
The game of Snake in $3$D is really just a game of Snake in $2$D with slightly different boundary conditions on the box. One can unfold a cube like this. On the inner squares, the place that the snake will go next is obvious, it's the same as it would be for the $2$D case. However, at the edge squares of the cube, you'll just have to look at the sides and see which ones correspond to which; I've drawn some examples there for reference. This is for the case of a $2\times 2$ grid, but it should be easy to generalize to $10\times 10$.