So, thanks to the properties of implicit surfaces, you can easily implement gradient descent algorithms that move a point to the surface of an implicit surface.
On particular the formula:
$\frac{-f(x,y,z)\nabla f(x,y,z)}{\nabla f(x,y,z)\cdot\nabla f(x,y,z)}$
Creates a gradient that points directly to the surface.
The information you have about the surface is it's implicit funciton and anything you can derive from it (i.e gradient, heat function....)
Say I have 2 points on the implicit surface $P_1$ and $P_2$
I want to move from $P_1$ to $P_2$ following 2 conditions. I must remain "on the surface" (so within some epsilon distance of the actual implicit surface)
And I want to do so in the shortest way possible.
Precondition 1 is much more important than precondition 2, in other words the function needs to push the point towards the surface more strongly than it moves the point along the shortest distance.
Anyone got any ideas?
EDIT:
Added information