Space curves: finding collisions and intersections

Ok, no specific example to pose, but I would just like to discuss the logic behind each.

So to find where each space curve collides, we want to find where the positions of two curves will be the exact same for a given time t. We do this by equating the corresponding parametric equations of one curve to the parametric equations of the second curve, and solving for the system of equations.

What is the logic behind finding where the curves intersect? The methods seem to involve substitution of the parametric equations of one space curve into the corresponding parametric equations of the other, and solving. I've also seen solutions in which corresponding parametric equations are equated and solved, except via using a different variable (say, 's' instead of 't') for one of the curves. So we are just looking to see where one curve can act as a function of the other? I'm a bit murky here... could I get an explanation as to the specific logic behind what we want to do?

I apologize if my math terminology sucks. :) Working to improve that.

Thanks.

Re: Space curves: finding collisions and intersections

When curves intersect, they have the same position, so the same (x, y, z) co-ordinates...