Why does the intersection of two parametric curves have different t results for each curve? I'm slightly confused after reading the problem posed here: Points of intersection of two parametric curves
Why is it that the "t" result at the intersection point of the two curves is not the same? I.e. the solution involves the answer "t1" for the first curve and "t2" for the second curve, but if the two curves are being made equal, why is t1 != t2?
Also, is there any way we can "manufacture" two different curves (say two parametric straight lines) that, at a specific t value for BOTH curves, yield the SAME point in space? How would you go about doing that? (given, for example, 3 points in space)

As for your second question, given two parametric curves $$C_1 :\ (a(t),b(t)) \\ C_2 :\ (c(t), d(t)) $$
they can be made to intersect at $t=t_0$ by choosing functions $a,b,c,d$ such that $$a(t_0) = c(t_0) \\ b(t_0)= d(t_0) $$
