Let's suppose that we have the following Bezier surface $$ P(t1,t2)= \sum_{i=0}^{3} { \sum_{j=0}^{3} p_{ij} { \varphi _{i}(t1) \varphi _{j}(t2) } } $$ Is there a way to determine if a specific Bezier curve, let's say: $$ Q(t)= \sum_{i=0}^{3} p_{i2} { \varphi _{1}(t) } $$ belongs to P(t1,t2)?
I know that in order to find if a specific point (x,y) belongs to a Bezier curve, we just need to find a value to that satisfies the equation P(t=to) = (x,y), but in this case is much more difficult.
Is there an easier way than trying to find values for t1 and t2 that satisfy the equation P(t1=t1x, t2 = t2y) = Q(t) ?