Suppose I have a line segment $L$ in 3D: $$x=a_1(1-t)+b_1t$$ $$y=a_2(1-t)+b_2t$$ $$z=(a_1^2+a_2^2-k_1^2)(1-t)+(b_1^2+b_2^2-k_2^2)t$$
Because $L$ is line segment then $0\leq t\leq 1$.
And defining paraboloid $P$ in 3D: $$P:z=2x^2+2y^2-1$$
Where $a_1,a_2,b_1,b_2,k_1,k_2$ are all variables and $k_1$ and $k_2$ are positive numbers.
I want to put some constraints on these variables such that $L$ and $P$ intersect or do not intersect.
I know by substituting $L$ into $P$ and then solving for $t$ where $0\leq t\leq 1$ is the solution of the intersection. But still i could not figure out about the constraints i am looking for.
Thanks for any suggestions.
If you only want to know if they intersect or not, you don't need to figure out where do they intersect...
The endpoints of your segment are at $(a_1, a_2, a_1^2+a_2^2-k_1^2)$ and $(b_1, b_2, b_1^2+b_2^2-k_2^2)$. If both endpoints are above, or both below the paraboloid, then the segment does not intersect the segment. For instance, the first endpoint is above the paraboloid if $a_1^2+a_2^2-k_1^2 > 2a_1^2+2a_2^2-1$. Working it out, both endpoints will be above the paraboloid if
$$k_1^2<1-a_1^2-a_2^2,$$ $$k_2^2<1-b_1^2-b_2^2,$$
You will have an intersection if only one of the above inequalities holds.