Given coordinates of two lines which intersect when one line is extended, how to find their intersection coordinates?

Question

I have two lines, each having starting and end coordinates $(x_1,y_1),(x_2,y_2)$ and $(u_1,v_1),(u_2,v_2)$, with

$(u_1,v_1) = (0,0)$

They are not necessarily intersecting yet, but definitely intersect when one line is extended.

Here is a sample scene

I would like to know how to find their intersection point $(s,t)$ only given their current coordinates.

Answer 1

Rewrite the line segments as line equations!

you then have:

\begin{cases} y = \frac{y_2-y_1}{x_2-x_1}(x_1+x) \\ v = \frac{v_2-v_1}{u_2-u_1}(u_1+u) \end{cases}

Solving this system of equations gives the cordinate of intersection

Answer 2

Working in homogeneous coordinates, let $p_1=(x_1,y_1,1)$, $p_2=(x_2,y_2,1)$, $q_1=(u_1,v_1,1)$ and $q_2=(u_2,v_2,1)$. Then computing the intersection point of the segments’ extensions is a matter of a few cross products: $(p_1\times p_2)\times(q_1\times q_2)$. Dehomogenize by dividing through by the last coordinate. If it’s equal to $0$, then the lines are parallel.