# Projective geometry - intersections

• Apr 1st 2013, 02:49 PM
Impo
Projective geometry - intersections
Hi,

Sorry if my post would be in the wrong section, but I didn't know where to put it as there's no topic called 'projective geometry'.

I'm working on the following problem. Suppose we're working in the projective space $\displaystyle \mathbb{P}^4$ where we have given the Plucker coordinates of a plane $\displaystyle \alpha: (a_0:a_1:a_2:a_3)$ and the Plucker coordinates of a line $\displaystyle l = (d:m)=(d_1:d_2:d_3:m_1:m_2:m_3)$. The question is, what are the coordinates of the intersection point?

I can represent the plane $\displaystyle \alpha$ analytically as $\displaystyle a_0x_0+a_1x_1+a_2x_2+a_3x_3=0$ which can be useful to work with, but I also want an analytically expression of the line, which I cannot find. I did some research and I found on wikipedia that the coordinates of the intersection point are given by: $\displaystyle (x_0:x)=(a\cdot d: a \times m - a_0d)$ where $\displaystyle x=(x_1:x_2:x_3)$ and $\displaystyle a=(a_1:a_2:a_3)$. Honestly, I don't know how they came to that answer as I have no analytical expression for the line $\displaystyle l$.

How do I have to approach this problem?

Many thanks,
Impo