Is the Intersection Point of Two Lines a Cross Product?

[orochimaru]

Hi,
i need enlightenment on this question of mine.

Suppose i have two lines, L and M.
L and M intersect at point X.
Then is X a cross product of L and M?

I read this from a pdf on Multiple View Geometry.
Here the link http://www.syseng.anu.edu.au/~hartley/Papers/CVPR99-tutorial/tut_4up.pdf
the eqn is on pg 8 top right corner slide.

Thanks in advance!

 

[TD]

Yes, that is correct.

For example, take the lines $x + y + 1 = 0$ and $-x + 2y = 0$ with vector representations (according to your pdf file) $\left( {1,1,1} \right)$ and $\left( {-1,2,0} \right)$.

The cross product is $\left( {1,1,1} \right) \times \left( { - 1,2,0} \right) = \left( {2,1, - 3} \right)$. Dividing by -3 to get the z-component to equal 1 gives the cartesian intersection point of $\left( { - \frac{2}{3}, - \frac{1}{3}} \right)$

 

[tacman]

Hi there,

The concept of a cross product is typically used in vector algebra to determine a vector that is perpendicular to two given vectors. It is not directly applicable to lines in a 2-dimensional space.

In the context of your question, it is not accurate to say that point X is a cross product of lines L and M. Instead, point X is the point of intersection between the two lines. This means that the two lines share the same coordinates at that particular point.

The equation you mentioned on page 8 of the PDF is a formula for finding the intersection point between two lines in a 2-dimensional space. It uses the concept of homogeneous coordinates, which is a way of representing points and lines in projective geometry.

I hope this helps clarify your question. If you have any further questions, please let me know.