Projective Co-ordinate Geometry

I am learning projective geometry in my computer vision course. So, we represent a co-ordinate point in an image as a homogeneous co-ordinate as $(x,y,1)$.

My professor says that if we are given two such points, $P(x_1,y_1,1)$ and $Q(x_2,y_2,1)$ then $P \times Q$ (vector product) represent the equation of this line.

Any explanation to this?

Solution 1:

In affine geometry you can express a line as

$$\left\{(x,y)\;\middle|\;ax+by+c=0\right\}$$

In homogeneous coordinates this becomes

$$\left\{(x,y,z)\;\middle|\;ax+by+cz=0\right\}$$

Now the coordinates of the line are this vector $(a,b,c)$. And the equation on the right side of the above set notation is a dot product between line $(a,b,c)$ and point $(x,y,z)$. So a point lies on a line if the dot product is zero. The dot product is zero if the vectors are perpendicular in $\mathbb R^3$.

The cross product in $\mathbb R^3$ is perpendicular to both its arguments. Therefore the resulting vector will represent a line which is incident to both the points from which you computed it.