what is the distance formula between two homogeneous coordinates

Question

As I know that the distance formula between two points in euclidean space is $\sqrt{(x-y)^2}$ will the same distance formula holds good for homogeneous coordinates ?

Answer 1

First some theoretical background. The projective plane all by itself does not have a concept of distance. In order to measure distances, you have to provide more structure. One typical approach is to consider the set of all points with a zero in their last coordinate to be the set of points at infinity. That is enough to compare lengths, i.e. to decide the ratio between one length and another length, as long as these two lengths are on the same line or on parallel lines. In order to compare lengths in different directions, you need to be able to characterize circles. One way to achieve this in the plane is by distinguishing the two ideal circle points $[1,\pm i,0]$. These points with complex coordinates on the line at infinity lie on every circle, and with circles you can change direction without changing length. The next thing people often do is pick one pair of points and define their length to be some constant, often one. Now you can measure arbitrary distances in arbitrary directions.

But how do you do that on the coordinate level? Well, one approach is to dehomogenize points and then apply the usual Euclidean formula. For the typical choice of coordinates, you divide the vector by the last coordinate, so that this last coordinate becomes one. Then you can take the difference of the two vectors and obtain a vector with a zero in the last coordinate. You take the usual norm of the vector, using the same formula as you did before. Since the last element of the difference is zero, it doesn't matter whether you drop the last coordinate (as some explanations have you do) or not. The distance measured in this fashion may be already correct, or it may need to be scaled by a constant depending on how you defined the reference length I mentioned earlier.

There are other ways to measure distances, and while some of them have nicer properties in some contexts, and in particular are easier to generalize to other geometries, the description above is probably the easiest for practical approaches.

Answer 2

Hint:

For two points $A=(x_a,y_a,z_a)$ and $B=(x_B,y_B,z_B)$ in an Euclidead space, write correctly the formula of the distance: $$ d(AB)=\sqrt{(x_A-x_B)^2+(y_A-y_B)^2+(z_A-z_B)^2} $$

and note that the homogenous coordinates of these points can always be written ( since they are not points at infinity) as $$ A=(x_a,y_a,z_a,1) \qquad B=(x_B,y_B,z_B,1) $$ so, also if you extend the formula to the four affine coordinates, you find the same result.