Is the homogeneous point $(0,4,0,0)$ a point at infinity?

Question

Can you please explain the geometric euclidean interpretation of homogeneous point $x= (0, 4, 0, 0) \in P$. Does it means a point at infinity?

Answer 1

To check whether a point in homogeneous coordinates is "at infinity", look at the last (*) number. If that last number is zero, it's a point at infinity; if it's non-zero, it's just a normal point.

In this case, the last coordinate is zero, so this is one of the points at infinity. Specifically, it's the limit of $\alpha \langle 0, 4, 0 \rangle$ as $\alpha$ goes to infinity.

Another way to think of it: you can convert from homogeneous to normal Cartesian coordinates by dividing everything by the last number. In this case, you'd be dividing by zero. So this is a point that can't be represented in Cartesian coordinates.

(*) As amd points out in the comments, some people put the "spare" coordinate first instead of last. I'm assuming last here, but your instructor or textbook should specify somewhere.