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?

Yes, it is a point at infinity. Namely, the point at infinity in the direction of the vector $(0,4,0)$. — GReyes, Mar 02 '19 at 19:34

Draconis · Accepted Answer · 2019-03-03T04:08:48.903

2

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.

edited Mar 03 '19 at 04:08

answered Mar 02 '19 at 19:37

Draconis

1,443

Thank you both. I am clear :) – Adnan Haider Noor Mar 02 '19 at 19:46
Some sources put the “extra” coordinate first instead of last. – amd Mar 03 '19 at 04:03
@amd Fair; added a note about that – Draconis Mar 03 '19 at 04:08

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

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