-1

I read from this answer and understand that the homogenous coordinates is of the form $[x, y, z]$ which represents by square bracket. Also read from this answer the homogenous coordinates is of the form $(x, y, z)$ which represents by first bracket.

My question is what is the right representations of homogenous coordinates or how both representations are right at the same time?

Gabriel
  • 4,513
S. M.
  • 1
  • 1
    Wikipedia's "Barycentric coordinate system" entry and Kimberling's Encyclopedia of Triangle Centers use the notation $x:y:z$. (I also happen to be using this notation in a note.) "Correct" mathematical notation is whatever the author decides best serves the reader. I'm personally trying to avoid "overloading" delimiters, and also seeking to reduce some visual clutter (my coords tend to be large rational expressions; giant surrounding parentheses can be off-putting). – Blue Dec 19 '21 at 23:54

1 Answers1

2

Different people may write projective coordinates as $(x,y,z)$, $(x:y:z)$, $[x,y,z]$, or even $[x:y:z]$. All have the same meaning, as long as the context makes it clear that those are indeed projective coordinates (the first option may be confused by affine coordinates, but the space where those coordinates live is usually clear from the context).

I understand that you wished another answer, but it just doesn't exist. There's no "best" notation or "universally accepted" notation. As long as everyone understands what is being said, which is very often the case, all is fine.

Gabriel
  • 4,513
  • 1
    Also, everything Lee Mosher said in your other post is perfect. – Gabriel Dec 19 '21 at 23:33
  • your statement "the first option may be confused by affine coordinates"-----The affine coordinates could be (x, y) , so how could we confuse, please give one example by which I can understand? – S. M. Dec 20 '21 at 05:46
  • I am confusing this thing which is: suppose $(x,y, z)$ is the representations of homogenous coordinates then it belongs to $\mathbb R^3$ but $[x,y, z]$ is the representations of homogenous coordinates then it belongs to $P^2.$ So my question is homogenous coordinates belongs to which? $\mathbb R^3$ or $P^2$? It's very confusing. Please help me. – S. M. Dec 20 '21 at 06:40
  • Let's forget about projective spaces for a moment and let's think about points in $\mathbb{R}^2$. I can denote such point by $(x,y)$. That's the usual notion. But I can also begin a book by writing something like "let $\to x,y \leftarrow$ denote a point of coordinates $x$ and $y$ in $\mathbb{R}^2$". Of course that's a funny way of denoting what you call $(x,y)$, but you can understand that it means the same thing because the author defined it to mean the same thing. – Gabriel Dec 20 '21 at 12:28
  • 1
    The same thing happens for projective coordinates. I prefer to write what you call $[x,y,z]$ as $[x:y:z]$. @Blue prefers $x:y:z$. Some other people may prefer $(x,y,z)$. That's fine, as long as the author makes it clear what it should mean in any case. – Gabriel Dec 20 '21 at 12:30
  • Final comment which may be helpful: when we're talking about homogeneous coordinates, we're always talking about projective spaces. – Gabriel Dec 20 '21 at 12:31
  • but Lee Mosher said the homogeneous coordinates are the point of $\mathbb R^3$.so what is right? – S. M. Dec 20 '21 at 12:37
  • He didn't; he said that a point in projective space is represented by points in $\mathbb{R}^3$.

    Let's be precise: a point in the projective space is a line passing through the origin in $\mathbb{R}^3$, right? But, since there's only one line which passes through two given points, a point $(x,y,z)\neq (0,0,0)$ in $\mathbb{R}^3$ determines a line (that is, a point of the projective space) which we denote by $[x:y:z]$. (Once again, you can denote this however you want, but let's fix this notation for now).

     – Gabriel Dec 20 '21 at 12:45
  • So... a point $(x,y,z)$ in $\mathbb{R}^3$ determines a point $[x:y:z]$ in the projective space. – Gabriel Dec 20 '21 at 12:46
  • Lee Mosher said that "ordered triple $(x,y,z)∈R^3$ is called "homogeneous coordinates" for the point $[x:y:z]∈P^2.$" That means he said $(x,y,z)$ belongs to $R^3$? – S. M. Dec 20 '21 at 13:32
  • Well... yes. We wrote $(x,y,z)\in\mathbb{R}^3$. I really don't get why you're focusing on this. That's just notation. Of course good notation is important, but it shouldn't be the main focus. – Gabriel Dec 20 '21 at 14:11
  • because, you said homogenous coordinates belongs to $P^2$ and from by Lee answer we see it's belongs to $\mathbb R^3$? – S. M. Dec 20 '21 at 14:15
  • 1
    I agree that this phrase that you cited is a little strange. Homogeneous coordinates define elements of $\mathbb{P}^2$. I'm absolutely sure that Lee understands perfectly well what he's saying; he just wrote a strange phrase. – Gabriel Dec 20 '21 at 14:19