How many coordinates does a point in $\mathbb{R}P^{n}$ have and which are the Homogenous Coordinates in $\mathbb{R}P^{n}$ ? To explain my question better. If we have a point p in the manifold $\mathbb{R}^{n+1}\smallsetminus\{0\}$ then we write p ,in a coordinate system of $\mathbb{R}^{n+1}\smallsetminus\{0\}$, as p = $(a_{0},a_{1}, \dots,a_{n})$ so we have n+1 coordinates in a n+1-manifold. Now if we take [p] = $\{q \in \mathbb{R}^{n+1}\smallsetminus\{0\} : p = tq,\,\, t \in \mathbb{R}^{\star}\}$, then [p] is in the manifold $\mathbb{R}P^{n}$. $\mathbb{R}P^{n}$ is a n dimensional manifold, but we note, [p] = $[(a_{0},a_{1}, \dots,a_{n})]$. So we have n+1 coordinates in a n-dimensoninal manifold. Is this correct? Is $[(a_{0},a_{1}, \dots,a_{n})]$ the homogenous coordinates of a point in $\mathbb{R}P^{n}$?.
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A straight line $y=mx+b$ in $\mathbb R^2$ consists of two coordinates even though it's 1-dimensional. The number of coordinates doesn't equal the dimension. – noctusraid Feb 18 '16 at 11:53
1 Answers
We do indeed say that these $n+1$ numbers are the "homogeneous coordinates" of $p$, but for most points $p$, any $n$ of these numbers, locally, form a local coordinate system in the sense of the definition of "manifold", so that $RP^n$ is $n$ dimensional, not $n+1$ dimensional. (The exception: around the point whose homogeneous coordinates are, say, (x, y, z) = (0, 0, 1), the coordinates $y$ and $z$ do not form a coordinate system.)
Now you might ask "How is it that 'homogeneous coordinates' are not actual 'coordinates' according to the definition of coordinates on a manifold?", and that's a good question: usually we think of adjectives as defining subsets: all red books are, at the very least, books. But in this case, the two terms probably originated independently, or maybe folks got sick of saying "homogeneous coordinate-like thingies", and this particular peculiarity arose.
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