I know that:

If one defines an equivalence relation on $\mathbb{R}^{n+1}-\{0\}$ by $$x\sim y \iff y=tx$$ for some nonzero real number $t$, where $x,y\in\mathbb{R}^{n+1}-\{0\}$, Then The real projective space $\mathbb{R} P^n$ is the quotient space of $\mathbb{R}^{n+1}-\{0\}$ by this equivalence relation.

Geometrically, two nonzero points in $\mathbb{R}^{n+1}$ are equivalent if and only if they lie on the same line through the origin, so $\mathbb{R} P^n$ can be interpreted as the set of all lines through the origin in $\mathbb{R}^{n+1}$. Each line through the origin in $\mathbb{R}^{n+1}$ meets the unit sphere $S^n$ in a pair of antipodal points, and conversely, a pair of antipodal points on $S^n$ determines a unique line through the origin (Figure). This suggests that we define an equivalence relation $\sim$ on $S^n$ by identifying antipodal points:

$$x\sim y \iff x=\pm y $$

We then have a bijection $\mathbb{R} P^n \leftrightarrow S^n/\sim$.

enter image description here

Now my question is:

Why is $\mathbb{R} P^n$ called projective space? What is projected and how? Is there another way to define the projective space so that in which projection is visible?

Thanks in advance.


The term originates with a different construction of the space than the ones you have outlind. Namely, this is the construction that "adds points at infinity". Briefly, we add points to $\mathbb{R}^n$ which correspond to pencils of parallel lines in $\mathbb{R}^n$. The intuitive picture is that of a "horizon" being added to a perspective drawing. The origin of the term "projective" is that 3-dimensional figures are being projected to a 2-dimensional image. For a drawing to look realistic, the artists had to develop precise rules for dealing with perspectives. These rules were eventually formalized into the construction of projective plane (or more generally, space) by adding points at infinity. The equivalence of the three constructions is proved in all the standard projective geometry textbooks.

  • $\begingroup$ when we take a sum or an integral over $\mathbb R$, do we have a concept of "point at infinity"? What does the notation $\infty$ mean at both ends? $\endgroup$ – Ooker Nov 6 '17 at 16:39

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