Visualizing the real projective plane and introducing homogeneous coordinates.

1. Visualizing the real projective plane

Let

\[ \mathcal A=(\mathbb R^2,\mathcal L_{\mathbb R^2},I_{\mathbb R^2}) \]

be the affine plane on \(\mathbb R^2\), also called the real affine plane.

The goal of this lecture is to understand how we can visualize its projective completion

\[ \mathcal A^\ast , \]

also called the real projective plane.

To help visualize this construction, you can click here for the Desmos 3D animation introduced in class.

2. Step 1: An isomorphic model for the real affine plane

Let

\[ \Hopen := \{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1,\ z>0\}. \]

This is the open northern hemisphere.

We identify a point \((x,y)\in \mathbb R^2\) with the point \((x,y,1)\in \mathbb R^3\). In other words, we identify the real affine plane with the plane \(z=1\) in \(\mathbb R^3\). This plane is tangent to the unit sphere at the north pole

\[ N=(0,0,1). \]

Now consider the segment from the origin \((0,0,0)\) to the point \((x,y,1)\). This segment intersects the open northern hemisphere in a unique point. We define

\[ f(x,y) := \frac {(x,y,1)}{\sqrt {x^2+y^2+1}}. \]

Thus we have a map

\[ f:\mathbb R^2\longrightarrow \Hopen . \]

Left as a calculus exercise.

More precisely, define

\[ \mathcal L_{\Hopen } := \{f(\ell ):\ell \in \mathcal L_{\mathbb R^2}\}. \]

Since lines in \(\Hopen \) are subsets of \(\Hopen \), we use the standard incidence relation of belonging:

\[ p\,I\,\ell \qquad \Longleftrightarrow \qquad p\in \ell . \]

This follows from the fact that \(f:\mathbb R^2\to \Hopen \) is bijective.

3. Step 2: Projective completion of the open northern hemisphere model

Let

\[ \Hclosed := \{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1,\ z\geq 0\}. \]

This is the closed northern hemisphere. It is the open northern hemisphere together with the equator

\[ \Equator := \{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1,\ z=0\}. \]

Therefore, an equivalence class of parallel lines in \(\mathcal L_{\Hopen }\) is determined by a pair

\[ \{p,-p\}, \]

where \(p\) is a point on the equator.

Thus, as a model of the projective completion of the real affine plane, we may consider the closed northern hemisphere with antipodal points on the equator identified.

4. Identifying antipodal points on the equator

We now make this rigorous. We define an equivalence relation on \(\Hclosed \) as follows.

If \(x\in \Hopen \), so \(x\) is in the open northern hemisphere, define

\[ x\sim y \qquad \Longleftrightarrow \qquad y=x. \]

Thus points in the open hemisphere are only equivalent to themselves.

If \(x\in \Hclosed \setminus \Hopen \), so \(x\) is on the equator, define

\[ x\sim y \qquad \Longleftrightarrow \qquad y=x \quad \text {or}\quad y=-x. \]

Thus points on the equator are identified with their antipodal points.

We denote the quotient set by

\[ \Hclosed /\!\sim . \]

5. The bijection from the projective completion to the quotient hemisphere

Let \(\Hopen ^\ast \) be the projective completion of the open northern hemisphere model. Recall that its points are

\[ \Hopen ^\ast = \Hopen \sqcup \mathcal L_{\Hopen }/\!\parallel . \]

Here \(\mathcal L_{\Hopen }/\!\parallel \) denotes the set of parallel classes of lines in \(\Hopen \).

We define a map

\[ F:\Hopen ^\ast \longrightarrow \Hclosed /\!\sim \]

as follows.

If \(x\in \Hopen \), define

\[ F(x)=[x]. \]

If \([\ell ]\) is a parallel class of lines in \(\Hopen \), then every line in this class, when continued to the closed northern hemisphere, meets the equator at the same antipodal pair \(\{p,-p\}\). Define

\[ F([\ell ])=[p]. \]

This is left as an exercise.

6. Mapping lines to lines

We now describe the corresponding lines.

Recall that the lines in \(\Hopen ^\ast \) are

\[ \mathcal L_{\Hopen }^\ast = \mathcal L_{\Hopen }\sqcup \{\ell _\infty \}. \]

Given a line \(\ell \in \mathcal L_{\Hopen }\), let \(\overline {\ell }\subseteq \Hclosed \) be the continuation of \(\ell \) to the closed northern hemisphere. This is a closed half-great circle, and it meets the equator at two antipodal points \(p\) and \(-p\).

Let

\[ \pi :\Hclosed \longrightarrow \Hclosed /\!\sim , \qquad x\longmapsto [x], \]

be the quotient map. Define

\[ G(\ell ):=\pi (\overline {\ell }). \]

Equivalently,

\[ G(\ell )=\{[x]:x\in \overline {\ell }\}. \]

Thus \(G(\ell )\) is the closed half-great circle corresponding to \(\ell \), with antipodal points identified on the equator.

For the line at infinity, define

\[ G(\ell _\infty ) := \{[x]:x\in \Equator \}. \]

This is the equator with antipodal points identified.

This is left as an exercise.

7. The sphere model

We can also use the whole sphere. Equip

\[ \Sph \]

with the antipodal equivalence relation:

\[ p\sim q \qquad \Longleftrightarrow \qquad q=p \quad \text {or}\quad q=-p. \]

We denote the quotient by

\[ \Sph /\{\pm 1\}. \]

There is a bijection

\[ \Hclosed /\!\sim \longrightarrow \Sph /\{\pm 1\}, \qquad [p]\longmapsto [p]. \]

As a result, we may define lines in \(\Sph /\{\pm 1\}\) as the images of lines in \(\Hclosed /\!\sim \), and we get an isomorphic model of the real projective plane.

8. Projective completion of \(F^2\)

We now describe an algebraic model of the projective completion of the affine plane over a field.

Let \(F\) be a field, for example

\[ F=\mathbb R,\quad \mathbb Q,\quad \mathbb C, \]

or any finite field.

9. Definition of \(\mathbb P^2(F)\)

Consider

\[ F^3\setminus \{(0,0,0)\}. \]

We define an equivalence relation on this set as follows.

Given two nonzero triples

\[ (x,y,z),(x',y',z')\in F^3\setminus \{(0,0,0)\}, \]

we say that

\[ (x,y,z)\sim (x',y',z') \]

if there exists \(\lambda \in F^\times =F\setminus \{0\}\) such that

\[ (x',y',z')=\lambda (x,y,z). \]

The equivalence class of \((x,y,z)\) is denoted by

\[ [x:y:z]. \]

Thus

\[ [x:y:z] = \{(\lambda x,\lambda y,\lambda z):\lambda \in F^\times \}. \]

10. Lines in \(\mathbb P^2(F)\)

The points of \(\mathbb P^2(F)\) are the equivalence classes

\[ [x:y:z]. \]

The lines of \(\mathbb P^2(F)\) are subsets of the form

\[ \ell _{a,b,c} := \{[x:y:z]\in \mathbb P^2(F): ax+by+cz=0\}, \]

where

\[ (a,b,c)\in F^3\setminus \{(0,0,0)\}. \]

Incidence is the standard incidence relation of belonging:

\[ p\,I\,\ell \qquad \Longleftrightarrow \qquad p\in \ell . \]

This incidence geometry is called the projective plane over \(F\).

11. \(\mathbb P^2(F)\) as a projective completion

To help visualize homogeneous coordinates and the projective completion of \(F^2\), you can click here for the Desmos 3D animation introduced in class.

Let

\[ \mathcal A=(F^2,\mathcal L_{F^2},I_{F^2}) \]
be the affine plane over \(F\), and let
\[ \mathcal A^\ast =(\mathcal P^\ast ,\mathcal L^\ast ,I^\ast ) \]
be its projective completion.

We define a map

\[ f:\mathcal P^\ast \longrightarrow \mathbb P^2(F) \]
as follows.

For an ordinary affine point \((x,y)\in F^2\), define

\[ f(x,y):=[x:y:1]. \]

Now let \([\ell ]\) be a point at infinity, where \(\ell \) is an affine line of the form

\[ ax+by+c=0. \]
Define
\[ f([\ell ]):=[-b:a:0]. \]

This is well-defined because every line parallel to \(\ell \) has the form

\[ ax+by+d=0 \]
for some \(d\in F\), and all such lines determine the same point
\[ [-b:a:0]. \]

This is left as an exercise.

We now define a map on lines

\[ g:\mathcal L^\ast \longrightarrow \mathcal L_{\mathbb P^2(F)}. \]

If \(\ell \in \mathcal L_{F^2}\) is the affine line

\[ ax+by+c=0, \]
define
\[ g(\ell ) := \{[x:y:z]\in \mathbb P^2(F): ax+by+cz=0\}. \]

For the line at infinity, define

\[ g(\ell _\infty ) := \{[x:y:z]\in \mathbb P^2(F): z=0\}. \]
Equivalently,
\[ g(\ell _\infty ) = \{[x:y:0]:(x,y)\neq (0,0)\}. \]

This is left as an exercise.

This is left as an exercise.

12. Another isomorphic model

There is another useful model of \(\mathbb P^2(F)\).

Define \(\widehat {\mathcal P}\) to be the set of lines in \(F^3\) through the origin. In other words, its elements are subsets of the form

\[ P = \{\lambda (x,y,z):\lambda \in F\}, \]

for some

\[ (x,y,z)\neq (0,0,0). \]

Define \(\widehat {\mathcal L}\) to be the set of planes in \(F^3\) through the origin. In other words, its elements are subsets of the form

\[ L = \{(x,y,z)\in F^3: ax+by+cz=0\}, \]

for some

\[ (a,b,c)\neq (0,0,0). \]

Incidence is inclusion:

\[ P\,I\,L \qquad \Longleftrightarrow \qquad P\subseteq L. \]

The map

\[ \mathbb P^2(F) \longrightarrow \widehat {\mathcal P}, \qquad [x:y:z] \longmapsto \{\lambda (x,y,z):\lambda \in F\}, \]

is bijective.

Together with the corresponding map sending projective lines to planes through the origin, this gives an isomorphic model of \(\mathbb P^2(F)\).