ℝP² hands-on: charts as canvases

1 · Points are rays of sight

$R P^{2}$ is the set of all lines through the origin of $R^{3}$ — each line a single point. Homogeneous coordinates $[a^{0} : a^{1} : a^{2}]$ name some direction vector, determined only up to a factor $λ \neq = 0$ :

[a^{0} : a^{1} : a^{2}] = [λ a^{0} : λ a^{1} : λ a^{2}] .

The λ slider shows exactly this: the arrow (the representative) slides along the line, the point stays put. The optional sphere $S^{2}$ is a reminder that every line meets it in a pair of antipodes — $R P^{2}$ is "the sphere with antipodes glued".

2 · To paint is to divide

The chart $φ_{0}$ is the construction of perspective painting — Alberti's "open window", Dürer's grid: eye at the origin, picture plane ${a^{0} = 1}$ (blue). A ray of sight with $a^{0} \neq = 0$ pierces the canvas in exactly one point. To find it, rescale the direction vector until its first coordinate equals 1:

(a^{0}, a^{1}, a^{2}) ⟼ (1, \frac{a ^{1}}{a ^{0}}, \frac{a ^{2}}{a ^{0}}) .

The last two entries are the painted image point — and that is the entire chart:

φ_{0} : U_{0} \to R^{2}, [a^{0} : a^{1} : a^{2}] ⟼ (\frac{a ^{1}}{a ^{0}}, \frac{a ^{2}}{a ^{0}}), U_{0} = {[a] a^{0} \neq = 0} .

The division is not an algebraic trick — it is the projection. And since $λa$ spans the same ray, $λ$ cancels on its own: well-definedness for free.

3 · The horizon of a chart

Rays parallel to the canvas ( $a^{0} = 0$ ) never paint anything. They form the line at infinity of $φ_{0}$ — a copy of $R P^{1}$ , and the exact analogue of the horizon in a painting: parallel rails converge to a vanishing point, the point of their common direction. Press ∞ φ₀ (or drag the red point there — it snaps near the line at infinity): the φ₀ panel reports ∞ while the other charts keep showing the very same point, completely unimpressed.

That is why several canvases are needed. Every point has at least one nonzero coordinate, hence

U_{0} \cup U_{1} \cup U_{2} = R P^{2}

— and fewer will not do: $R P^{2}$ is compact, so a single chart to $R^{2}$ can never exist.

4 · Changing charts is re-projecting

A ray in $U_{0} \cap U_{2}$ pierces both canvases — both markers are visible at once. The transition map slides the image point along its ray from one canvas to the other:

φ_{0} \circ φ_{2}^{- 1} (u, v) = φ_{0} ([u : v : 1]) = (\frac{v}{u}, \frac{1}{u}), u \neq = 0.

Smooth where defined — the three canvases form an atlas, and $R P^{2}$ becomes a smooth 2-manifold.

5 · Conic sections: one figure, three appearances

A circle on ${a^{2} = 1}$ spans a cone of rays. Other canvases cut the same cone — as an ellipse, a parabola or a hyperbola, depending on whether the circle avoids, touches or crosses the horizon of $φ_{0}$ . Projectively it is a single object.

Try it

Move the λ slider: the coordinates change, the point does not.
Press ∞ φ₀: chart φ₀ loses the point, φ₁ and φ₂ keep it.
Switch on the canvases a¹, a² and click [1:0:0]: origin for φ₀ — point at infinity for the other two.
Choose Circle and slide its center towards 0: ellipse → parabola → hyperbola in the φ₀ panel.
Choose Line: a straight stroke in every chart — through infinity if necessary.
Click directly on a canvas: you pick the point by choosing its image.

One dimension down, the construction gives itself away completely: the chart of $R P^{1}$ is $φ_{0} ([a^{0} : a^{1}]) = a^{1} / a^{0}$ — simply the slope of the line. A vertical line, slope "∞"? Take the other canvas.

$φ_{i}$ divides by $a^{i}$ because dividing is projecting. Charts are canvases — and the line at infinity is the horizon: everything parallel to the canvas.

The visualization runs entirely in your browser — built with three.js (WebGL) and KaTeX.