The Spin Chronicles (Part 18): Introducing Klein absolute

 “... there is something still lacking in this picture—the 'conformal infinity,' the 'absolutes' of projective geometry. Points where parallel lines meet...” That’s what I claimed in the previous post. It seems, however, that the idea of parallels meeting "somewhere" doesn’t sit well with the otherwise grounded tone of our series on the Clifford geometric algebra of space. Naturally, this stirred the curiosity of one particular reader (Bjab), who came armed with questions sharper than Occam's razor:

• "What does 'meet' mean? (Different) Parallel lines don't have common points."
• "What does 'intersect' mean?
• "Meridians are not circles. They are semicircles."
• "Are they really (in the common sense of the word) intersecting, or are those circles just tangent to each other?"

Ah, questions! The perpetual call to adventure for anyone foolish—or ambitious—enough to write about math. Since the cosmos obliges, so must I. Today, we’re taking a small detour (I really intend a small one) to tackle these questions head-on. And as luck would have it, this diversion ties neatly into our main subject, because it showcases the adaptability of the mathematical tools we’ve been exploring. By repurposing the same algebraic framework, we’ll reveal not just the versatility of mathematics but its elegance as well. Prepare for a deeper dive into the sublime weirdness of projective and conformal geometry!

Circles, Spheres, and Stereographic Projections

Let’s ease into this with a concept we all know and love: the humble circle. Actually, we should start with the slightly more ambitious sphere—it’s a better stand-in for the idea of "space." But hey, 3D geometry is cumbersome to scribble on a napkin or screen, so let’s stick to 2D for now. Circles are easier to draw, and, more importantly, we can rely on algebraic tools we’ve already mastered.

Circles are easier to draw

For this introductory phase, we’ll forego Clifford algebra. Don’t worry, it’ll return when we shift from static descriptions to dynamic transformations—that’s where Clifford algebra shines. After all, it’s not just about understanding objects but also how they morph, twist, and turn under various operations. These transformations form groups, and the Clifford group, as you’ll see later, is the MVP of this show. But for now, let's lay the groundwork with simpler tools.

Enter the Klein absolute. This concept will guide us into the realms of projective and conformal geometry, which are surprisingly practical—think computer vision and robotics. The journey begins with a stereographic projection on the 2D plane, using Cartesian coordinates x=(x₁,x₂).

Ready? Here’s the formula for (inverse) stereographic projection through the South Pole (because why not switch hemispheres for variety?):

σ(x) = (2x, (1-x·x))/(1+x·x).

Looks innocent enough, right? But let’s imagine we’re afflicted with a peculiar condition—denominatophobia. Yes, it’s a (fictional) fear of denominators. Even when they’re calm and friendly, like the one above, they still make us squirm. Now, as long as x is a real vector, there’s no reason to panic. But what if x develops a taste for the imaginary? After all, we’ve seen this happen before—constructing the Clifford algebra of space brought us to complex Minkowski spacetime! And who’s to say we won’t one day construct a Clifford algebra of a Clifford algebra? (Math is an infinite playground.)

To calm our nerves, we’ll take preventive measures. Mathematics offers a clever remedy for denominatophobia: projective space.

Projective Space: Fearless Math for Fearful Minds

We introduce a fourth coordinate, x₀, alongside x₁,x₂,x₃.
Initially, we set x₀ = 1. This embeds our 2D point into 4D space:

x ⟼ ( 1, 2x/(1+x·x), (1-x·x)/(1+x·x) ).

In projective geometry, equivalence reigns supreme. Two points ζ and ζ′
 are equivalent (ζ ∼ζ′) if there exists a scalar λ>0 such that ζ′ = λζ. These equivalence classes—essentially half-lines in R⁴—are denoted by [ζ]. With this setup, we can rewrite the embedding as:

[( 1, 2x/(1+x·x), (1-x·x)/(1+x·x) )] = [((1+x·x)/2, x, (1-x·x)/2 )].

Victory! The denominators are gone, leaving us free to bask in mathematical serenity. But wait—there’s more.

The Bonus: A Light Cone Emerges
The point ζ = ((1+x·x)/2, x, (1-x·x)/2 ) lies on the light cone of Minkowski space. Let’s verify this. Define:

ζ₀ = (1+x·x)/2,

ζ₁ = x₁,

ζ₂ = x₂,

ζ₃ = (1-x·x)/2.

Then, check the relation:

(ζ₀)²- (ζ₁)² - (ζ2)² - (ζ₃)² = 0.

Exercise 1: Prove this formula.

This light cone is the so-called Klein absolute for our 2D plane. But don’t pack up just yet—there’s more excitement in store. In the next post, circles and parallel lines will make their dramatic entrance. For now, the stage is set, the props are in place, and the mathematical actors are ready for action.

References:

[1[ Wilson, Mitchell A, "Meeting at a far meridian",  "Встреча на далеком меридиане" -  Movie (1977)  here.

[2] Cecil, Thomas E. "Lie Sphere Geometry", Springer 2000

P.S. A schematic drawing taken from Ref. [2] (I have a somewhat different notation):