Lie Sphere Geometry Part 10: Planes

 Spheres of infinite radius, those that cross the infinity point,  are just planes. Spheres are mysterious, planes are mysterious too, perhaps even more so.

Even more so...

This post is a straightforward continuation of Lie Sphere Geometry: Part 9: Spheres of negative radius. The stereographic projection of the oriented sphere

S_t(m) = {x∈S³: x·m = cos(t)},     0<t<π ,      (0)

leads to the following equation (Eq. (4) of Part 9) for its image in R³:

y²(m⁰+cos(t)) - 2m'·y= m⁰ - cos(t),                (1)

where

m '= m¹e₁+m²e₂+m³e₃.                (2)

We have already considered the case of m⁰+cos(t) ≠ 0. We consider now the case of m⁰+cos(t) = 0, i.e. m⁰ = -cos(t). Eq (1) then becomes

m'·y = cos(t).                 (3)     

Definition 1. With n∈S²⊂R³ and h∈R, the oriented plane Π_h(n) with unit normal n and signed height h, is a pair (n,h) and  associated with it the set [Π_h(n)] defined as:

[Π_h(n)] = {y∈R³: y·n = h}.                (4)

Remark. To see the meaning of h, suppose we orient the Cartesian axes of R³ so that n is the unit vector in the z-direction. Then h is the value of the z-coordinate at which the plane intersects the z-axis. Thus the name "signed height". Notice that while Π_h(n)  and Π_-h(-n) define the same set, these are two different oriented planes, as they have opposite normals.

Proposition 1. The image of the oriented sphere S_t(m)⊂S³ , containing -e₀, under the stereographic projection x⟼y  is the oriented Π_h(n') , where n' = m'/sin(t), h = cot(t).

Proof.  Comparing Eq. (3) with (4) we get instantly that n' has to be ε=±1 times the normalized vector m'. Now, (m⁰)²+m'²=1, and m⁰ = -cos(t). It follows that m'² = 1- cos²(t) = sin(t)². Since 0<t<π, we deduce that ||m'|| = sin(t).

Thus n'=ε m'/sin(t), h = ε cot(t). In Part 9 we have calculated (Eq. (11)) the image n'' of n:

n''(x)ⁱ = (1+x⁰)^-1(mⁱ/sin(t) -xⁱ(m⁰+cos(t))/(sin(t)(1+x⁰)) ,

which now reduces to

n''(x)ⁱ = (1+x⁰)^-1(mⁱ/sin(t)).

Since 1+x⁰ > 0, we deduce that ε = 1.

⎕

In the next post we will deal with the inverse stereographic projection of oriented spheres and planes.

P.S. 24-04-25 17:41 Today, Laura posted her first article on Substack: 
Letters From the Edge of Reality

Is AI modeled on an Alien Control System? Or Why X is Not a Free Speech Platform

P.S. 25-04-25 12:03 Yesterday in a comment Bjab wrote:

"Consider three spheres St(m), St-ε(m) and St+ε(m) where ε is a small number and m is a fixed vector. Let St(m) pass through -e0. Then St-ε(m) and St+ε(m) do not pass through -e0.

All three spheres have the same direction (insidnes).

Now let's look at the projections of these three spheres.

The projection of the sphere St-ε(m) has the opposite direction (insidnes) than the projection of the sphere St+ε(m). Therefore, the projection of the sphere St(m) cannot have a direction."

Here is an illustration with three circles. I have shown normals for St-ε(m) and St+ε(m)  and their stereographic projections, using the formulas from the posts:

I didn't draw the normal to the middle line, but it is easy to guess what it would be.

Coordinates on the picture are x1=x,x2=y,x0=z. 

In coordinates x0,x1,x2, I take t=π/2, ε=π/4, m={0,-1,0}.

P.S. 25-04-25 19:26 Yesterday during Valdimirov's seminar “Fundamental physics basics” A. Yu,  Sevalnikov presented a talk on "Ten Aspects of Understanding Time: Philosophical Implications". One slide of his talk attracted my special attention:

I strongly disagree with his statement about "events" in quantum theory. Contemplating writing to the Author in order to explain the situation in quantum physics with "events". After all I published a number of papers in physics journals about "events" and "time of events".