Lie Sphere Geometry Part 3: oriented circles and Seneca

 “It is of course required of a man that he should benefit his fellow-men — many if he can; if not, a few; if not a few, those who are nearest; if not these, himself. For when he renders himself useful to others, he engages in public affairs.” (Seneca, On Leisure 3.5)

... man that he should benefit his fellow-men 

Seneca was a stoic. Having this in mind let us continue from Lie Sphere Geometry Part 3: oriented circles. We have considered there circles S_t(m), where m is a point on the unit sphere S² in R³, and t varies between 0 and 2π. In the animation we have endowed each of this circles with a unit normal vector field n(t,ϕ). For t=0 and t=π, the circle shrinks to a point and the normal vectors n(t,ϕ) point in different directions for different values of ϕ. 

 n(3.2,ϕ), ϕ = k π/4, k=0,...,7

Thus points have their orientations undefined.

We could think that all oriented circles (including points) are parametrized by m∈S², and 0 ≤ t < 2π. However, looking at the animation of the previous post we can easily visualize the fact that a circle starting at m=(1,0,0) and  t = 3π/2 is exactly the same, including its normal vector field, as the circle that starts at m=(-1,0,0) and t=π/2.  The first one collapsed to a point (-1,0,0) after t=π, and starts expanding again, the second one simply starts at (-1,0,0). More generally circles (m,t) and (-m,t+π mod 2π) are exactly the same, including their orientations. (Can you see it?)

It follows that the set of all oriented circles on S² is nothing else but

(S²⨉S¹)/Z₂,

where Z₂ = (+1,-1) acts by

(-1)(m,t) = (-m,t+π mod 2π).

Taking quotient by Z₂ may lead to non-orientable surfaces like Mobius strip or Klein bottle. Such surfaces need higher dimensions to embed them in. And that is our plan for the future posts. We will discuss the manifold of all oriented circles using projective geometry. We will add not just one or two, but three extra dimensions!

Lie Sphere Geometry Part 3: oriented circles

 This post is a continuation of Lie Sphere Geometry Part 2: unoriented circles. We will move to oriented circles now. Here I am trying to follow the exposition of this subject as it is described in Ch. 15. 1, Oriented circles in S³ of [1]. Instead of S³ I am taking S² first. I say "I am trying to follow" instead of "I am following". The reason for that is that I am not entirely happy with my understanding of the exposition in this source. 

 I am not entirely happy with my understanding of the exposition in this source.

Of course there are other sources discussing the Lie Sphere Geometry, but each author has a different approach, different angle of approach, different main concepts, and I have chosen Jensen as best fitting to our purpose, at least for now. In this post we will discuss oriented circles on the sphere S². Let us recall the definition from the previous post.

S² = {x∈R³:  x² = 1}

For every m∈S², and for every r∈[0,π] the unoriented circle S_r(m)is defined as

S_r(m) = {x∈S²:  x·m = cos r}.

We have then

S_r(m)=S_π-r(-m).

We now extend this definition of S_r(m) any r∈[0,2π]. Of course it makes sense, but S_r(m) = S_2π-r(m). To remove this degeneracy, and following Ref. [1],  we define the oriented circle as follows:

Definition. An oriented circle in S² is a circle S_r(m), together with a choice of  continuous unit normal vector field on it.

Ref. [1] expands this definition as follows.

We can use the radius to define an orientation of S_r(m) as follows. Recall that x∈S_r(m) if and only if x·m = cos(r). Let n∈T_xS² be the unit vector satisfying the equation

m = cos(r)x + sin(r)n.         (0)

Except for the cases r = aπ for any integer a, this uniquely determines n for each point x on the locus. The cases r = aπ, a any integer, are the point spheres m and -m, and there is no condition on n in

n(x) = (m - cos(r) x)/sin(r).                (1)

We consider point spheres to be in the set of all oriented spheres, but without orientation. Thus, the oriented spheres with
center m are parametrized by their radius r satisfying 0 ≤ r < 2π.

I am having some problem with the formula (1) as r approaches 0. The reasoning above was causing me some stomach problems. Therefore I suggest a somewhat different approach. The rest of this post is my own thinking about the subject.

So, let us consider the family of circles for a fixed m. In order to easily visualize thi family let us take m = (1,0,0). I will also use the letter t instead of r. That is because I want to treat the radius as a parameter of some kind of a dynamics. You will see it below.

The parametric equations x(t,ϕ) of the family are then:

x(t,ϕ) = cos(t),
y(t,ϕ) = sin(t)cos(ϕ),
z(t,ϕ) = sin(t)sin(ϕ).

Here ϕ is a parameter along the circle. For t=0 we have the point m=(1,0,0). We notice that

x(t,ϕ)² =  x(t,ϕ)²+y(t,ϕ)²+z(t,ϕ)² ≡1,

so that the circles of constant t are indeed all on S².

Now, for a fixed ϕ calculate the partial derivative ∂x(t,ϕ)/∂t with respect to t. We will call it the "velocity vector field" and denote it n(t,ϕ):

n(t,ϕ) = ∂x(t,ϕ)/∂t,

or, explicitly

n_x(t,ϕ) = - sin(t),
n_y(t,ϕ) = cos(t)cos(ϕ),
n_z(t,ϕ) = cos(t)sin(ϕ).

We see that n(t,ϕ)² = 1, so we have a field of unit vectors. Moreover n(t,ϕ)is tangent to the sphere at x(t,ϕ), as it is tangent to the great circle of constant  ϕ.. Moreover, it is perpendicular to the circle of constant t. Let us calculate the numerator of (1) in our parametrization:

m_x - cos(t)x(t,ϕ) = 1 -cos²(t) = sin²(t) = -sin(t)n_x(t,ϕ) ,
m_y - cos(t)y(t,ϕ) = 0 - cos(t)sin(t)cos(ϕ) = - sin(t)n_y(t,ϕ) ,
m_z - cos(t)z(t,ϕ) = 0 - cos(t)sin(t)sin(ϕ). = - sin(t)n_z(t,ϕ) .

Comparing with (1) we see that our n(t,ϕ) coincides with -n(x) of (1), but it makes a perfect sense also for t=0 and t=π.

Simulation of n(t,ϕ) for varying t, and ϕ=0, π/2,π,3π/2. 

To be continued ...

References

[1] G.R. Jensen et al., Surfaces in Classical Geometries, Springer 2016.