# Geometry of curves on the sphere

Let P be a finite set of points on the unit sphere $$S^2$$ such that

• for every $$p\in P$$, there exists a closed curve $$\gamma_p \subset S^2$$ which has a self intersection at $$p$$ and passes through $$-p$$. Moreover, every plane passing through $$p$$ and the origin intersect $$\gamma_p$$ in at most 2 points, excluding $$-p$$ (see the graph below by Neil Strickland).

• At the self intersection point $$p$$, $$\gamma_p$$ is orthogonal to itself.

• $$\gamma_{p_1} \cap \gamma_{p_2} \subset P$$ for every $$p_1,p_2 \in P$$

• If $$A_i \subset S^2$$ be such that $$\partial A_i \subset \gamma_{p_i}$$, $$p_i\in P$$, i=1,2, then neither $$A_1 \subset A_2$$ nor $$A_2 \subset A_1$$.

• $$P \subset \gamma_p$$ for all $$p\in P$$.

Can we prove that $$P$$ has either one or infinite elements? It seems that more than one element leads to infinite number of elements in $$P$$. It is easy to see that if $$P$$ has more than one point, then it has at least 6 points.

Please see the picture provided by Neil Strickland for a visualization of $$\gamma_p$$.

• Please write couple of words on motivation. Sep 5, 2017 at 20:51
• Is it OK to assume that $\gamma_p$ is smooth? Sep 5, 2017 at 23:01
• Yes, $\gamma_p$ is indeed smooth. Sep 5, 2017 at 23:24
• I find it hard to imagine even two of these curves satisfying all assumptions. Could you draw a picture with two or three curves, please? Sep 6, 2017 at 16:26
• No, just excluding -p. It is indeed at most three points including all points. Jan 21, 2023 at 14:25

This is just a comment, really. One can write a nice formula for a family of curves of the indicated type, as follows. Let $p$, $q$ and $r$ be an orthonormal basis of $\mathbb{R}^3$, and suppose that $a>0$. Put $$C = \{x\in S^2: (q.x)\,(r.x)=a((q-r).x)(1 - p.x)\}$$
This can be parametrised as $$\gamma(t) = \frac{(a^2(c-s)^2-s^2c^2)p+2asc(c-s)(cq+sr)}{a^2(c-s)^2+s^2c^2},$$ where $s=\sin(t/2)$ and $c=\cos(t/2)$. This passes through $p$ with derivative $q/a$ at $t=0$ and through $p$ again with derivative $r/a$ at $t=\pi$. It also passes through $-p$ with derivative $-2a(q+r)$ at $t=\pi/2$. Maple code is as follows:
g := unapply(subs({s = sin(t/2),c=cos(t/2)},((a^2*( c-s)^2-s^2*c^2) *~