A question about straight lines tangent to a sphere in 3-dimensional Euclidean space Let E(3) be 3-dimensional Euclidean space with its standard metric and let S(2) be the 2-dimensional boundary of a fixed ball of E(3) whose radius is positive. Does there exist a set L of straight lines which satisfies the following conditions? (1) Each straight line in L is a subset of E(3) that is tangent to S(2). (2) Each point of S(2) belongs to exactly one straight line in L. (3) No pair of distinct straight lines in L is co-planar.
 A: I believe such a set of lines can be constructed by transfinite recursion, using an argument similar to the one in this answer on MathOverflow.
Let $\alpha$ be the smallest ordinal of cardinality $\lvert S \rvert$. Fix a bijection between $\alpha$ and $S$, so that each ordinal $\beta \lt \alpha$ corresponds to a point $x_\beta \in S$. Let $\gamma$ be an ordinal $\lt \alpha$ and suppose that we already have a set of lines $\{ l_\beta \mid \beta \lt \gamma \}$ such that


*

*$l_\beta$ is tangent to $S$ at $x_\beta$ for all $\beta$;

*$l_\beta$ and $l_{\beta'}$ are not coplanar whenever $\beta \neq \beta'$.


The set of lines that are tangent to $S$ at $x_\lambda$ has cardinality $2^{\aleph_0}$. The lines that are coplanar to at least one of the $l_\beta$ form a subset of cardinality at most $\lvert \gamma \rvert$; indeed, for each $\beta \lt \gamma$, there is exactly one line that is both tangent to $S$ at $x_\gamma$ and coplanar to $l_\beta$. Since $\lvert \gamma \rvert \lt \lvert S \rvert = 2^{\aleph_0}$, it follows that it is possible to choose a line $l_\gamma$ tangent to $S$ at $x_\gamma$ without creating a pair of coplanar lines.
