Degrees of freedom of a line in 3D

Question

what are the degrees of freedom of a line in 3D? (defend your argument plz)

Answer 1

Let's consider directed lines.

First you can choose the direction of your line $\ell$. This amounts to choosing a point ${\bf u}$ on the two-dimensional unit sphere $S^2\subset{\mathbb R}^3$, so that we have two degrees of freedom for that.

Given ${\bf u}$, let $$H_{\bf u}:=\bigl\{{\bf x}\in{\mathbb R}^3\bigm|{\bf u}\cdot{\bf x}=0\bigr\}$$ be the plane orthogonal to ${\bf u}$ through the origin. We can now choose the point ${\bf p}\in H_{\bf u}$ where our line $\ell$ intersects $H_{\bf u}$. Since $H_{\bf u}$ is two-dimensional this amounts to another two degrees of freedom.

Now $\ell$ is completely determined by ${\bf u}$ and ${\bf p}$, so there are four degrees of freedom when choosing a directed line in ${\mathbb R}^3$.

If you are caring about undirected lines then two antipodal points $\pm{\bf u}\in S^2$ determine the same line; but this does not change the number of degrees of freedom.