I would like to know if it is possible to fill $\mathbb{R}^3$ with lines with the following two properties:

(1) Every point $x \in \mathbb{R}^3$ is contained in precisely one line.

(2) Every neighborhood of every point is pierced by lines whose directions fill out the
sphere of possible line orientations, in this sense:
For every point $x$ and every $\epsilon > 0$, the lines that pass through a point
in the ball $B_\epsilon(x)$ of radius $\epsilon$ centered on $x$ have the property
that, were they all translated to pass through the origin, the closure of the set of points that
constitutes their intersection with an origin-centered sphere $S$, fills out $S$ completely.
This image below is meant to suggest the idea:

I am sure there is a more concise way to phrase the second condition; apologies for my
ungainly formulation. I want to be able to find every line orientation within a neighborhood of
every point.

Perhaps condition (2) is not possible to achieve in conjunction with (1). But I don't see an argument. Any ideas/insights/pointers would be appreciated—Thanks!