Definition of homotopy of slope fields

Question

I can't come up with a correct definition of homotopic slope fields (on $\mathbb{R}^2$). Idea is clear - almost the same as vector field homotopy, but problem with defining slope as a function (case when field contains lines in each direction).

Answer 1

A continuous slope field on a plane is a continuous map into the quotient space $S^1/i$ where $i:S^1\to S^1$ is the antipodal map. This quotient space is actually $S^1$ again.

Explicitly, you can identify a line through the origin $(0,0)$ with a point on the circle $x^2+y^2 = y$: horizontal line corresponds to $(0,0)$, while other lines are mapped to the point of intersection with the circle, other than $(0,0)$.

Note that if you disallow vertical lines, then the slope field becomes a map into $S^1$ minus a point, which is contractible, so there are no interesting homotopy classes then.