Similarly to the definition of the distance of closest approach (DCA) for two arbitrary ellipses1 we define the DCA for an arbitrary ellipse and a line segment.

This distance is used in some models to calculate the repulsive force between a pedestrian and a wall2. The quantity we are looking for is $$l$$ with:

$$P$$ is the nearest point on $$[AB]$$ to the ellipse. See also notes of P. Bourke3.

Knowing $$\alpha$$ we can easily calculate $$r$$. To solve the above mentioned equation
one has to find the quantity $$d$$, which would be the necessary amount to translate $$[AB]$$ along the direction of $$OP$$ such that it becomes tangential to the ellipse.

We have

The parametric definition of the line segment $$[AB]$$ is:

$$x_{R^\prime}$$ is on the ellipse, which implies

or

with

In case the point $$R$$ is known the solution of Eq. (2) is

However, in general $$R$$ is not known. Developing Eq. (1) with respect to $$u$$ yields

with

with the substitutions $$x_{BA}=x_B-x_A$$ and $$y_{BA} = y_B - y_A$$.

Since the line $$(AB)$$ is tangential to the ellipse we get

Supposing that $$P, A$$ and $$B$$ are not collinear, we solve (3) and get
For $$d$$ we calculate $$u$$ as
If the inequality $$0 \le u \le 1$$ does not hold or $$P, A$$ and $$B$$ are collinear, then this would mean the point $$R$$ is an end point of $$[AB]$$, i.e. $$A$$ or $$B$$. In that case we solve (2) twice with $$R:=A$$ and $$R:=B$$ and get $$d= \min(|d_A|, |d_B|)$$.