Distance between two straight lines

The distance between two straight lines, $$r$$ and $$s$$, is the minimal distance between any point of $$r$$ and any point of $$s$$.

• If the straight lines are secant or coincidental, their distance is obviously zero. Namely $$d (r, s) = 0$$.
• If the straight lines are parallel, the distance between $$r$$ and $$s$$ is the distance from a point of any of both straight lines to any other.

To find the analytical expression of the distance from $$r$$ to $$s$$, we will suppose that we have $$r: Ax + By + C = 0$$ and $$s: Ax + By + C' = 0$$. As the straight lines need to have parallel director vectors, we can suppose that they do have the same, which is why $$A = A'$$ and $$B = B'$$.

As the straight lines cannot be coincidental, we will obviously have $$C\neq C'$$.

Let $$P =(p_1,p_2)$$ now be a point belonging to the straight line $$r$$. Then we have: $$$\displaystyle d(r,s)=d(P,s)=\frac{|A\cdot p_1+B\cdot p_2+C'|}{\sqrt{A^2+b^2}}$$$ But since $$P$$ belongs to the straight line $$r$$ we have $$$A\cdot a_1+B\cdot a_2+C=0 \Leftarrow A\cdot a_1+B\cdot a_2=-C$$$ substituting, $$$d(r,s)=d(P,S)=\displaystyle \frac{|C'-C|}{\sqrt{A^2+B^2}}$$$