如何从与另一个给定线段相交的一组点中找到最短的线（2个端点就可以了）

Question

Say I got a bunch of points(red points) and a blue line segment , i want to find a line(2 points) which is intersect with the blue line segment and the sum of two end points' perpendicular distance to the blue line segment is the shortest distance.

P in this picture above is CD and the blue line segment intersect point.

I only know that they should be on the blue line's different side (left/right or above/below).

But there are still serveral variants can make this go south.Like AB and CE in this picture.

Edit: At first i thought i only need to make sure both of two end points' projection point lies on the blue line segment .But if I do so,there might be not so many good points for this case.I might just throw the only solution away.

Answer 1

Sketch of algorithm:

1. Calculate all perp. distances including orientation (left or right of dividing line L).
2. Devide points in two lists (left or right).
3. Sort both lists by perp. distance.
4. Now build pairs from both lists, sorted by minimal sum of perp. distance.
5. Choose the first (smallest sum) pair with a connecting line that intersects L as result.