How can I remove unnecessary nodes from my path?

Question

How do I remove the unnecessary nodes from the path to make it as short as possible? Idealy I'd like the result path to look like the green line. The red thingy is not walkable.

        var newPath = new List<Vector3>();
        var nodes = p.vectorPath;
        int lastCp = 0;

        List<int> toRemove = new List<int>();

        for (int i = 2; i < nodes.Count; i++)
        {
            if (IsPassable(nodes[lastCp], nodes[i]))
            {
                toRemove.Add(i - 1);
            }
            else
            {
                lastCp = i - 1;
            }
        }

        for (int i = 0; i < nodes.Count; i++)
        {
            if (!toRemove.Contains(i))
            {
                newPath.Add(nodes[i]);
            }
        }

Here's my attempt to solve the problem and it usualy works well but sometimes if I stand next to obstacle I get the orange line in the picture above.

I'm assuming the path has at least 3 nodes.

nodes is a list of Vector3 (the positions of grey dots on picture), IsPassable returns true if the path between two given Vector3 is passable.

newPath idealy should contain two points in the case of example path above.

EDIT:

I know why the algorithm doesn't work in this case (look at @Draco18s answer below). What I don't know is how to fix it so it always returns the shortest path regardless of start and target location.

Answer 1

Your method works just as you've programmed it: that is, nodes are being removed. The problem with getting the orange path instead of the yellow sometimes has to do with the fact that you trimmed Node3 (Node0 being the first, where the character is) and so the first non-trimmed node is Node10 at the corner with the orange line. Node11 isn't reachable from Node0 because of the obstacle, so your final path is 0 --> 10 --> 19 whereas the yellow line is 0 --> 3 --> 19.

It is because 0 --> 4 is uninterrupted (IsPassable(0,4) returns true), which causes the removal of Node3.

If you were to trim in reverse you would have similar problems with a destination very close to an obstacle, while the example case here would work beautifully.