Finding when shapes overlap and lines intersect but exclude points/edges

Question

I've been looking for a c# algorithm to help determine if shapes and lines intersect but want to exclude boundary overlaps. Here's an example:

My true problem is finding if polygons intersect but exclude boundary overlaps. I broke down the problem to look at the individual lines of that polygon and check each line segment from the shape1 against each line segment from shape2.

A: Is this an efficient/correct way to do the checking?

B: How do I ignore when they are on the boundary as illustrated above?

Here's some code I found here - I've been testing for line segments but it fails on fig 2, when the end point of the 2nd line segment finishes on the end point of the 1st segment.

private bool DoesLineIntersect(Vector2 start1, Vector2 end1, Vector2 start2, Vector2 end2)
{
    float denom = ((end1.x - start1.x) * (end2.y - start2.y)) - ((end1.y - start1.y) * (end2.x - start2.x));

    //  AB & CD are parallel 
    if (denom == 0)
        return false;

    float numer = ((start1.y - start2.y) * (end2.x - start2.x)) - ((start1.x - start2.x) * (end2.y - start2.y));

    float r = numer / denom;

    float numer2 = ((start1.y - start2.y) * (end1.x - start1.x)) - ((start1.x - start2.x) * (end1.y - start1.y));

    float s = numer2 / denom;

    if ((r < 0 || r > 1) || (s < 0 || s > 1))
        return false;

    return true;
}

Answer 1

You need to change your final check to also check for equality. So change the line

if ((r < 0 || r > 1) || (s < 0 || s > 1)) 

to

if ((r <= 0 || r >= 1) || (s <= 0 || s >= 1))