Question

I have a two segments AB and CD (in red). These two segments are facing each others. They are not completely parallel but will never be perpendicular to each others either.

From that, I need to find the two normals of these segments (in blue) that oppose each others (i.e. the two normals are outside ABCD). I know how to calculate the normals of the segments but obviously each segment has two normals, and I cannot figure out how to programmatically select the ones I need. Any suggestion?

Answer 1

Calculate the vector v between the midpoints of the two segments, pointing from AB to CD. Now the projection of the desired normal to AB onto v must be negative and the projection of the desired normal to CD onto v must be positive. So just calculate the normals, check against v, and negate the normals if needed to make them satisfy the condition.

Here it is in Python:

# use complex numbers to define minimal 2d vector datatype
def vec2d(x,y): return complex(x,y)
def rot90(v): return 1j * v
def inner_prod(u, v): return (u * v.conjugate()).real

def outward_normals(a, b, c, d):
    n1 = rot90(b - a)
    n2 = rot90(d - c)
    mid = (c + d - a - b) / 2
    if inner_prod(n1, mid) > 0:
        n1 = -n1
    if inner_prod(n2, mid) < 0:
        n2 = -n2
    return n1, n2

Note that I assume the endpoints define lines meeting the conditions in the problem. Nor do I check for the edge case when the lines have the same midpoint; the notion of “outside” doesn’t apply in that case.