The shortest line between two lines in 3D (based on http://paulbourke.net/geometry/pointlineplane/ (Paul Bourke 1988))

nearestPtsBetweenLines.py

# [START] The shortest line between two lines in 3D ###
#
# credits to the following sources:
#   - http://paulbourke.net/geometry/pointlineplane/   (Paul Bourke 1988)
#   - code from  http://paulbourke.net/geometry/pointlineplane/source.mel   (Dan Wills [dan@rsp.com.au] 2003)

def d_function(m, n, o, p):
    return ((m.x - n.x) * (o.x - p.x) + (m.y - n.y) * (o.y - p.y) + (m.z - n.z) * (o.z - p.z))

def interpolate(a, b, factor):
    return b * factor + (1 - factor) * a

def nearestPtsBetweenLinesPts(lineA_pt0, lineA_pt1, lineB_pt0, lineB_pt1, mustLieOnSegments=True):
    p1, p2, p3, p4 = lineA_pt0, lineA_pt1, lineB_pt0, lineB_pt1
    mu_A = (d_function(p1, p3, p4, p3) * d_function(p4, p3, p2, p1) - d_function(p1, p3, p2, p1) * d_function(p4, p3, p4, p3)) / \
           (d_function(p2, p1, p2, p1) * d_function(p4, p3, p4, p3) - d_function(p4, p3, p2, p1) * d_function(p4, p3, p2, p1))
    mu_B = (d_function(p1, p3, p4, p3) + mu_A * d_function(p4, p3, p2, p1)) / d_function(p4, p3, p4, p3)

    # clamp the values between [0..1], constraining the solution to lie on the original curves
    if mustLieOnSegments and ((mu_A < 0 or mu_A > 1) or (mu_B < 0 or mu_B > 1)):
        #mu_A = clamp(mu_A, 0, 1)
        #mu_B = clamp(mu_B, 0, 1)
        raise RuntimeError("[nearestPtsBetweenLinesPts] No constrained solution found.")

    # get the actual points by linearly interpolating
    pt_A = interpolate(p1, p2, mu_A)
    pt_B = interpolate(p3, p4, mu_B)

    return pt_A, pt_B

def nearestPtsBetweenLines(lineA, lineB, mustLieOnSegments=True, **kwargs):
    linesPts = convertToPts((lineA.ep[0], lineA.ep[-1], lineB.ep[0], lineB.ep[-1]))
    pt_A, pt_B = nearestPtsBetweenLinesPts(*linesPts, mustLieOnSegments=mustLieOnSegments)
    return pt_A, pt_B

def intersectLinesWrapper_Bourke(crv1, crv2, **kwargs):
    res = nearestPtsBetweenLines(crv1, crv2, **kwargs)
    return res[0]

# [END] The shortest line between two lines in 3D ###