In your example with the cuboid, let's say the curve goes along the line defined by y=2, x=0 [recall that y=2 defines a plane; you need another equation to restrict that down to a line]. Given a point, say, (5, 15, 2), find a point on the line which must be of the form (0, 2, z) for any z such that the tangent to the line at (0, 2, z) is orthogonal to the line connecting (0, 2, z) and (5, 15, 2). All tangents to this particular line are (0, 0, 1), since the line is increasing along the z axis.

Then we have the condition that D = (5, 15, 2) - (0, 2, z) = (5, 13, 2-z) is perpendicular to T = (0, 0, 1). When does this happen? When D dot T = 0, or

(5, 13, 2-z)*(0, 0, 1) = 5*0+13*0+(2-z)*1 = 2-z = 0 -> z = 2.

Thus we check the point (0, 2, 2) assuming my algebra is miraculously good at 4am.


Let's say that the square which is swept along the z axis is actually a circle of radius 4, with the center of the circle at (0, 2, z) as it moves along the path. Then the distance between (5, 15, 2) and (0, 2, 2) [about 14] is the radius between the cross section's center and the point in question. Since 14 > 4, we must be outside of the circular cross section, meaning this point must be outside of the volume.

Note that this assumes the loop is actually swept through (0, 2, 2) [in fact, I checked all z]. If I were better at drawing I'd make a picture, but I do think this method works.

I'd be happy to clear up any confusion as it arises.