Improved test for closed loops in surface intersections

The following two theorems are proven: If two nonsingular curve segments intersect twice, if the range of normal vectors of each segment does not vary more than 90°, and if each curve is C¹ smooth, then there exists a line which is perpendicular to both curve segments simultaneously. If two nonsingular surface patches intersect in a closed loop, if the dot product between any normal vector on one surface and any other normal vector on either surface is never zero, and if the normal vector is uniquely defined at every point on each surface region, then there exists a line which is perpendicular to both surface patches simultaneously. The second theorem is of value because it provides a criterion for eliminating a robustness limitation that arises in computing surface intersections using the marching method; namely, assuring that all branches of the intersection curve have been found. If all lines are computed that are perpendicular to both patches, and the patches are subdivided at the points at which those lines intersect the patches, then there can be no closed loops in the intersection between the set of patches obtained by subdividing the first patch and the set patches obtained by subdividing the second patch.