By examining the derivation of the equations in the preceding section, we can interpret the stationary point condition in terms of the concept of collinear normal points. This idea is used in [376,209,210] to detect closed loops in surface-surface intersections (see (5.100)). Simply stated, two points on two surfaces are said to be collinear normal points if their associated normal vectors lie on the same line.
It is possible to interpret the conditions in (7.15) geometrically. Notice that the line joining and must be orthogonal to the two partial derivative vectors on at and to the two partial derivative vectors on at . As long as these derivatives do not degenerate, they span the tangent planes to the two surfaces. Therefore the normals to these two surfaces must lie on the line joining the two points, and and are collinear normal points as long as (7.15) are satisfied. The difference between (5.100) and (7.15) is that when two surfaces transversally intersect, i.e. = and at the points of intersection, the first set of equations of (5.100) are not satisfied, while (7.15) are automatically satisfied.
Similar geometrical interpretations may be made for other distance problems. In the point-surface distance problem, (7.9) states that for a point to be a stationary point, both partial derivatives to the surface at that point must be orthogonal to the line joining and . If these derivatives do not degenerate, an equivalent statement is that the normal to the surface at is collinear with the line joining the point and the surface. In other words point is an orthogonal projection of onto a surface . Pegna and Wolter [306] derived a set of differential equations to orthogonally project a space curve onto parametric surfaces as well as implicit surfaces and solved them efficiently as an initial value problem. These methods were also applied in [3] for inspection of sculptured surfaces.