An Algorithm for Locating and Displaying the Intersection of Two Arbitrary Surfaces

Given two surfaces in three-dimensional Euclidean space R³, specified by two twice continuously differentiable functions f(p)=0 and g(p)=0, where p denotes a point in R³, the authors construct a third-order autonomous system of ordinary differential equations (ODEs), p´=Z(p), using the gradients of f and g. The domain of Z(h) is the intersection of the domains of f and g. If the surfaces defined by f=0 and g=0 have an intersection S in this domain, then S is an invariant manifold of this ODE system. That is, trajectories with initial conditions on the intersection of the surfaces flow forever along that intersection. Furthermore, the ODE system is constructed so that solution trajectories starting near S approach S asymptotically. The above construction can be used in concert with a (stiff) ODE initial-value-problem integration scheme and general-purpose software than displays three-dimensional phase portraits of autonomous ODE systems to draw out the loci where any two surfaces intersect. This technique is used to display the 3-D shape of a single surface of interest, f(p)=0.