Unified Representations of Nonlinear Splines

Golomb and Jerome's framework is modified and extended. The new framework is more general since it also handles interpolants which are not allowed to “slide” at the nodes. The space of interpolants of variable length is shown to be a smooth manifold. If the length is fixed, and there are no nodes, then the space of interpolants is a manifold. When there is at least one node, and at least one node is not on the line segment between the endpoints, then the space of interpolants of fixed length is a smooth manifold. Sufficient conditions are given which ensure the space of interpolants continues to be a smooth manifold in the presence of additional constraints such as clamping and pinning. A new fundamental finite-dimensional equation is derived. When it is solved it yields all nonlinear splines, and every nonlinear spline appears in this way. An important feature is that the same symbolic equation is used for all possible combinations of the constraints considered. It is shown how to take the solutions of the fundamental equation and use them to express the corresponding nonlinear splines in terms of a pair of elliptic functions. An inequality is derived that specifies which elliptic function appears along each section of the spline. The nonlinear splines are in a unified way shown to beC²for all possible combinations of the constraints considered.