CHAPTER VI - Nonlinear Splines

Abstract

This chapter focuses on nonlinear splines. Particularly, the parametric cubic spline curve of cumulative chord length is one kind of large-deflection spline that is used in geometric design and has the advantages of simple computation and shape fairness. The splines discussed in the chapter do not form linear spaces; even their continuity equations are nonlinear. For that reason, they are called nonlinear spline curves. The main property of these curves is their non-linearity. The second property of these curves is their clear geometric meaning. An interpolatory spline that satisfies various conditions is defined as a geometric spline: (1) between any two adjacent knots, it is a Cornu spiral, and (2) the whole spline is GC2-continuous, that is, its tangent line and curvature are continuous at the joints. The reason why such splines are called geometric splines is that their equations consist only of geometric quantities such as curvature and arc length and thus, they differ from the splines represented by functional equations in coordinates system.