Λ-Spline Curves With Range Dimension d

Abstract

All forms of splines we shall consider for 2D-functions and 3D-space curves will be Λ-splines as now introduced. Let Λ be a class of vector- valued functions which mapRto R^d, such that no two functions in Λ are identical on [r ₀, r _n−1]. Given the real values r₀ ≤ r₁ ≤ … ≤ r_{n −1}, the parametric function s(t) defined on the interval [r₀, r_{n −1}] is called a Λ- spline of join order k with respect to r₀, r₁,…, r_{n −1} if s(t) = g _i (t − _r i−1) for r_{i −1} ≤ t ≤ r _i and 1 ≤ i ≤ n − 1, where each function g _i is a function in Λ, and if s is C^k continuous at r₁ … , r_{n −2}, i.e., s is continuous at r₁,…, r_{n −2} and s has at least k successive derivatives that are continuous at r₁,… , r_{n −2}. The Λ-spline function s is also said to be C^k-joined at the points s(r₁),… , s(r_{n −2}), called the join points of s.