Generation of spherical non-uniform rational basis spline curves and its application in five-axis machining

Abstract

A method of generating spherical non-uniform rational basis spline curves based on De Boor’s algorithm is presented in this article. The spherical curve preserves many good properties from non-uniform rational basis spline curves in Euclidean space, such as local modification property, convex hull property, rotation invariant property, knot insertion property, and so on. Construction of closed spherical non-uniform rational basis spline curve will be discussed too. Furthermore, a progressive iterative approximation scheme is proposed to approximate the given points on unit sphere using spherical non-uniform rational basis spline curves. The convergence and curve continuity will be discussed. Since the analytical expression of the spherical non-uniform rational basis spline curve is messy, numerical differentiation is used to test its continuity at interior knots. Finally, several examples are presented to verify the effectiveness of the proposed method. The proposed method is applied on tool path generation of five-axis machining to avoid interference by adjusting fewer directions and obtain smooth tool path simultaneously.

GLS‐PIA: n‐Dimensional Spherical B‐Spline Curve Fitting based on Geodesic Least Square with Adaptive Knot Placement

“…Crouch extended the De Casteljau algorithm to S n manifolds and Lie groups [CKL99]. Huang used the method of knot insertion with De Boor to calculate NURBS curves on the sphere [HGFL15]. This method directly utilizes the properties of the sphere manifold and achieves direct and efficient spherical spline construction computation.…”

“…Data become points on spheres after nomalization. Therefore, B‐spline curves on spherical surfaces of different dimensions are essential tools for solving a wide range of problems, including spherical curve modeling, rigid motion description, and temporal data description in shape space [AMAS16, Sho85, HGFL15, SDK*12, KDLS20, HLGQ05]. Specifically, B‐spline curves on S 2 are applied in the field of spherical modeling, such as fitting Earth remote sensing data and constructing vector maps of the Earth [AMAS16, AS19].…”

“…B‐spline curves on S 3 are applied in describing inertial navigation, particularly in the description of rigid motion. Rigid motion includes two parts: translation and rotation, and the rotation motion can be described using curves on S 3 [Sho85, HGFL15]. B‐spline curves on S n are used to describe temporal data in shape space [SDK*12].…”