C1 rational interpolation of spherical motions with rational rotation-minimizing directed frames

doi:10.1016/j.cagd.2012.05.001

Computer Aided Geometric Design

Volume 30, Issue 1, January 2013, Pages 159-173

https://doi.org/10.1016/j.cagd.2012.05.001 Get rights and content

Abstract

An interpolation method for constructing rational curves on the unit sphere with rational directed rotation-minimizing frames is presented. This type of curves is useful, for instance, in the description of smoothly varying camera motions. The proposed scheme uses as input data the initial and the final curve positions and tangents on the sphere, together with the associated end frame orientations. Both the curve and the rotation-minimizing directed frame produced by the scheme are $C^{1}$ continuous and are rational of degree 8, which, under suitable mild assumptions on angular velocity directions, can be reduced to 6 if only $G^{1}$ continuity is required.

Highlights

► A scheme for constructing rational rotation-minimizing spherical motions had been developed. ► The method is based on interpolation of initial/final curve positions, orientations, and compatible motion directions. ► The resulting curve segments possess $C^{1}$ continuity and therefore can be used to construct a $C^{1}$ spline by interpolation. ► The degree of obtained curves is usually eight, as well as the degree of the associated rational rotation-minimizing frames.

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Citation Excerpt :
Such frames have possible applications to endoscopic imaging or surgery, in which a miniature camera mounted on a flexible tube is inserted through a small incision, and directed along a curved path to image or sample tissue at a specific location in the body cavity. See [21,33] for further development of the directed RMF algorithms. Notwithstanding the considerable progress recently made in elucidating the characterization, properties, and construction algorithms for RRMF curves, the results surveyed herein indicate that significant gaps in the understanding and utilization of these curves still remain.
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Highlights

References (15)

Euler–Rodrigues frames on spatial Pythagorean-hodograph curves

Comput. Aided Geom. Design

Quintic space curves with rational rotation-minimizing frames

Comput. Aided Geom. Design

Two moving coordinate frames for sweeping along a 3D trajectory

Comput. Aided Geom. Design

Construction of rational curves with rational rotation-minimizing frames via Möbius transformations

Theoretical Kinematics

Curves and Surfaces for CAGD: A Practical Guide

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Construction of G<sup>2</sup> spatial interpolants with prescribed arc lengths

Geometric interpolation of ER frames with G<sup>2</sup> Pythagorean-hodograph curves of degree 7

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C<sup>1</sup> interpolation by rational biarcs with rational rotation minimizing directed frames

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