research-article

Computation of rotation minimizing frames

Authors:

Yang LiuAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 27, Issue 1

Article No.: 2, Pages 1 - 18

https://doi.org/10.1145/1330511.1330513

Published: 20 March 2008 Publication History

Abstract

Due to its minimal twist, the rotation minimizing frame (RMF) is widely used in computer graphics, including sweep or blending surface modeling, motion design and control in computer animation and robotics, streamline visualization, and tool path planning in CAD/CAM. We present a novel simple and efficient method for accurate and stable computation of RMF of a curve in 3D. This method, called the double reflection method, uses two reflections to compute each frame from its preceding one to yield a sequence of frames to approximate an exact RMF. The double reflection method has the fourth order global approximation error, thus it is much more accurate than the two currently prevailing methods with the second order approximation error—the projection method by Klok and the rotation method by Bloomenthal, while all these methods have nearly the same per-frame computational cost. Furthermore, the double reflection method is much simpler and faster than using the standard fourth order Runge-Kutta method to integrate the defining ODE of the RMF, though they have the same accuracy. We also investigate further properties and extensions of the double reflection method, and discuss the variational principles in design moving frames with boundary conditions, based on RMF.

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Djordjevic JNesovic E(2024)On bishop frame of a partially null curve in Minkowski space-time E41Filomat10.2298/FIL2404439D38:4(1439-1449)Online publication date: 2024
https://doi.org/10.2298/FIL2404439D
Abdel-Aziz HSerry HEl-Adawy FSaad M(2024)Geometry and evolution of Hasimoto surface in Minkowski 3-spacePLOS ONE10.1371/journal.pone.029431019:1(e0294310)Online publication date: 2-Jan-2024
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Index Terms

Computation of rotation minimizing frames
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
    2. Numerical analysis
      1. Numerical differentiation
2. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 27, Issue 1

March 2008

135 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/1330511

Issue’s Table of Contents

Copyright © 2008 ACM.

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Publication History

Published: 20 March 2008

Accepted: 01 October 2007

Revised: 01 August 2007

Received: 01 October 2006

Published in TOG Volume 27, Issue 1

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Ministry of Science and Technology of the People's Republic of China

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https://doi.org/10.2298/FIL2404439D
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