Interpolating Solid Orientations with a C2-Continuous B-Spline Quaternion Curve

  • Wenbing Ge
  • Zhangjin Huang
  • Guoping Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4469)

Abstract

An algorithm is presented to construct a C2-continuous B-spline quaternion curve which interpolates a given sequence of unit quaternions on the rotation group SO(3). We present a method to extend a B-spline interpolation curve to SO(3). The problem is essentially to find the quaternion control points of the quaternion B-spline interpolation curve. Although the associated constraint equation is non-linear, we can get the accurate quaternion control points according to two additional rules for quaternion computations in S3. In addition, we provide a point insertion method to construct interpolation curves that have local modification property. The effectiveness of the algorithm is verified by applying it to some examples.

Keywords

C2-continuous interpolation quaternion  B-spline curve computer animation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barr, A., Currin, B., Gabril, S., Hughes, J.: Smooth interpolation of orientations with angular velocity constraints using quaternions. In: Computer Graphics (Proc. of SIGGRAPH’92), pp. 313–320 (1992)Google Scholar
  2. 2.
    Dyn, N., Levin, D., Gregory, J.: 4-point interpolatory subdivision scheme for curve design. Computer Aided Geometric Design 4, 257–268 (1988)CrossRefGoogle Scholar
  3. 3.
    Gregory, M.N.: v-Quaternion Splines for the Smooth Interpolation of Orientations. IEEE Transactions on Visualization and Computer Graphics 10(2) (2004)Google Scholar
  4. 4.
    Lee, J., Shin, S.Y.: General Construction of Time-Domain Filters for Orientation Data. IEEE Transactions on Visualization and Computer Graphics 8(2) (2002)Google Scholar
  5. 5.
    Schlag, J.: Using geometric constructions to interpolate orientation with quaternions. In: Graphics GEMS II, pp. 377–380. Academic Press, London (1992)Google Scholar
  6. 6.
    Shoemake, K.: Animating rotation with quaternion curves. In: Computer Graphics (Proc. of SIGGRAPH’85), pp. 245–254 (1985)Google Scholar
  7. 7.
    Kenjiro, T.M.: Unit quaternion integral curve: A new type of fair free-form curves. Compueter AIded Geometric Design 17, 39–58 (2000)CrossRefGoogle Scholar
  8. 8.
    Kim, M.J., Kim, M.S.: a C2-continous B-spline Quaternion Curve Interpolating a Given Sequence of Solid Orientations. In: Computer Animation ’95. Proceedings (1995)Google Scholar
  9. 9.
    Kim, M.J., Kim, M.S., Shin, S.Y.: A General Construction Scheme for Unit Quaternion Curves with Simple High Order Derivatives. In: Computer Graphics (Proceedings of SIGGRAPH 95) vol. 29, pp. 369-376 (August 1995)Google Scholar
  10. 10.
    Kim, M.S., Nam, K.W.: Interpolation solid orientations with circular blending quatern-ion curves. Computer-Aided Design 27(5), 385–398 (1995)MATHCrossRefGoogle Scholar
  11. 11.
    Ramamoorthi, R., Alan, H.B.: Fast construction of accurate quaternion splines. In: Proceedings of the 24th annual conference on Computer graphics and interactive techniques, pp. 287-292 (1997)Google Scholar
  12. 12.
    Samuel, R.B., Jay, P.F.: Spherical averages and applications to spherical splines and interpolation. ACM Transactions on Graphics 20(2) (2001)Google Scholar
  13. 13.
    Wang, W., Joe, B.: Orientation interpolation in quaternion space using spherical biarcs. In: Proc. Of Graphics Interface’93, pp. 23–32 (1993)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Wenbing Ge
    • 1
  • Zhangjin Huang
    • 1
  • Guoping Wang
    • 1
  1. 1.Dept. of Computer Science and Technology, Peking University, BeijingChina

Personalised recommendations