Interactive Control of Geometric Shape Morphing based on Minkowski Sum

민코프스키 덧셈 연산에 근거한 기하 도형의 모핑 제어 방법

  • Lee, J.-H. ;
  • Lee, J. Y. ;
  • Kim, H. ;
  • Kim, H. S.
  • Published : 2002.12.01

Abstract

Geometric shape morphing is an interesting geometric operation that interpolates two geometric shapes to generate in-betweens. It is well known that Minkowski operations can be used to test and build collision-free motion paths and to modify shapes in digital image processing. In this paper, we present a new geometric modeling technique to control the morphing on geometric shapes based on Minkowski sum. The basic idea develops from the linear interpolation on two geometric shapes where the traditional algebraic sum is replaced by Minkowski sum. We extend this scheme into a Bezier-like control structure with multiple control shapes, which enables the interactive control over the intermediate shapes during the morphing sequence as in the traditional CAGD curve/surface editing. Moreover, we apply the theory of blossoming to our control structure, whereby our control structure becomes even more flexible and general. In this paper, we present mathematical models of control structure, their properties, and computational issues with examples.

Keywords

References

  1. Adobe Illustrator 9.0 Adobe Systems Inc.
  2. International Journal of Computational Geometry & Applications v.5 no.4 Computing Minkowski Sums of Plane Curves A. Kaul;R. T. Farouki https://doi.org/10.1142/S0218195995000258
  3. Discrete Geometric Shapes: Matching, Interpolation, and Approximation in Handbook of Computational Geometry H. Alt;L.J. Cuibas;J.-R. Sack(eds.);J. Urrutia(eds.)
  4. Computational Geometry v.21 no.1;2 Polygon decomposition for efficient construction of Minkowski sums P.K. Agarwal;E. Flato;D. Halperin https://doi.org/10.1016/S0925-7721(01)00041-4
  5. ACM Transactions on Graphics v.17 no.2 Three-Dimensional Distance Field Metamorphosis D. Cohen-Or;D. Levein, D;A. Solomovici https://doi.org/10.1145/274363.274366
  6. Curves and Surfaces for CAGD(5th edition) G. Farin
  7. Computer Graphics Forum v.15 no.3 Blob Metamorphosis based on Minkowski Sums E. Galin;S. Akkouche https://doi.org/10.1111/1467-8659.1530143
  8. CVGIP v.44 A Mathematical Model for Shape Description Using Minkowski Operators P.K. Ghosh
  9. Mathematical Methods for Curve and Surfaces Spatial Rational Motions and Their Application in Computer Aided Geometric Design B. Juttler;M. Daehlen(Eds.);T. Lyche(Eds.);L.L. Schumaker(Eds.)
  10. GMIP v.60 no.2 Polynomial/Rational Approximation of Minkowski Sum Boundary Curves I.-K. Lee;M. S. Kim;G. Elber
  11. The Visual Computer v.16 no.3;4 Polygonal boundary approximation for a 2D general sweep based on envelope and Boolean operations J.-H. Lee;S.J. Hong;M.-S. Kim https://doi.org/10.1007/s003710050209
  12. Technical Reports, ETRI Notes on Minkowski Sum and Difference Operations: Properties and Applications J.-H. Lee
  13. EUROGRAPHICS '94 AGRELs and BIPs: Metamorphosis as a Bezier curve in the space of polyhedra J. Rossignac;A. Kaul
  14. SIGGRAPH 99 Shape Transformation Using Variational Implicit Functions G.Turk;J. O'Brien
  15. Warping and Morphing of Graphical Objects J. Gomes;L. Darsa;B. Costa;L. Velho