Haptic Exploration of Mathematical Knots

  • Hui Zhang
  • Sidharth Thakur
  • Andrew J. Hanson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4841)


We present a novel multi-modal haptic interface for sketching and exploring the structure and properties of mathematical knots. Our interface derives from the familiar pencil-and-paper process of drawing 2D knot diagrams to facilitate the creation and exploration of mathematical knots; however, with a touch-based interface, users can also leverage their physical intuition by seeing, touching, and feeling the knots. The pure haptic component provides an intuitive interaction model for exploring knots, focusing on resolving the apparent conflict between the continuous structure of the actual knot and the visual discontinuities at occlusion boundaries. The auditory component adds redundant cues that emphasize the traditional knot crossings, where the haptic proxy crosses a visual disruption in the graphics image. Our paradigm enhances and extends traditional 2D sketching methods by exploiting both touch and sound to assist in building clearer mental models of geometry such as knot structures.


Force Feedback Haptic Device Curve Segment Default Representation Haptic Interface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Livingston, C.: Knot Theory. In: Livingston, C. (ed.) The Carus Mathematical Monographs. Mathematical Association of America, Washington, U.S, vol. 24 (1993)Google Scholar
  2. 2.
    Scharein, R.G.: Interactive Topological Drawing. PhD thesis, Department of Computer Science, The University of British Columbia (1998)Google Scholar
  3. 3.
    Cohen, J., Markosian, L., Zeleznik, R., Hughes, J., Barzel, R.: An interface for sketching 3d curves. In: SI3D 1999. Proceedings of the 1999 symposium on Interactive 3D graphics, pp. 17–21. ACM Press, New York, NY, USA (1999)CrossRefGoogle Scholar
  4. 4.
    Snibbe, S., Anderson, S., Verplank, B.: Springs and constraints for 3d drawing. In: Proceedings of the Third Phantom Users Group Workshop, Dedham, MA (1998)Google Scholar
  5. 5.
    Brown, J., Latombe, J.C., Montgomery, K.: Real-time knot-tying simulation. The Visual Computer 20, 165–179 (2004)CrossRefGoogle Scholar
  6. 6.
    Wang, F., Burdet, E., Dhanik, A., Poston, T., Teo, C.L.: Dynamic thread for real-time knot-tying. In: WHC 2005. Proceedings of the First Joint Eurohaptics Conference and Symposium on Haptic Interfaces for Virtual Environment and Teleoperator Systems, pp. 507–508. IEEE Computer Society Press, Washington, DC, USA (2005)Google Scholar
  7. 7.
    Phillips, J., Ladd, A.M., Kavraki, L.E.: Simulated knot tying. In: ICRA, pp. 841–846. IEEE Computer Society Press, Los Alamitos (2002)Google Scholar
  8. 8.
    Hanson, A.J., Zhang, H.: Multimodal exploration of the fourth dimension. In: Proceedings of IEEE Visualization, pp. 263–270. IEEE Computer Society Press, Los Alamitos (2005)CrossRefGoogle Scholar
  9. 9.
    Baxter, W.V., Scheib, V., Lin, M.C.: dAb: Interactive haptic painting with 3D virtual brushes. In: Fiume, E. (ed.) SIGGRAPH 2001. Computer Graphics Proceedings, ACM SIGGRAPH, pp. 461–468. ACM Press, New York (2001)CrossRefGoogle Scholar
  10. 10.
    Okamura, A.M.: Haptic Exploration of Unknown Objects. PhD thesis, Stanford University, Department of Mechanical Engineering, California, USA (2000)Google Scholar
  11. 11.
    Okamura, A.M., Cutkosky, M.R.: Feature detection for haptic exploration with robotic fingers. The International Journal of Robotics Research 20, 925–938 (2001)CrossRefGoogle Scholar
  12. 12.
    Kim, L., Sukhatme, G., Desbrun, M.: A haptic-rendering technique based on hybrid surface representation. IEEE Comput. Graph. Appl. 24, 66–75 (2004)Google Scholar
  13. 13.
    Yu, W., Ramloll, R., Brewster, S.: Haptic graphs for blind computer users. In: Brewster, S., Murray-Smith, R. (eds.) Haptic Human-Computer Interaction. LNCS, vol. 2058, pp. 41–51. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  14. 14.
    Zahariev, M.A., MacKenzie, C.L.: Auditory, graphical and haptic contact cues for a reach, grasp, and place task in an augmented environment. In: ICMI 2003. Proc. of 5th Intl. Conf on Multimodal Interfaces, pp. 273–276. ACM Press, New York (2003)CrossRefGoogle Scholar
  15. 15.
    Kennedy, J.M.: Optics and haptics: The picture. In: The conference on Multimodality of Human Communication: Theory, Problems and Applications, University of Toronto (2002)Google Scholar
  16. 16.
    Forsyth, B.: Intelligent support of interactive manual control: Design, implementation and evaluation of look-ahead haptic guidance. Master’s thesis, The University of British Columbia (2004)Google Scholar
  17. 17.
    Park, J.G., Niemeyer, G.: Haptic rendering with predictive representation of local geometry. In: HAPTICS, pp. 331–338 (2004)Google Scholar
  18. 18.
    Reynolds, C.W.: Steering behaviors for autonomous characters. In: The proceedings of the 1999 Game Developers Conference, pp. 763–782 (1999)Google Scholar
  19. 19.
    SensAble, Inc.: 3D Touch SDK OpenHaptics Toolkit Programmer’s Guide (2004)Google Scholar
  20. 20.
    Larsson, T., Akenine-Möller, T.: Collision detection for continuously deforming bodies. In: Eurographics 2001. Short Presentations, Manchester, Eurographics Association 325–333 (2001)Google Scholar
  21. 21.
    Gottschalk, S., Lin, M.C., Manocha, D.: OBBTree: A hierarchical structure for rapid interference detection. Computer Graphics 30, 171–180 (1996)Google Scholar
  22. 22.
    Haeberli, P.: Dynadraw. Grafica OBSCURA (1989)Google Scholar
  23. 23.
    Hanson, A.J., Ma, H.: Space walking. In: Proceedings of Visualization 1995, pp. 126–133. IEEE Computer Society Press, Los Alamitos (1995)CrossRefGoogle Scholar
  24. 24.
    Kamada, T., Kawai, S.: A simple method for computing general position in displaying three-dimensional objects. Computer Vision, Graphics, and Image Processing 41, 43–56 (1988)CrossRefGoogle Scholar
  25. 25.
    Hlavác, V., Leonardis, A., Werner, T.: Automatic selection of reference views for image-based scene representations. In: ECCV (1), pp. 526–535 (1996)Google Scholar
  26. 26.
    Vázquez, P.P., Feixas, M., Sbert, M., Heidrich, W.: Viewpoint selection using viewpoint entropy. In: VMV 2001. Proceedings of the Vision Modeling and Visualization Conference 2001, Aka GmbH, pp. 273–280 (2001)Google Scholar
  27. 27.
    Vázquez, P.P., Feixas, M., Sbert, M., Llobet, A.: Viewpoint entropy: a new tool for obtaining good views of molecules. In: VISSYM 2002. Proceedings of the symposium on Data Visualisation 2002, Aire-la-Ville, Switzerland, Switzerland, Eurographics Association, pp. 183–188 (2002)Google Scholar
  28. 28.
    Gray, A.: Modern Differential Geometry of Curves and Surfaces. CRC Press, Inc., Boca Raton, FL (1993)zbMATHGoogle Scholar
  29. 29.
    Bishop, R.L.: There is more than one way to frame a curve. American Mathematical Monthly 82, 246–251 (1975)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Hui Zhang
    • 1
  • Sidharth Thakur
    • 1
  • Andrew J. Hanson
    • 1
  1. 1.Computer Science Department, Indiana University 

Personalised recommendations