“Design of solids with free-form surfaces” by Chiyokura and Kimura

  • ©Hiroaki Chiyokura and Fumihiko Kimura




    Design of solids with free-form surfaces



    We propose a unified method of generating a wide range of three dimensional objects from polyhedra to solids with free-form surfaces. Modeling systems for polyhedra and systems for free-form surfaces have been developed independently in the past because of the difference in their underlying theory and practices. However, this is not desirable for a designer. So in this paper, we have shown a method in which a wide range of shapes are generated in one system by using local modifications. Local modifications are procedures used to change the shape of solids locally. The construction and the modification of three dimensional shapes by these procedures are natural and easy for a designer in many cases. The implementation of these procedures in a computer is easy and their execution does not require much time. Our method to construct a solid with free-form surfaces consists of following three phases. 1) A solid which serves as a basis of free-form shape design is generated by local modifications. Edges of this solid are straight lines but its faces are not necessarily flat planes. 2) From this model, a curve model which adequately represents the characteristics of a free-form shape is generated. 3) Surface equations interpolating over the curve model are generated. We have made a geometric modeling system MODIF. Using this system, a complicated solid with free-form surfaces can be designed easily. MODIF can generate color shaded pictures and cutter path data for making a real object model by NC machining tool.


    1. Barnhill,R.E., Brown,J.H. and Klucewicz,I.M.: A New Twist in Computer Aided Geometric Design, Computer Graphics and Image Processing 8, pp. 78-91, 1978.
    2. Baumgart,B.G.: GEOMED-A GEOMETRIC EDITOR, Stanford Artificial Intelligence Laboratory Memo AIM-232, Computer Science Department Report no. cs-414, May 1974.
    3. Bezier,P.E.: Numerical Control – Mathematics and Applications, John Wiley and Sons, London, 1972.
    4. Braid,I.C., Hillyard,R.C. and Stroud,I.A.: Stepwise Construction of Polyhedra in Geometric Modelling, Mathematical Methods in Computer Graphics and Design (Ed. by K.W.Brodlie) Academic Press, pp. 123-141, 1980.
    5. Coons,S.A.: Surfaces for Computer Aided Design of Space Forms, MIT Project MAC TR-41, June 1967.
    6. Catmull,E. and Clark,J.: Recursively Generated B-spline Surfaces on Arbitrary Topological Meshes, Computer Aided Design, pp. 350-355, vol. 10, no.6, November, 1978.
    7. Doo,D.: A Subdivision Algorithm for Smoothing Down Irregular Shaped Polyhedrons, Proc. Conf. Interactive Technique in CAD,. pp. 157-165, IEEE Computer Society 78CH1289-8C, 1978.
    8. Faux,I.D. and Pratt,M.J.: Computational Geometry for Design and Manufacture, Chapter 7, ELLIS HORWOOD LIMITED, 1979.
    9. Forrest,A.R.: On Coons and Other Methods for the Representation of Curved Surfaces, Computer Graphics and Image Processing 1, pp. 341-359, 1972.
    10. Forrest,A.R.: A Unified Approach to Geometric Modelling, SIGGRAPH 78 Proceedings, pp. 264-269, 1978.
    11. Gregory,J.A.: Smooth Interpolation Without Twist Constraints, in Computer Aided Geometric Design (R. E. Barnhill and R. F. Riesenfeld, Eds.), pp. 71-87, Academic Press, New York, 1974.
    12. Hosaka,M. and Kimura,F.: An Interactive Geometrical Design System with Handwriting Input, Information Processing 77, pp. 167-172, North-Holland, Amsterdam, 1977.
    13. Hosaka,M. and Kimura,F.: Synthesis Methods of Curves and Surfaces in Interactive CAD, Proc. Conf. Interactive Technique in CAD, pp. 151-156, IEEE Computer Society 78CH1289-8C, 1978.
    14. Requicha,A.G.: Representation for Rigid Solids: Theory, Method and Systems, Computing Surveys, vol.12, no.4, pp. 437-464, December 1980.
    15. Riesenfeld,R.F.: Applications of B-spline Approximation to Geometric Problems of Computer Aided Design, Ph.D. thesis, Syracuse University, May 1973.

ACM Digital Library Publication:

Overview Page: