Advertisement

Advanced Applications of Theory of Surfaces

  • Mamoru Hosaka
Part of the Computer Graphics — Systems and Applications book series (COMPUTER GRAPH.)

Abstract

Special topics on the geometric properties of surfaces are treated in this chapter. They are developed from the fundamental theory of surfaces which has been explained in the previous chapter. Knowledge of these topics is useful for treating free-form surfaces in advanced problems. First we discuss the umbilics and lines of curvature. On a free-form surface, there are points and regions which have the special characters inherent to its shape, and whose locations do not depend on the coordinate system adopted. The lines of curvature make orthogonal nets on a surface, and the pattern they form exhibits inherent features of the surface. The umbilics are singular points, or curves or regions seeing from the lines of curvature. On the free-form surface the umbilics appear more frequently than our expectation. There are other curves on the surface which depend not only on its inherent features, but also on its orientation with respect to its observers or its environments or its surface physical properties. These curves are useful for describing or evaluating the objects from engineering and aesthetic criteria.

Keywords

Principal Curvature Characteristic Curf Intersection Curve Advance Application Contour Curve 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Coxeter, H.S.M.: Introduction to geometry. 2nd edition, John Wiley 1969Google Scholar
  2. [2]
    Hosaka, M.: *Theory of curve and surface synthesis and their smooth fitting. J. IPS Japan 10 (3): 121–131, 1969Google Scholar
  3. Hosaka, M.: English abstract, Information Processing in Japan 9: 60–68, 1969MathSciNetMATHGoogle Scholar
  4. [3]
    Enomoto, H. et al.: *Computer experiment on global properties of structure lines of images using graphic display and its consideration). J. IPS Japan 17 (7): 641–649, 1976Google Scholar
  5. [4]
    Kajiya, J.T.: Ray tracing parametric patches. Computer Graphics (Proc. Siggraph’82) 16 (3): 224–254, 1984Google Scholar
  6. [5]
    Sederberg, T.W., Anderson, D.C., Goldman, R.N.: Implicit representation of parametric curves and surfaces. Computer Vision, Graphics and Image Processing 28: 72–84, 1984CrossRefGoogle Scholar
  7. [6]
    Goldman, R.N., Sederberg, T.W., Anderson, D.C.: Vector elimination: a technique for implicitization, inversion and intersection of planar parametric rational polynomial curves. Computer Aided Geometric Design 1 (4): 327–356, 1984MATHCrossRefGoogle Scholar
  8. [7]
    Sederberg, T.W.: Planar piecewise algebraic curves. Computer Aided Geometric Design 1 (3): 241–255, 1984MATHCrossRefGoogle Scholar
  9. [8]
    Poeschle, R.: Detecting surface irregularities using isophotes. Computer Aided Geometric Design 1 (2): 163–168, 1984CrossRefGoogle Scholar
  10. [9]
    Tiller, W., Hanson, E.G.: Offsets of two-dimensional profiles. IEEE Computer Graphics and Applications 9: 36–46, 1984CrossRefGoogle Scholar
  11. [10]
    Satterfield, S.D., Rogers, D.F.: A procedure for generating contour lines from a B spline surface. IEEE Computer Graphics and Applications 4: 71–75, 1985CrossRefGoogle Scholar
  12. [11]
    Farouki, R., Rajan, V.: Exact offset procedures for simple solids. Computer Aided Geometric Design 2 (4): 257–294, 1985MATHCrossRefGoogle Scholar
  13. [12]
    Beck, J.M., Farouki, R.T., Hind, J.K.: Surface analysis methods. IEEE Computer Graphics and Applications 6 (12): 18–36, 1986CrossRefGoogle Scholar
  14. [13]
    Klok, F.: Two moving coordinate frames for sweeping along 3D trajectory. Computer Aided Geometric Design 3 (3): 217–229, 1986MathSciNetMATHCrossRefGoogle Scholar
  15. [14]
    Farouki, R.: The approximation of non-degenerate offset surface. Computer Aided Geometric Design 2 (1): 15–44, 1986CrossRefGoogle Scholar
  16. [15]
    Rossignac, J.R., Requicha, A.A.G.: Offsetting operations in solid modelling. Computer Aided Geometric Design 3 (2): 129–148, 1986MATHCrossRefGoogle Scholar
  17. [16]
    Stoer, J., Bulirsch, R.: Introduction to numerical analysis. New York: Springer-Verlag 1980Google Scholar
  18. [17]
    Press, W. H., Flannery, B.P., Teukolsky, S.A., Vetterling W.T.: Numerical Recipies in C Cambridge: Cambridge University Press 1988Google Scholar
  19. [18]
    Higashi, M., Kushimoto, T., Hosaka, M.: On formulation and display for visualizing features and evaluating quality of free-form surfaces. In: Vandoni, C.E., Duce, D.A.(eds): Proc. Eurographics’90 1990, pp. 299–309Google Scholar
  20. [19]
    Love, A.E.H.: Treatise on the mathematical theory of elasticity. Cambridge: Cambridge University Press 1934, pp. 401–410Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Mamoru Hosaka
    • 1
  1. 1.Tokyo Denki UniversityChiyoda-ku, TokyoJapan

Personalised recommendations