Efficient Raster-Graphing of Bivariate Functions by Incremental Methods

  • Wolfgang G. Kramarczyk
Conference paper
Part of the NATO ASI Series book series (volume 40)


The graph of a bivariate function z=f(x,y) is a surface in a three-dimensional space. Suppose that we have ascribed discrete intensities or pseudo-colors i=1,2,..,ncolors to the graph in such a manner that
$$i(x,y) = l + \left[ {\frac{{f(x,y) - b}}{{\Delta z}}} \right],$$
where [ ] means the nearest integer function, Δz is a positive discretization step and b is a lower bound of f(x,y) on a rectangle with opposite corners at (xmin,Ymin) and (xmax,Ymax). i=0 is reserved for the background color. Suppose further that we have sampled these intensities on a grid (xk,y1), where
$$ \begin{array}{*{20}{c}} {{x_k} = {x_{\min }} + k\Delta x{\text{ }},k = o,1, \ldots ,{n_{cols}} - 1,{\text{ }}\Delta x = \frac{{{x_{\max }} - {x_{\min }}}}{{{n_{cols}} - 1}},} \\ {{y_1} = {y_{\min }} + 1\Delta y{\text{ }},1 = o,1, \ldots ,{n_{rows}} - 1,\Delta y = \frac{{{y_{\max }} - {y_{\min }}}}{{{n_{rows}} - 1}},} \end{array} $$
and stored them in a two-dimensional array. Such an array may be considered to be a raster-image of the graph viewed from the top. Let us call it the source image.


Source Image Vertical Segment Horizontal Coordinate Bivariate Function Orthographic Projection 
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. [l]
    Müller S.C., Plesser Th., Hess B., Two-Dimensional Spectrophotometry and Pseudo-Color Representation of Chemical Reaction Patterns, Naturwissenschaften 73 (1986) 165–179CrossRefGoogle Scholar
  2. [2]
    MÜller S.C., Plesser Th., Hess B., Three-Dimensional Representation of Chemical Gradients, Biophys. Chem. 26 (1987) 357–365CrossRefGoogle Scholar
  3. [3]
    Anderson D.P., Hidden Line Elimination in Projected Grid Surfaces, ACM Trans. Graph. 1 (1982) 274–288CrossRefGoogle Scholar
  4. [4]
    Boiler D.J., Efficient Hidden Line Removal for Surface Plots Utilising Raster Graphics, in Fundamental Algorithms for Computer Graphics, edited Earnshaw R.A. (Springer-Verlag, 1985), pp. 603–615CrossRefGoogle Scholar
  5. [5]
    Bresenham J.E., Algorithm for Computer Control of a Digital Plotter, IBM Systems J. 4 (1965) 25–30CrossRefGoogle Scholar
  6. [6]
    Skala V., Hidden-Line Processor, CSTR/209-03-84, Computer Science Dept., Technical University, Plzeñ 1984Google Scholar
  7. [7]
    Skala V., An Interesting Modification to the Bresenham Algorithm for Hidden-Line Solution, in Fundamental Algorithms for Computer Graphics, edited Earnshaw R.A. (Springer-Verlag, 1985), pp. 593–601CrossRefGoogle Scholar
  8. [8]
    Rogers D.F., Procedural Elements for Computer Graphics (Mc Graw -Hill 1985)Google Scholar
  9. [9]
    Watkins S.L., Algorithm 483, Masked Three-Dimensional Plot Program with Rotations, Comm. ACM 17 (1974) 520–523CrossRefGoogle Scholar
  10. [10]
    Field D., Incremental Linear Interpolation, ACM Trans, Graph, 4 (1985) 1–11CrossRefMATHMathSciNetGoogle Scholar
  11. [11]
    Isner J.F., A Fortran Programming Methodology Based on Data Abstraction, Comm, ACM 25 (1982) 686–697CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Wolfgang G. Kramarczyk
    • 1
  1. 1.Max-Planck-Institut für ErnährungsphysiologieDortmund 1Federal Republic of Germany

Personalised recommendations