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Theoretical and Linguistic Methods for Describing Straight Lines

  • Conference paper
Fundamental Algorithms for Computer Graphics

Part of the book series: NATO ASI Series ((SSE,volume 17))

Abstract

Two distinct approaches exist for the generation of a straight line in an arbitrary direction on a lattice, structural algorithms and conditional algorithms. A survey of these approaches is included in this paper.

Structural algorithms are of great theoretical value. From numbertheory the Farey-series can be used for directions on a lattice with a rational slope. With continued fractions it is also possible to approximate irrational numbers.

Knowing these properties it is possible to determine the set of all lines corresponding to a given chaincode for a segment of a line. This is useful for the estimation of properties of a line like length, slope and intercept. Research of this kind is related to pattern analysis and picture processing.

The structural algorithm can be presented by linguistic methods, for instance a context-free programmed grammar and a TOL-grammar, a variant of the Lindenmayer grammar with the important property that in each derivation each relevant symboltype in a string is rewritten at the same time by the same rule.

The principle of the conditional algorithm is more practical: given two points that determine the line, connected points on the lattice with minimal distance to the real line have to be selected. This method is very important for plotters and computer graphics displays.

The conditional algorithm can be presented by linguistic methods too, by using a programmed grammar with a tail. The tail has a bookkeeping function. Consequentially the grammar is context-sensitive.

Structural and conditional methods both generate lines satisfying the chord property, which is a conditional property. A structural property of a digitized straight line is spacing the least occurring type of chainelement as uniformly as possible. It is shown that this can be built into a conditional method. So an integration between both methods is achieved.

Finally some remarks on progress in the science of line drawing are made.

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Author information

Authors and Affiliations

  1. Department of Economic Affairs and Planning, Catholic University of Nijmegen, 6500 HC, Nijmegen, The Netherlands

    Reyer Brons

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  1. Reyer Brons
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Editor information

Editors and Affiliations

  1. University Computing Service, University of Leeds, LS2 9JT, Leeds, UK

    Rae A. Earnshaw

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© 1985 Springer-Verlag Berlin Heidelberg

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Brons, R. (1985). Theoretical and Linguistic Methods for Describing Straight Lines. In: Earnshaw, R.A. (eds) Fundamental Algorithms for Computer Graphics. NATO ASI Series, vol 17. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-84574-1_1

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  • DOI: https://doi.org/10.1007/978-3-642-84574-1_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54397-8

  • Online ISBN: 978-3-642-84574-1

  • eBook Packages: Springer Book Archive

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