Abstract
Bresenham's algorithm minimizes error in drawing lines on integer grid points; leap year calculations, surprisingly, are a generalization. We compare the two calculations, explicate the pattern, and discuss the connection of the leap year/line pattern with integer division and Euclid's algorithm for computing the greatest common divisor.
- Allouche, J.-P. and Shallit, J. 2003. Automatic Sequences. Cambridge University Press, Cambridge, U.K.Google Scholar
- Berstel, J. 1990. Tracé de droites, fractions continues et morphismes itérés. In Mots: Mélanges Offerts à M.-P. Schützenberger, M. Lothaire, Ed. Editions Hermès, Paris, France, 298--309.Google Scholar
- Bresenham, J. E. 1965. Algorithm for computer control of a digital plotter. IBM Syst. J. 4, 1, 25--30.Google Scholar
- Brons, R. 1974. Linguistic methods for the description of a straight line on a grid. Comput. Graph. Image Process. 3, 1, 48--62.Google Scholar
- Castle, C. M. A. and Pitteway, M. L. V. 1987. An efficient structural technique for encoding 'best-fit' straight lines. Comput. J. 30, 2, 168--175. Google Scholar
- Fraenkel, A. S., Mushkin, M., and Tassa, U. 1978. Determination of {n θ} by its sequence of differences. Can. Math. Bull. 21, 441--446.Google Scholar
- Graham, R. L., Knuth, D. E., and Patashnik, O. 1994. Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA. Google Scholar
- Knuth, D. E. 1998. The Art of Computer Programming (Volume 2: Seminumerical Algorithms), 3rd ed. Addison-Wesley, Reading, MA. Google Scholar
- Pitteway, M. L. V. 1985. The relationship between Euclid's algorithm and run-length encoding. In Fundamental Algorithms for Computer Graphics, R. A. Earnshaw, Ed. Springer, Berlin, Germany, 105--111.Google Scholar
- Reingold, E. M. and Dershowitz, N. 2001. Calendrical Calculations: The Millennium Edition. Cambridge University Press, Cambridge, U.K. Google Scholar
- Rockett, A. M. and Szüsz, P. 1992. Continued Fractions. World Scientific, Singapore.Google Scholar
- Shallit, J. 1994. Pierce expansions and rules for the determination of leap years. Fibonacci Quart. 32, 5, 416--423.Google Scholar
- Sproull, R. F. 1982. Using program transformations to derive line-drawing algorithms. ACM Trans. Graph. 1, 4, 259--273. Google Scholar
- Troesch, A. 1998. Droites discrètes et calendriers. Math. Inform. et Sci. Humaines 141, 36, 11--41.Google Scholar
Index Terms
- Line drawing, leap years, and Euclid
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