Jeff Tupper
ABSTRACT
This paper presents a series of new algorithms for reliably graphing two-dimensional implicit equations and inequalities. A clear standard for interpreting the graphs generated by two-dimensional graphing software is introduced and used to evaluate the presented algorithms. The first approach presented uses a standard interval arithmetic library. This approach is shown to be faulty; an analysis of the failure reveals a limitation of standard interval arithmetic. Subsequent algorithms are developed in parallel with improvements and extensions to the interval arithmetic used by the graphing algorithms. Graphs exhibiting a variety of mathematical and artistic phenomena are shown to be graphed correctly by the presented algorithms. A brief comparison of the final algorithm presented to other graphing algorithms is included.
- ABK95.Ron Avitzur,Olaf Bachmann, and Norbert Kajler. From Honest to Intelligent Plotting.In A.H.M.Levelt,editor, Proc.of the International Symp.on Symbolic and Algebraic Computat on (ISSAC '95),Montreal,Canada , pages 32 - 41.ACM Press,1995. Google Scholar
Digital Library
- Arn83.Dennis S.Arnon.Topologically Reliable Display of Algebraic Curves.Computer Graphics (SIGGRAPH 83 Conference Proceed ngs),17(3):219 - 227,July 1983. Google ScholarDigital Library
- CS93.J.Comba and J.Stolfi Affine Arithmetic an its Applications to Computer Graphics.In Anais do VI Symposio Bras le ro de Computa, computacao Grafia e Processamento de Imagens (SIBGRAPI '93),pages 9 - 18,1993.Google Scholar
- Don00.Jack J.Dongarra. Performance of Various Computers Using Standar Linear Equations Software. Technical Report CS-89-85,University of Tennessee, 2000. Google ScholarDigital Library
- DWM00.Jack Dongarra, Ree Wade, and Paul McMahan. Linpack Benchmark - Java Version. http ://www.netlib.org /benchmark/linpackjava, 2000.Google Scholar
- Fat92.R.Fateman. Honest Plotting, Global Extrema an Interval Arithmetic.In P.S.Wang, editor, Proc. of the International Symp. on Symbolic and Algebra c Computat on (ISSAC '92), Berkeley, USA , pages 216 - 223. ACM Press, 1992. Google ScholarDigital Library
- Han80.Eldon Hansen. Global Optimization Using Interval Analysis - The Multi-Dimensional Case. Numerische Mathematik ,34(3):247 - 270, 1980.Google ScholarDigital Library
- HQvE00.Timothy J. Hickey, Zhe Qiu, and Maarten H. van Emden. Interval Constraint Plotting for Interactive Visual Exploration of Implicitly Defined Relations. Reliable Computing ,6(1):81 - 92, 2000.Google Scholar
- IEE85.IEEE Task P754.ANSI/IEEE 754-1985, Standard for Binary Floating-Point Arithmetic .IEEE, New York, NY, USA, August 1985. Revise 1990. A preliminary draft was published in the January 1980 issue of IEEE Computer,together with several companion articles.Also stan ardize as IEC 60559 (1989-01)B nary .oating-po nt arithmetic for microprocessor systems .Google Scholar
- Kah68.W.M.Kahan.A More Complete Interval Arithmetic.Lecture notes prepare for a summer course at the University of Michigan,June 17 - 21,1968.Google Scholar
- Met99.Metrowerks.Macintosh Linpack Benchmark. http://www.metrowerks.com/benchmarks/desktop/ mac_linpack.html , 1999.Google Scholar
- Mic00.Tom Michiels.http://www.cs.kuleuven.ac.be/~tomm/ bench.html , 2000.Google Scholar
- Moo66.R.E.Moore.Interval Analysis . Prentice Hall, dEnglewood Cliffs, New Jersey, 1966.Google Scholar
- Moo79.R.E.Moore. Methods and Applications of Interval Analysis. SIAM, Philadelphia, 1979. Google ScholarDigital Library
- Ped.Pedagoguery Software Inc.GrafEq TM . http://www.peda.com/grafeq .Google Scholar
- RR88.H. Ratschek an J. Rokne. New Computer Methods for Global Optimization .Ellis Horwoo Ltd.,Chichester, 1988. Google ScholarDigital Library
- Sny92a.John M. Snyder.Generative Modeling for Computer Graphics and CAD:Symbol c Shape Des gn Us ng Interval Analysis .Aca emic Press,San Diego,1992. Google ScholarDigital Library
- Sny92b.John M. Snyder. Interval Analysis for Computer Graphics. Computer Graphics (SIGGRAPH 92 Conference Proceedings), 26(2):121 - 130, July 1992. Google ScholarDigital Library
- Tau93.Gabriel Taubin. An Accurate Algorithm for Rasterizing Algebraic Curves.In Second Symposium on Solid Modeling . ACM SIGGRAPH and IEEE Computer Society,May 1993. Google ScholarDigital Library
- Tau94.Gabriel Taubin.Distance Approximations for Rasterizing Implicit Curves.ACM Transactions on Graphics , 13(1):3 - 42,January 1994. Google ScholarDigital Library
- Tup96.Jeffrey Allen Tupper.Graphing Equations with Generalize Interval Arithmetic.Master 's thesis,University of Toronto,1996.Google Scholar
Index Terms
Reliable two-dimensional graphing methods for mathematical formulae with two free variables
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