Journal of Systems Science and Complexity

, Volume 32, Issue 1, pp 150–157 | Cite as

A Proposal for the Automatic Computation of Envelopes of Families of Plane Curves

  • Francisco BotanaEmail author
  • Tomás Recio


The idea of envelope of a family of plane curves is an elementary notion in differential geometry. As such, its implementation in dynamic geometry environments is quite universal (Cabri, The Geometer’s Sketchpad, Cinderella, GeoGebra,...). Nevertheless, most of these programs return, when computing certain envelopes, both some spurious solutions and the curves that truly fit in the intuitive definition of envelope. The precise distinction between spurious and genuine parts has not been made before: This paper proposes such distinction in an algorithmic way, ready for its implementation in interactive geometry systems, allowing a finer classification of the different parts resulting from the current, advanced approach to envelope computation and, thus, yielding a more precise output, free from extraneous components.


Automated deduction in geometry dynamic geometry envelopes parametric polynomial systems symbolic computation 


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With the occasion of ProfessorWen-tsunWu’s Centennial Birthday, we would like to honor, through this humble contribution, the decisive role of Professor Wentsun Wu’s work in the development of symbolic computation methods for the automatization of geometric reasoning.


  1. [1]
    Abánades M A, Botana F, Montes A, et al., An algebraic taxonomy for locus computation in Dynamic Geometry, Comp. Aided Des., 2014, 56: 22–33.CrossRefGoogle Scholar
  2. [2]
    Botana F and Recio T, Some issues on the automatic computation of plane envelopes in interactive environments, Math. Comp. Simul., 2016, 125: 115–125.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Botana F and Recio T, Computing envelopes in dynamic geometry environments, Ann. Math. Artif. Intel., 2017, 80: 3–20.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Bruce J W and Giblin P J, Curves and Singularities, Cambridge University Press, Cambridge, 1984.zbMATHGoogle Scholar
  5. [5]
    Montes A and Wibmer M, Gröbner bases for polynomial systems with parameters, J. Symbol. Comput., 2010, 45: 1391–1425.CrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Depto. de Matemática Aplicada IUniversidad de VigoPontevedraSpain
  2. 2.Depto. de Matemáticas, Estadística y ComputaciónUniversidad de CantabriaSantanderSpain

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