A Study on Fuzzy Triangle and Fuzzy Trigonometric Properties

  • Debdas GhoshEmail author
  • Debjani Chakraborty
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 253)


This paper investigates fuzzy triangle, fuzzy triangular properties, and fuzzy trigonometry. A fuzzy triangle on the plane is constructed by three fuzzy points as its vertices. Using the proposed fuzzy triangle, basic fuzzy trigonometric functions are investigated. The extension principle and the concepts of same and inverse points in fuzzy geometry are used to define all the proposed ideas. It is shown that some well-known trigonometric identities for crisp angles may not hold with proper equality for fuzzy angles.


Fuzzy number Fuzzy point Same points Inverse points Fuzzy angle Fuzzy triangle Extension principle 

AMS Subject Classification



  1. 1.
    Bogomolny, A.: On the perimeter and area of fuzzy sets. Fuzzy Sets Syst. 23, 257–269 (1987)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Buckley, J.J., Eslami, E.: Fuzzy plane geometry II: circles and polygons. Fuzzy Sets Syst. 87, 79–185 (1997)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Buckley, J.J., Eslami, E.: An Introduction to Fuzzy Logic and Fuzzy Systems. Physica-Verlag, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Chakraborty, D., Ghosh, D.: Analytical fuzzy plane geometry II. Fuzzy Sets Syst. 243, 84–109 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chaudhuri, B.B.: Some shape definitions in of space fuzzy geometry. Pattern Recogn. Lett. 12, 531–535 (1991)CrossRefGoogle Scholar
  6. 6.
    Ghosh, D., Chakraborty, D.: Analytical fuzzy plane geometry I. Fuzzy Sets Syst. 209, 66–83 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Imran, B.M., Beg, M.M.S.: Estimation of \(f\)-similarity in \(f\)-triangles using fis. In: Meghanathan, N., Chaki, N., Nagamalai, D. (eds.) CCSIT 2012, Part III LNICST, vol. 86, pp. 290–299. Springer, Heidelberg (2012)Google Scholar
  8. 8.
    Imran, B.M., Beg, M.M.S.: Elements of sketching with words. In: Hu, X. (ed.) IEEE International Conference on Granular Computing, pp. 241–246. IEEE Computer Society, San Jose, California, USA (2010)Google Scholar
  9. 9.
    Li, Q., Guo, S.: Fuzzy geometric object modelling. Fuzzy Inf. Eng. (ICFIE) ASC 40, 551–563 (2007)CrossRefGoogle Scholar
  10. 10.
    Liu, H., Coghill, G.M.: Fuzzy qualitative trigonometry. Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, Hawaii, USA vol. 2, pp. 1291–1296 (2005)Google Scholar
  11. 11.
    Pham, B.: Representation of fuzzy shapes. In: Arcelli C., et al. (eds.) IWVF4, LNCS, Vol. 2059, pp. 239–248. Springer, Heidelberg (2001)Google Scholar
  12. 12.
    Rosenfeld, A.: Fuzzy plane geometry: triangles. Pattern Recogn. Lett. 15, 1261–1264 (1994)CrossRefGoogle Scholar
  13. 13.
    Rosenfeld, A.: Fuzzy geometry: an updated overview. Inf. Sci. 110, 127–133 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rosenfeld, A., Haber, S.: The perimeter of a fuzzy set. Pattern Recogn. 18, 125–130 (1985)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zadeh, L.A.: Toward extended fuzzy logic-a first step. Fuzzy Sets Syst. 160, 3175–3181 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIndian Institute of Technology (BHU)VaranasiIndia
  2. 2. Department of Mathematics Indian Institute of Technology KharagpurKharagpurIndia

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