Journal of Integrative Natural Science (통합자연과학논문집)

The Basic Science Institute Chosun University (조선대학교 기초과학연구원)
권호 표지
DOI QR코드

DOI QR Code

Envelope of the Wallace-Simson Lines with Signed Angle ${\alpha}$

  • Bae, Sung Chul (Department of Math-Education, Korea University, Anam Campus) ;
  • Ahn, Young Joon (Department of Math-Education, Chosun University)
  • Received : 2011.12.30
  • Accepted : 2012.03.27
  • Published : 2012.03.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we show that for any triangle and any point on the circumcircle the envelope of the Wallace-Simson lines with signed angle ${\alpha}$ is a parabola. The proof is obtained naturally using polar coordinates. We also present the reparametrization of the envelope which is a linear normal curve.

Keywords

References

  1. S. Adu-Saymeh and M. Hahha, "Triangle centers with linear intercepts and linear subangles", Forum Geom., Vol. 5, pp. 33-36, 2005.
  2. Y. J. Ahn and C. M. Hoffmann, "Approximate convolution with pairs of cubic Bezier curve", Comp. Aided Geom. Desi,. Vol. 28, No. 6, pp. 357-367, 2011. https://doi.org/10.1016/j.cagd.2011.06.006
  3. Y. J. Ahn, C. M. Hoffmann, and Y. S. Kim, "Curvature-continuous offset approximation based on circle approximation using biguadratic Bezier curves", Comp. Aided Desi. Vol. 43, No. 8, pp. 1011-1017, 2010.
  4. Y. J. Ahn and J. H. Lee, "Orthocenters of triangle on the unit hypersphere", International Journal of Mathematical Education in Science and Technology, Vol. 39, pp. 419-421, 2008. https://doi.org/10.1080/00207390701734242
  5. S. C. Bae, "Reproof of generalized Simson theorem. Master thesis", Chosun University, 2009.
  6. O. Giering, "Affine and projective generalization of wallace lines", J. Geom. Graph., Vol. 1, pp. 119- 133, 1997.
  7. Miguel de Guzman, "An extension of the wallacesimson theorem: Project-ing in arbitrary directions", Amer. Math. Mont., Vol. 106, pp. 574-580, 1999. https://doi.org/10.2307/2589470
  8. Miguel de Guzman, "The envelope of the wallacesimson lines of a triangle. a simple proof of the steiner theorem on the deltoid", Rev. R. Acad. Cien. Serie A. Mat., Vol. 95, pp. 57-64, 2001.
  9. P. Pech, "On the simson-wallace theorem and its generalizations", J. Geom. Graph, Vol. 9, pp. 141-153, 2005.
  10. M. Peternell and T.Steiner, "Minkowski sum bou ndary surfaces of 3D objects", Graph. Models, Vol. 69, pp. 180-190, 2007. https://doi.org/10.1016/j.gmod.2007.01.001
  11. M. Sampoli, M. Peternell, and B. Juttler, "Exact parameterization of con-volution surfaces and rational surfaces with linear normals", Comp. Aided Geom. Desi., Vol. 23, pp. 179-192, 2006. https://doi.org/10.1016/j.cagd.2005.07.001
  12. Y. Zhihong, "Proof of Longuerre's theorem and its extensions by the method of polar coordinates", Paci. J. Math., Vol. 176, pp. 581-585, 1996. https://doi.org/10.2140/pjm.1996.176.581