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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 105283, 9 pages
Research Article

A Method to Solve the Limitations in Drawing External Rays of the Mandelbrot Set

1Instituto de Tecnologías Físicas y de la Información (ITEFI), Consejo Superior de Investigaciones Científicas (CSIC), Serrano 144, 28006 Madrid, Spain
2Department of Mathematics and Computer Science, “Avram Iancu” University, Ilie Măcelaru 1A, 400380 Cluj-Napoca, Romania
3Romanian Institute of Science and Technology, Cireşilor 29, 400487 Cluj-Napoca, Romania

Received 9 July 2013; Accepted 4 August 2013

Academic Editor: Guanrong (Ron) Chen

Copyright © 2013 M. Romera et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The external rays of the Mandelbrot set are a valuable graphic tool in order to study this set. They are drawn using computer programs starting from the Böttcher coordinate. However, the drawing of an external ray cannot be completed because it reaches a point from which the drawing tool cannot continue drawing. This point is influenced by the resolution of the standard for floating-point computation used by the drawing program. The IEEE 754 Standard for Floating-Point Arithmetic is the most widely used standard for floating-point computation, and we analyze the possibilities of the quadruple 128 bits format of the current IEEE 754-2008 Standard in order to draw external rays. When the drawing is not possible, due to a lack of resolution of this standard, we introduce a method to draw external rays based on the escape lines and Bézier curves.