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Technical Briefs

# Bobillier Formula for One Parameter Motions in the Complex Plane

[+] Author and Article Information
Soley Ersoy

Department of Mathematics,Faculty of Arts and Sciences,  Sakarya University, Sakarya 54187, Turkeysersoy@sakarya.edu.tr

Nurten Bayrak

Department of Mathematics, Graduate School of Natural and Applied Sciences,  Yildiz Technical University, Yildiz Central Campus, Istanbul, 34349 Turkeynbayrak@yildiz.edu.tr

J. Mechanisms Robotics 4(2), 024501 (Apr 12, 2012) (4 pages) doi:10.1115/1.4006195 History: Received December 21, 2011; Revised January 09, 2012; Published April 10, 2012; Online April 12, 2012

## Abstract

This is a brief note expanding on the aspect of Fayet (2002, “Bobillier Formula as a Fundamental Law in Planar Motion,” Z. Angew. Math. Mech., 82 (3), pp. 207–210), which investigates the Bobillier formula by considering the properties up to the second order planar motion. In this note, the complex number forms of the Euler Savary formula for the radius of curvature of the trajectory of a point in the moving complex plane during one parameter planar motion are taken into consideration and using the geometrical interpretation of the Euler Savary formula, Bobillier formula is established for one parameter planar motions in the complex plane. Moreover, a direct way is chosen to obtain Bobillier formula without using the Euler Savary formula in the complex plane. As a consequence, the Euler Savary given in the complex plane will appear as a particular case of Bobillier formula.

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## Figures

Figure 1

IMi→, IMi*→, 1≤i≤3, vectors

Figure 2

eiθj,  1≤j≤3, vectors

Figure 3

Qj , 1 ≤ j ≤ 3 points

Figure 4

(IQ→1-IQ→2)   and   (IQ→2-IQ→3) vectors

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