# Euler-Savary formula for the planar homothetic motions

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```					   Scientia Magna
Vol. 5 (2009), No. 4, 18-25

Euler-Savary formula for the planar homothetic
motions
M. A. Gungor, S. Ersoy and M. Tosun

Department of Mathematic, Faculty of Arts and Science, Sakarya University,
u
Sakarya, 54187, T¨rkiye
E-mail: agungor@sakarya.edu.tr sersoy@sakarya.edu.tr tosun@sakarya.edu.tr

Abstract The homothetic motion in 2-dimensional Euclidean space E 2 , the relation between
the velocities of this motion and geometric results for the pole curves were studied in ref [1].
In this paper a canonical relative system of any plane with respect to other planes are given.
Therefore in a homothetic motion E/E , Euler-Savary formula giving the relation between
the curvature of trajectory curves drawn in the ﬁxed plane E by the points of the moving
plane E is obtained. In the special case of homothetic scale h identically equal to 1, we get
the Euler-Savary formula which was given by Muller [5]. Finally some geometrical results are
reached using Euler-Savary formula.
Keywords Euler-Savary formula, homothetic motion, kinematics.

§1. Introduction
This study deals with instantaneous geometric plane kinematics. Which is the study, for
a certain instant during a continuous motion, of the diﬀerential geometric properties? So if we
are interested in, for instance, the path of a point we study its tangent, its curvature, and so
on. The best way to deal with this subject analytically is to introduce canonical coordinate
systems and to make use of the concept of instantaneous invariants. The circumstance of the
motion being restricted to a plane simpliﬁes considerably the general theory.
To investigate to geometry of the motion of a line or a point in the motion of plane is
important in the study of planar kinematics or planar mechanisms or in physics. The geometry
of such a motion of a point or a line has a number of applications in geometric modeling
and model-based manufacturing of the mechanical products or in the design of robotic motions.
These are speciﬁcally used to generate geometric models of shell-type objects and thick surfaces,
[2]-[4].
Muller considered one and two parameter planar motions and gave the relation Va = Vf + Vr
between these motions’ absolute, sliding and relative velocities [5]. Mathematicians had worked
widely the curvature problems in the planar motion in the 18th and 19th centuries. At the
end of these works the radius of the arc’s curvature was calculated by using the Euler-S
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Description: The homothetic motion in 2-dimensional Euclidean space E^sup 2^, the relation between the velocities of this motion and geometric results for the pole curves were studied in ref [1]. In this paper a canonical relative system of any plane with respect to other planes are given. Therefore in a homothetic motion E/E', Euler-Savary formula giving the relation between the curvature of trajectory curves drawn in the fixed plane E' by the points of the moving plane E is obtained. In the special case of homothetic scale h identically equal to 1, we get the Euler-Savary formula which was given by Muller [5]. Finally some geometrical results are reached using Euler-Savary formula. [PUBLICATION ABSTRACT]
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