Adv. Appl. Cliﬀord Algebras 27 (2017), 2873–2884
2017 Springer International Publishing
published online February 20, 2017
Applied Cliﬀord Algebras
A Note on Evolution of Curves
in the Minkowski Spaces
Onder G¨okmen Yıldız and Murat Tosun
Communicated by Wolfgang Spr¨ossig
Abstract. In this paper, we study the evolution of non-null curve in
n-dimensional Minkowski Space. We express evolution equation of the
Frenet frame by matrix equation. We obtain integrability conditions for
the evolutions. Finally, we give examples of evolutions.
Mathematics Subject Classiﬁcation. Primary 53C44; Secondary 53A05,
Keywords. Evolution of curves, Curve ﬂows, Minkowski Space.
In diﬀerential geometry, there are countless study of curved spaces or shapes
in which time did not play a role. Recently, contrary to what is known geo-
metricians have made great improvement in understanding the curved spaces
that envolve in time. Among them, an envolving curve have in many applica-
tions such as engineering, computer vision [9,11], computer animation and
even structural mechanics . Also, there exists many physical applications
(see [3, 5,6]).
The envolving curve can be thought of as a family of curves parametrized
by time. The time evolution of a curve generated by its corresponding ﬂow
so we shall also refer to curve evolutions as ﬂows throughout this paper.
Firstly, Kwon and Park studied inextensible ﬂows of curves and developable
surfaces in Euclidean 3-space . Following them, inextensible ﬂows of curves
are studied in many diﬀerent spaces. For example, G¨urb¨uz have examined
inextensible ﬂows of spacelike, timelike and null curves in andU¸cum et
al. have studied inextensible ﬂows for null curves in E
 and Yıldız et
al. have studied inextensible ﬂows of curves according to Darboux frame in
This work was completed with the support of our T