Abstract
In this study, we consider the kinematics of the hyperbolic plane and define the notions of inflection curve, circling-point curve, cubic of twice stationary curvature curve, and cubic of thrice stationary curve. We also obtain Cartesian and parametric equations of these curves and illustrate some special cases. Finally, we investigate the hyperbolic Ball point, the ordinary and the sixth-order Burmester points in the case of a finite instant pole.
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References
1.
Freudenstein
F.
Higher path-curvature analysis in plane kinematics . J Manuf Sci Eng 1965 ; 87 : 184 –190 .
2.
Freudenstein
F
Sandor
GN.
On the Burmester points of a plane . J Appl Mech 1961 ; 28 : 41 –49 .
3.
Veldkamp
GR.
Some remarks on higher curvature theory . J Manuf Sci Eng 1967 ; 89 : 84 –86 .
4.
Kamphuis
HJ.
Application of spherical instantaneous kinematics to the spherical slider-crank mechanism . J Mech 1969 ; 4 : 43 –56 .
5.
Roth
B
Yang
AT.
Higher-order path curvature in spherical kinematics . J Manuf Sci Eng 1973 ; 95 : 612 –616 .
6.
Stachel
H.
Bemerkungen zur sphärischen kreispunktkurve . In: 2nd Austrian Geometry Colloquium , 1984 .
7.
Chiang
CH.
Kinematics of spherical mechanisms . Taiwan : McGraw-Hill , 1996 .
8.
Ting
KL
Wang
SC.
Fourth and fifth order double Burmester points and the highest attainable order of straight lines . J Mech Des 1991 ; 113 : 213 –219 .
9.
Özçelik
Z
Şaka
Z
. Ball and Burmester points in spherical kinematics and their special cases . Forsch Ingenieurwes 2010 ; 74 : 111 –122 .
10.
Özçelik
Z.
Design of spherical mechanisms by using instantaneous invariants . PhD Thesis, Selçuk University , Turkey , 2008 .
11.
Bottema
O
Roth
B.
Theoretical kinematics . Oxford : North-Holland Publishing Company , 1979 .
12.
Garnier
R.
Cours de cinématique, géométrie et cinématique cayleyennes . Paris : Gauthier-Villars , 1951 .
13.
Frank
H.
Zur ebenen hyperbolischen kinematik . Elem Math 1971 ; 26 : 121 –131 .
14.
Gunn
C.
Geometry, kinematics, and rigid body mechanics in Cayley–Klein geometries . PhD Thesis, Technical University of Berlin , Germany , 2011 .
15.
Weinstein
T.
An introduction to Lorentz surfaces . Berlin : Walter de Gruyter , 1996 .
16.
Coxeter
HSM.
Non-Euclidean geometry . Toronto : University of Toronto Press , 1965 .
17.
Sossinsky
AB.
Geometries . Providence : American Mathematical Society , 2012 .
18.
Uğurlu
HH
Topal
A
. Relation between Darboux instantaneous rotation vectors of curves on a timelike surface . Math Comput Appl 1996 ; 1 : 149 –157 .
19.
Izumiya
S
Pei
DH
Sano
T
. Evolutes of hyperbolic plane curves . Acta Math Sin 2004 ; 20 : 543 –550 .