Generalized constant ratio surfaces and quaternions
Keywords:
Generalized constant ratio surfaces, kinematic, rotation matrices, real quaternions, unit quaternions.Abstract
In this study, we show that the generalized constant ratio surfaces can be obtained by the quaternion product of a unitquaternion and a pure quaternion. Also, the quaternion product of the unit quaternion and the position vector of the generalized constant ratio surface as a pure quaternion is a generalized constant ratio surface. Then we give some results about the generalized constant ratio surfaces by using the quaternions.
References
Babaarslan, M. & Yayli, Y. (2012). A new approach to constant slope surfaces with quaternion. ISRN Geometry (ISRN GEOM.), Vol. 2012, Article ID 126358, DOI: 10.5402/2012/126358, 8 Pages.
Bekar, M. & Yayl, Y. (2013). Involutions of complexified quaternions
and split quaternions. Advances in Applied Clifford Algebras, 23, pp. 283-299.
Dillen, F. Fastenakels, J. Van der Veken, J. & Vrancken, L. (2007).
Constant angle surfaces in . Monatshefte fr Mathematik,
(2):89-96.
Dillen, F. & Munteanu, M.I. (2009). Constant angle surfaces in
Bulletin of the Brazilian Mathematical Society, 40(1):85-97.
Dldl, M. (2010). Two and three dimensional regions from homothetic motions. Applied Mathematics ENotes, 10:86-93.
Fu, Y. & Munteanu, M.I. (2014). Generalized constant ratio surfaces
in . Bulletin of the Brazilian Mathematical Society, 45(1):73-90.
Hamilton, W.R. (1844). On quaternions; or on a new system of
imaginaries in algebra. London, Edinburgh, and Dublin Philosophical Magazine and Journal of science, 25(3):489-495.
Koenderink, J. (1990). Solid shape, MIT Press, Cambridge, MA.
Munteanu, M.I. (2010). From golden spirals to constant slope surfaces. Journal of Mathematical Physics, 51(7), Article ID 073507, 9 pages.
Munteanu, M.I. & Nistor, A.I. (2009). A new approach on constant
angle surfaces in . Turkish Journal of Mathematics, doi:10.3906/mat-0802-32, 33(1):169-178.
Shoemake, K. (1985). Animating rotation with quaternion curves.
In Proceedings of the Proceedings of the 12th Annual Conference on
Computer Graphics and Interactive Techniques (SIG-GRAPH 85),
ACM, New York, NY, USA, vol. 19, pp. 245-254.
Ward, J.P. (1997). Quaternions and Cayley numbers algebra
and applications, Kluwer Academic Publishers, Dordrecht, The
Netherlands.
