Generating hyperbolical rotation matrix for a given hyperboloid

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• 作者：Hakan Simsek ; ^{hakansimsek@akdeniz.edu.tr" class="auth_mail" title="E-mail the corresponding author} ; Mustafa Ö ; zdemir ^{mozdemir@akdeniz.edu.tr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A63 ; 15A66 ; 53A17 ; 53A35 ; 53B30 ; 70B05 ; 70B10 ; 70E17
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：496
• 期：Complete
• 页码：221-245
• 全文大小：547 K

文摘

Hyperbolic rotation is hyperbolically the motion of a smooth object on general hyperboloids given by −a₁x²+a₂y²+a₃z²=±λ$-a_1{x}^2+a_2{y}^2+a_3{z}^2=\pm \lambda$, λ∈R⁺$\lambda \in {\mathbb{R}}^{+}$. In this paper, we investigate the hyperbolical rotation matrices in order to get the motion of a point about a fixed point or axis on the general hyperboloids by defining the Lorentzian Scalar Product Space ${\mathbb{R}}_{a_1{a}_2{a}_3}^{2,1}$ such that the general hyperboloids are the pseudo-spheres of ${\mathbb{R}}_{a_1{a}_2{a}_3}^{2,1}$. We adapt the Rodrigues, Cayley, and Householder methods to ${\mathbb{R}}_{a_1{a}_2{a}_3}^{2,1}$ and define hyperbolic split quaternions to obtain an hyperbolical rotation matrix.