# The Holditch-Type Theorem for the Polar Moment of Inertia of the Orbit Curve in the Generalized Complex Plane

Research paper by Tülay Erişir, Mehmet Ali Güngör; Murat Tosun

Indexed on: 09 Nov '16Published on: 01 Dec '16Published in: Advances in Applied Clifford Algebras

#### Abstract

Abstract In this study, we first calculate the polar moment of inertia of orbit curves under one-parameter planar motion in the generalized complex plane $${{\mathbb{C}_p}}$$ and then give the Holditch-type theorem for $${{\mathbb{C}_p}}$$ : When the fixed points $${X}$$ and $${Y}$$ on the moving plane $${{\mathbb{K}_p} \subset {\mathbb{C}_p}}$$ trace the same curve $${k}$$ with the polar moment of inertia $${{T_X}}$$ , the different point $${Z}$$ on this line segment $${XY}$$ traces another curve $${{k_Z}}$$ with the polar moment of inertia $${{T_Z}}$$ during the one-parameter planar motion in the fixed plane $${{\mathbb{K}'_p} \subset {\mathbb{C}_p}}$$ . Thus, we obtain that the difference between the polar moments of inertia of these curves $${( {{T_Z} - {T_X}} )}$$ depends on the only the $${p}$$ -distances of this points and $${p}$$ -rotation angle of the motion, $${{T_X} - {T_Z} = {\delta _p}ab.}$$ AbstractIn this study, we first calculate the polar moment of inertia of orbit curves under one-parameter planar motion in the generalized complex plane $${{\mathbb{C}_p}}$$ and then give the Holditch-type theorem for $${{\mathbb{C}_p}}$$ : When the fixed points $${X}$$ and $${Y}$$ on the moving plane $${{\mathbb{K}_p} \subset {\mathbb{C}_p}}$$ trace the same curve $${k}$$ with the polar moment of inertia $${{T_X}}$$ , the different point $${Z}$$ on this line segment $${XY}$$ traces another curve $${{k_Z}}$$ with the polar moment of inertia $${{T_Z}}$$ during the one-parameter planar motion in the fixed plane $${{\mathbb{K}'_p} \subset {\mathbb{C}_p}}$$ . Thus, we obtain that the difference between the polar moments of inertia of these curves $${( {{T_Z} - {T_X}} )}$$ depends on the only the $${p}$$ -distances of this points and $${p}$$ -rotation angle of the motion, $${{T_X} - {T_Z} = {\delta _p}ab.}$$ $${{\mathbb{C}_p}}$$ $${{\mathbb{C}_p}}$$ $${{\mathbb{C}_p}}$$ $${{\mathbb{C}_p}}$$ $${X}$$ $${X}$$ $${Y}$$ $${Y}$$ $${{\mathbb{K}_p} \subset {\mathbb{C}_p}}$$ $${{\mathbb{K}_p} \subset {\mathbb{C}_p}}$$ $${k}$$ $${k}$$ $${{T_X}}$$ $${{T_X}}$$ $${Z}$$ $${Z}$$ $${XY}$$ $${XY}$$ $${{k_Z}}$$ $${{k_Z}}$$ $${{T_Z}}$$ $${{T_Z}}$$ $${{\mathbb{K}'_p} \subset {\mathbb{C}_p}}$$ $${{\mathbb{K}'_p} \subset {\mathbb{C}_p}}$$ $${( {{T_Z} - {T_X}} )}$$ $${( {{T_Z} - {T_X}} )}$$ $${p}$$ $${p}$$ $${p}$$ $${p}$$ $${{T_X} - {T_Z} = {\delta _p}ab.}$$ $${{T_X} - {T_Z} = {\delta _p}ab.}$$