The Cauchy-Length Formula and Holditch Theorem in the Generalized Complex Plane C_p

Abstract

In this paper, firstly, we calculate Cauchy-length formula for the one-parameter planar motion in generalized complex plane C p which is generalization of the complex, dual and hyperbolic planes. Then, we give the length of the enveloping trajectories of lines C p. In addition, we prove the Holditch theorem for the non-linear three points with the aid of the length of the enveloping trajectories in C p. So, the Holditch theorem for the linear three points which is given by Erişir et al. in C p is generalized for trajectories drawn by the non-linear three points in generalized complex plane C p .

Holditch-Type Theorem for Non-Linear Points in Generalized Complex Plane $\mathbb{C}_{p}$

“…Then, they calculated the polar moment of inertia of trajectories under the one-parameter planar motion and proved Holditch-type theorem in C p , [4]. Moreover, Erişir and Güngör gave the Cauchy-length formula and proved Holditch theorem for non-linear points in C p , [22]. Now, using the above studies, we give some operations on this system.…”

“…Now, we mention Cauchy formula in C p which is used in this study. This formula in C p was studied by Erişir and Güngör in [22]. Let g be a line in the branch I of C p .…”

“…is the length of orthogonal projection of the line segment Q 1 Q 2 of the moving pole curve (Q) on the line g. Moreover,q = h − q 1 cos pψ p + pq 2 sin pψ p is distance of the pole point Q to the line g in the generalized complex plane in C p , [22]. In addition, the following theorem can be given.…”