A generalization of the Holditch Theorem for the planar homothetic motions

In the literature, Holditch theorem was obtained under periodic rotation and translation motions in [H. Holditch, Geometrical theorem, Q. J. Pure Appl. Math. 2 (1858) 38] or periodic shear and translation motions in [O. Röschel, Der satz von Holditch in der isotropen ebene. Abh. Braunschweig. Wiss. Ges. 36 (1984) 27–32]. In this paper, by introducing the projection of a vector onto a plane, scalar area, area vector of a surface, we investigate Holditch theorem under periodic rotation, translation and shear motions. We give two interpretations for the Holditch type theorem in Galilean space.

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The Steiner formula and the Holditch theorem for the homothetic motions on the planar kinematics

Mechanism and Machine Theory ◽

10.1016/s0094-114x(98)00028-7 ◽

1999 ◽

Vol 34 (1) ◽

pp. 1-6 ◽

Cited By ~ 11

Author(s):

A. Tutar ◽

N. Kuruoǧlu

Keyword(s):

Holditch Theorem ◽

Steiner Formula

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Application of Holditch Theorem to Bezier and Quadratic Uniform B-Spline Curves

Mathematical and Computational Applications ◽

10.3390/mca6030169 ◽

2001 ◽

Vol 6 (3) ◽

pp. 169-176 ◽

Cited By ~ 1

Author(s):

A. Caliskan ◽

B. Tantay

Keyword(s):

B Spline ◽

Holditch Theorem ◽

Spline Curves

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A New Construction of Holditch Theorem for Homothetic Motions in C p

Advances in Mathematical Physics ◽

10.1155/2021/5583457 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Tülay Erişir

Keyword(s):

Complex Plane ◽

Complex Number ◽

Number System ◽

Geometric Representation ◽

Area Formula ◽

Holditch Theorem ◽

Generalized Complex ◽

New Construction ◽

The One ◽

The Relationship

In this study, the planar kinematics has been studied in a generalized complex plane which is a geometric representation of the generalized complex number system. Firstly, the planar kinematic formulas with one parameter for homothetic motions in the generalized complex plane have been mentioned briefly. Then, the Steiner area formula given areas of the trajectories drawn by the points taken in a generalized complex plane have been obtained during the one-parameter planar homothetic motion. Finally, the Holditch theorem, which gives the relationship between these areas of trajectories, has been expressed for homothetic motions in a generalized complex plane. So, this theorem obtained in this study is the most general form of all Holditch theorems obtained so far.

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