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Gülsüm Yeliz ŞENTÜRK, Salim YÜCE
A METHOD FOR THE INVOLUTE-EVOLUTE OFFSETS OF RULED SURFACES
 
Abstract: Introduction: Differential geometry of the surface is a fundamental subject of the geometry. A ruled surface is a special type of surface and can always be easily parameterized. These surfaces can be described by moving a straight line along a chosen curve. Therefore, the equation of the ruled surface can be written as where is a curve which is called the base curve or the directrix of the ruled surface and the curve is also called the (spherical) indicatrix vector of the ruled surface. These surfaces are very useful in many areas of sciences for instance Computer-Aided Manufacturing (CAM), Computer-Aided Geometric Design (CAGD), geometric modeling and kinematics. In many industrial applications such as welding, cutting, painting, screwing, the movement of the tool (the robot end effector motion) is determined by a ruled surface. From past to present, the geometers have defined some different offsets of curves (surfaces) for example the involute-evolute, Bertrand, Mannheim and Smarandache. Offsets of curves (surfaces) generally more complicated than their progenitor curve (surfaces). The involute-evolute offsets of a curve were discovered by Christian Huygens in 1673. An evolute offsets of a given curve is some curves that always remains perpendicular to the tangent line to the progenitor curve. In this case, the progenitor curve is called an involute. These curves also have many applications in gear industry and business. Moreover, the frame fields constitute an important subject while examining the differential properties of curves and surfaces. Properties of ruled surfaces and their offset surfaces according to frame fields have been examined in Euclidean and non-Euclidean spaces. The ruled surfaces with Darboux frame and the theory of Bertrand offsets of the ruled surfaces with Darboux frame are defined by G. Y. Şentürk and S. Yüce (2015, 2017). Purpose: The purpose of this study is to define the involute-evolute offsets of ruled surfaces according to the Darboux Frame and to give their characteristic properties as a striction curve, distribution parameter and orthogonal trajectory. Scope: We can define the involute-evolute offsets of any ruled surfaces. Limitations: The study is limited to Euclidean 3-space and using the Darboux frame. Method: Using the Darboux frame of progenitor ruled surface’s base curve, we can give characteristic properties of the involute-evolute offset ruled surfaces. Findings: Let $\varphi \left( {s,v} \right)$ with Darboux frame $\left\{ \mathbf{T},\mathbf{g},\mathbf{n}\right\} ~$ and ${\varphi ^*}\left( {s,v} \right)$ with Darboux frame $\left\{ \mathbf{T}^*,\mathbf{g}^*,\mathbf{n}^*\right\} ~$ be two ruled surfaces in $\mathbb{E}^3$. $\varphi \left( {s,v} \right)$ is said to be an involute offset of ${\varphi ^*}\left( {s,v} \right)$ or ${\varphi ^*}\left( {s,v}\right)$ is said to be an evolute offset of $\varphi \left( {s,v} \right)$, if there exists a one-to-one correspondence between their points such that $\mathbf{T}$ of $\varphi \left( {s,v} \right)$ and $\mathbf{g}^*$ of ${\varphi ^*}\left( {s,v} \right)$ are linearly dependent. Conclusion: In this study, using Darboux frame {T,g,n} of ruled surface $\varphi(s,v)$ , the involute-evolute offsets $\varphi^*(s,v) $ with Darboux frame $\left\{ \mathbf{T}^*,\mathbf{g}^*,\mathbf{n}^*\right\}$ of $\varphi(s,v)$ are defined. Characteristic properties of $\varphi^*(s,v) $ as a striction curve, distribution parameter and orthogonal trajectory are investigated using the Darboux frame. The distribution parameters of ruled surfaces ${\varphi _{{\mathbf{T^*}}}},{\varphi _{{\mathbf{g^*}}}}$ and ${\varphi _{{\mathbf{n^*}}}}$ are given. By using Darboux frame of the surfaces we have given the relations between the instantaneous Pfaffian vectors of motions $H/H^{'}$ and $H^*/H^{*'}$, where $H=\left\{ \mathbf{T},\mathbf{g},\mathbf{n}\right\} ~$ be the moving space along the base curve of $\varphi(s,v)$, $H^*=\left\{ \mathbf{T}^*,\mathbf{g}^*,\mathbf{n}^*\right\} ~$ be the moving space along the base curve of $\varphi^*(s,v)$, $H'$ and $H^{*'}$ be fixed Euclidean spaces.

Anahtar Kelimeler: Darboux Frame, Involute, Evolute, Offset, Ruled Surface



 


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