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A REMARK ON CANAL AND TUBE HYPERSURFACES IN E^4
 
The envelope of a one-parameter set of spheres, centred at the spine curve $\alpha(u)$ with a radius function $r(u)$ is called a canal surface. If the radius function $r(u)$ is constant, the canal surface is called tube (tubular, pipe) surface. Canal surfaces have extensive applications in computer aided geometric design [1]-[2]. The geometrical properties of canal surfaces in $\mathbb{E}^3$ are examined in [3]-[5]. There are many studies in the literature related to canal and tube surfaces in [6]-[27]. Besides, the canal hypersurfaces are examined in [28]-[33]. This study deals with the canal and tube hypersurfaces by taking a special parametrisation and examines them by a different point of view in 4-dimensional Euclidean space $\mathbb{E}^4$. Moreover, the coefficients of the first and second fundamental form, the matrix of the shape operator, the Gaussian curvature and the mean curvature of tube hypersurface are investigated for the spine curve $\alpha(u)=\left(\alpha_1(u),\alpha_2(u),\alpha_3(u),\alpha_4(u)\right)$ by using the geometrical properties given in [34]-[38]. Also, the curvatures are examined both the spine curve is a space curve and a planar curve. To condition to be a minimal tube hypersurface is given. Finally, the graphs of the projections of the tube hypersurfaces using different radius functions are presented in $\mathbb{E}^4$.

Anahtar Kelimeler: Canal hypersurface, Tube hypersurface, Gaussian curvature, Mean curvature



 


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