Differential geometry of non-transversal intersection curves of three implicit hypersurfaces in Euclidean 4-space

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• 作者：O. Alé ; ssio^a ; ^{osmar@matematica.uftm.edu.br" class="auth_mail" title="E-mail the corresponding author} ; M. Dü ; ldü ; l^b ; ^{mduldul@yildiz.edu.tr" class="auth_mail" title="E-mail the corresponding author} ; B. Uyar Dü ; ldü ; l^c ; ^{buduldul@yildiz.edu.tr" class="auth_mail" title="E-mail the corresponding author} ; Nassar H. Abdel-All^d ; ^{nhabdeal2002@yahoo.com" class="auth_mail" title="E-mail the corresponding author} ; Sayed Abdel-Naeim Badr^d ; ^{sayed_badr@ymail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：53A04 ; 53A05
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：308
• 期：Complete
• 页码：20-38
• 全文大小：497 K

文摘

The aim of this paper is to compute all the Frenet apparatus of non-transversal intersection curves (hyper-curves) of three implicit hypersurfaces in Euclidean 4-space. The tangential direction at a transversal intersection point can be computed easily, but at a non-transversal intersection point, it is difficult to calculate even the tangent vector. If three normal vectors are parallel at a point, the intersection is “tangential intersection”; and if three normal vectors are not parallel but are linearly dependent at a point, we have “almost tangential” intersection at the intersection point. We give algorithms for each case to find the Frenet vectors class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S037704271630228X&_mathId=si2.gif&_user=111111111&_pii=S037704271630228X&_rdoc=1&_issn=03770427&md5=ee4ce13b3e6de1fbac5889765121ce54">class="imgLazyJSB inlineImage" height="15" width="83" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S037704271630228X-si2.gif">class="mathContainer hidden">class="mathCode">$(t,n,b_1,b_2)$ and the curvatures class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S037704271630228X&_mathId=si3.gif&_user=111111111&_pii=S037704271630228X&_rdoc=1&_issn=03770427&md5=ca9316fd424e415f44da905a61fafda7" title="Click to view the MathML source">(k₁,k₂,k₃)class="mathContainer hidden">class="mathCode">$(k_1,k_2,k_3)$ of the non-transversal intersection curve.