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            <front>

                <journal-meta>
                                    <journal-id></journal-id>
            <journal-title-group>
                                                <journal-title></journal-title>
            </journal-title-group>
                            <issn pub-type="ppub">2667-419X</issn>
                                                                                                        <publisher>
                    <publisher-name>Eskisehir Technical University</publisher-name>
                </publisher>
                    </journal-meta>
                <article-meta>
                                        <article-id pub-id-type="doi">10.20290/estubtdb.1673455</article-id>
                                                                <article-categories>
                                            <subj-group  xml:lang="en">
                                                            <subject>Algebraic and Differential Geometry</subject>
                                                    </subj-group>
                                            <subj-group  xml:lang="tr">
                                                            <subject>Cebirsel ve Diferansiyel Geometri</subject>
                                                    </subj-group>
                                    </article-categories>
                                                                                                                                                        <title-group>
                                                                                                                        <article-title>SHAPE OPERATORS OF A DIRECTIONAL TUBULAR SURFACE IN 4-SPACE</article-title>
                                                                                                                                                                                                <trans-title-group xml:lang="tr">
                                    <trans-title>SHAPE OPERATORS OF A DIRECTIONAL TUBULAR SURFACE IN 4-SPACE</trans-title>
                                </trans-title-group>
                                                                                                    </title-group>
            
                                                    <contrib-group content-type="authors">
                                                                        <contrib contrib-type="author">
                                                                    <contrib-id contrib-id-type="orcid">
                                        https://orcid.org/0000-0003-4067-3034</contrib-id>
                                                                <name>
                                    <surname>Yağbasan</surname>
                                    <given-names>Başak</given-names>
                                </name>
                                                                    <aff>ESKİŞEHİR OSMANGAZİ ÜNİVERSİTESİ</aff>
                                                            </contrib>
                                                    <contrib contrib-type="author">
                                                                    <contrib-id contrib-id-type="orcid">
                                        https://orcid.org/0000-0002-3247-5727</contrib-id>
                                                                <name>
                                    <surname>Ekici</surname>
                                    <given-names>Cumali</given-names>
                                </name>
                                                                    <aff>ESKİŞEHİR OSMANGAZİ ÜNİVERSİTESİ</aff>
                                                            </contrib>
                                                                                </contrib-group>
                        
                                        <pub-date pub-type="pub" iso-8601-date="20250825">
                    <day>08</day>
                    <month>25</month>
                    <year>2025</year>
                </pub-date>
                                        <volume>13</volume>
                                        <issue>2</issue>
                                        <fpage>109</fpage>
                                        <lpage>121</lpage>
                        
                        <history>
                                    <date date-type="received" iso-8601-date="20250410">
                        <day>04</day>
                        <month>10</month>
                        <year>2025</year>
                    </date>
                                                    <date date-type="accepted" iso-8601-date="20250613">
                        <day>06</day>
                        <month>13</month>
                        <year>2025</year>
                    </date>
                            </history>
                                        <permissions>
                    <copyright-statement>Copyright © 2010, </copyright-statement>
                    <copyright-year>2010</copyright-year>
                    <copyright-holder></copyright-holder>
                </permissions>
            
                                                                                                <abstract><p>This paper examines a tubular surface, a specific example of a canal surface, in 4-dimensional Euclidean space. In the plane stretched by the quasi-frame vectors B_q and C_q, this surface is established by the motion of a circle with a constant radius that uses each point on the curve a(t) as its center. Using the general equation provided in Euclidean 4-space, the first and second partial derivatives are determined. The Gram-Schmidt technique was used to derive the surface&#039;s first unit normal vector field U_1, and second unit normal vector field U_2, using the acquired partial derivatives. Using quasi-vectors, the tubular surface&#039;s first and second fundamental form coefficients were found. Furthermore, the shape operator matrices for the tubular surface&#039;s the unit normal vector fields U_1 and U_2 were acquired. We have found algebraic invariants of the shape operator, Gaussian curvature, and mean curvature. For a thorough understanding of the obtained theoretical calculations, an example of a directional tubular surface, the equation of the tubular surface has been parametrized using quasi-frame vectors and quasi-frame curvatures for a given space curve in 4-dimensional Euclidean space.</p></abstract>
                                                                                                                                    <trans-abstract xml:lang="tr">
                            <p>This paper examines a tubular surface, a specific example of a canal surface, in 4-dimensional Euclidean space. In the plane stretched by the quasi-frame vectors B_q and C_q, this surface is established by the motion of a circle with a constant radius that uses each point on the curve a(t) as its center. Using the general equation provided in Euclidean 4-space, the first and second partial derivatives are determined. The Gram-Schmidt technique was used to derive the surface&#039;s first unit normal vector field U_1, and second unit normal vector field U_2, using the acquired partial derivatives. Using quasi-vectors, the tubular surface&#039;s first and second fundamental form coefficients were found. Furthermore, the shape operator matrices for the tubular surface&#039;s the unit normal vector fields U_1 and U_2 were acquired. We have found algebraic invariants of the shape operator, Gaussian curvature, and mean curvature. For a thorough understanding of the obtained theoretical calculations, an example of a directional tubular surface, the equation of the tubular surface has been parametrized using quasi-frame vectors and quasi-frame curvatures for a given space curve in 4-dimensional Euclidean space.</p></trans-abstract>
                                                            
            
                                                            <kwd-group>
                                                    <kwd>Euclidean Space</kwd>
                                                    <kwd>  Quasi-frame</kwd>
                                                    <kwd>  Tubular Surface</kwd>
                                                    <kwd>  Shape Operator</kwd>
                                            </kwd-group>
                                                        
                                                                            <kwd-group xml:lang="tr">
                                                    <kwd>Euclidean Space</kwd>
                                                    <kwd>  Quasi-frame</kwd>
                                                    <kwd>  Tubular Surface</kwd>
                                                    <kwd>  Shape Operator</kwd>
                                            </kwd-group>
                                                                                                            </article-meta>
    </front>
    <back>
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