Geodesic
Ruled and Quadric Surfaces Satisfying ΔIIIx = Λ x
Journal for geometry and graphics, Tome 20 (2016) no. 2, pp. 147-157.

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We consider ruled and quadric surfaces in the 3-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form III, i.e., their position vector x satisfies the relation ΔIIIx = Λ x where Λ is a square matrix of order 3. We show that helicoids and spheres are the only classes of surfaces mentioned above satisfying this equation.
Classification : 53A05, 47A75
Mots-clés : Surfaces in Euclidean space, surfaces of coordinate finite type, Beltrami operator 
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     author = {H. Al-Zoubi and S. Stamatakis },
     title = {Ruled and {Quadric} {Surfaces} {Satisfying} {\ensuremath{\Delta}\protect\textsuperscript{III}x} = {\ensuremath{\Lambda}} x},
     journal = {Journal for geometry and graphics},
     pages = {147--157},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2016},
     url = {http://geodesic.mathdoc.fr/item/JGG_2016_20_2_JGG_2016_20_2_a0/}
}
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H. Al-Zoubi; S. Stamatakis . Ruled and Quadric Surfaces Satisfying ΔIIIx = Λ x. Journal for geometry and graphics, Tome 20 (2016) no. 2, pp. 147-157. http://geodesic.mathdoc.fr/item/JGG_2016_20_2_JGG_2016_20_2_a0/