C o m m u n .Fa c .S c i.U n iv .A n k .S e rie s A 1 Vo lu m e 5 9 , N u m b e r 2 , P a g e s 3 7 -5 1 (2 0 1 0 ) IS S N 1 3 0 3 -5 9 9 1 A CHARACTERIZATION OF CYLINDRICAL HELIX STRIP

The spherical helix and the strip are respectively proved firstly by Scofield and Sabuncuoglu and Hacısalihoglu. In this paper helix strip on sphere is investigated by using characteristics of spherical helix and strip. Firstly using strip after helixand finally spherical helix. So spherical helix strips are obtained. Furhermore Joachimsthal Theorem and Terquem Theorem are investigated when the strip and helix strips which lie on the sphere and given a characterization about spherical helix strips.

In this paper Euclidean metric induced by μ : M → G be the mean curvature of Tubular surfaces by Bishop frame are computed and the curvatures in differential geometry that are so important for computing the Helfrich and Willmore energy of the tubular surfaces by bishop frame and giving some theorems are seen. Even though these calculations are very important to prove that the curvatures are very important in differential geometry, in actually these calculations are clearly important as mathematical physics. Sabit Ortalama Eğrilikli Bishop Çatılı Tubular Yüzeylerin EnerjileriAnahtar kelimeler Tubular yüzeyler; Eğrilikler; Bishop çatısı; Willmore enerji Öz Bu makalede μ : M → G Öklid metrikli Bishop çatılı Tubular yüzeylerin Helfrich ve Willmore enerjileri hesaplandı and Bishop çatılı Tubular yüzeylerin Helfrich ve Willmore enerjilerinin bulunmasında ve bazı teoremlerin verilmesinde diferensiyel geometrideki eğriliklerinin ne denli önemli olduğu görüldü. Bu hesaplamalar her ne kadar diferensiyel geometri için eğriliklerin önemli olduğunu ispatlasa da aslında bu hesaplamalar matematiksel fizik için de önem arz etmektedir.

The energy is one of the most important source for people and the other living things. We know that the energy of sun is the most energy source in universe. The sun is the source of the life. When the science and technology began to progressed, people found a lot of types of energy, but most of them are from the nature. For example energy of sun is naturel but energy of electric is produced by sun and using a lot of other nature sources (wind, water,…etc). The amount and capasity of the energy of the electric, water, wind,…etc. are calculated. For calculating energy we have to use mathematics and physics. Geometricians calculated the functions and energies of the curves by using Willmore and Horn' method lately. In this paper the Wilmore function of the σ-Bezier curves in H 3 and Gauss-Bonnet theorem of the σ-Bezier curves are calculated by using Willmore method.