Parallel Surfaces to Normal Ruled Surfaces of General Helices in the Sol Space Sol

In this paper, parallel surfaces to normal ruled surface of general helices in the Sol 3 are studied. Also, explicit parametric equations of parallel surfaces to normal ruled surface of general helices in the Sol 3 are found.


Introduction
Ruled surfaces have been popular in architecture.Structural elegance, these and many other contributions are in contrast to recent free-form architecture.Applied mathematics and in particular geometry have initiated the implementation of comprehensive frameworks for modeling and mastering the complexity of today's architectural needs shapes in an optimal sense by ruled surfaces.
In this paper, parallel surfaces to normal ruled surface of general helices in the Sol 3 are studied.Also, explicit parametric equations of parallel surfaces to normal ruled surface of general helices in the Sol 3 are found.

Riemannian Structure of Sol Space Sol 3
Sol space, one of Thurston's eight 3-dimensional geometries, can be viewed as R 3 provided with Riemannian metric where (x, y, z) are the standard coordinates in R 3 .

Talat Körpınar and Essin Turhan
Note that the Sol metric can also be written as: where and the orthonormal basis dual to the 1-forms is where the (i, j)-element in the table above equals ∇ ei e j for the basis Lie brackets can be easily computed as: The isometry group of Sol 3 has dimension 3. The connected component of the identity is generated by the following three families of isometries: x, e c y, z + c .

General Helices in Sol Space Sol 3
Assume that {T, N, B} be the Frenet frame field along γ.Then, the Frenet frame satisfies the following Frenet-Serret equations: Parallel Surfaces to Normal Ruled Surfaces of General Helices 247 where κ is the curvature of γ and τ its torsion and With respect to the orthonormal basis {e 1 , e 2 , e 3 }, it is possible to write Theorem 3.1.( [14]) Let γ : I −→ Sol 3 be a unit speed non-geodesic general helix.Then, the parametric equations of γ are   The purpose of this section is to study parallel surfaces to normal ruled surfaces of general helices in Sol 3 .
The normal ruled surface of γ is A parallel surface to normal surface of γ is a parametrized surface where ̥ is a constant.Now, we can prove the following interesting main result.

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Parallel Surfaces to Normal Ruled Surfaces of General Helices in Sol 3

Fig. 2 , 3 :
Fig. 2,3: The equations of normal surface and its parallel surface are illustrated colour Blue, Yellow , respectively.