Mehmet Önder

DIFFERENTIAL EQUATIONS CHARACTERIZING TIMELIKE AND SPACELIKE CURVES OF CONSTANT BREADTH IN MINKOWSKI 3-SPACE E 1 3

In this paper, we give the differential equations characterizing the timelike and spacelike curves of constant breadth in Minkowski 3-space . Furthermore, we give a criterion for a timelike or spacelike curve to be the curve of constant breadth in . As an example, the obtained results are applied to the case = const. and = const., and are discussed.

On the Developable Mannheim Offsets of Timelike Ruled Surfaces

In this paper, using the classifications of timelike and spacelike ruled surfaces, we study the Mannheim offsets of timelike ruled surfaces in the Minkowski 3-space. First, we define the Mannheim offsets of a timelike ruled surface by considering the Lorentzian casual character of the offset surface. We obtain that the Lorentzian casual character of the Mannheim offset of a timelike ruled surface may be timelike or spacelike. Furthermore, we give characterizations for developable Mannheim offsets of a timelike ruled surface.

Dual Darboux Frame of a Timelike Ruled Surface and Darboux Approach to Mannheim Offsets of Timelike Ruled Surfaces

In this paper, we introduce the dual geodesic trihedron (dual Darboux frame) of a timelike ruled surface. By the aid of the E. Study Mapping, we consider timelike ruled surfaces as dual hyperbolic spherical curves and define the Mannheim offsets of timelike ruled surfaces by means of dual Darboux frame. We obtain the relationships between invariants of Mannheim timelike surface offsets. Furthermore, we give the conditions for these surface offsets to be developable.

Some Results and Characterizations for Mannheim Offsets of the Ruled Surfaces

abstract: In this study, we give dual characterizations for Mannheim offsets of ruled surfaces in terms of their integral invariants and obtain a new characterization for the Mannheim offsets of a developable surface, i.e., we show that the striction lines of developable Mannheim offset surfaces are Mannheim partner curves. Furthermore, we obtain the relationships between the area of projections of spherical images for Mannheim offsets of ruled surfaces and their integral invariants.

On the invariants of Mannheim offsets of timelike ruled surfaces with spacelike rulings

In this paper, we give the characterizations for Mannheim offsets of timelike ruled surfaces with spacelike rulings in dual Lorentzian space . We obtain the relations between terms of their integral invariants and also we give new characterization for the Mannheim offsets of developable timelike ruled surface. Moreover, we find relations between the area of projections of spherical images for Mannheim offsets of timelike ruled surfaces and their integral invariants.

Eikonal slant helices and Eikonal Darboux helices in 3-dimensional Riemannian manifold

In this study, we give the definitions and characterizations of Eikonal slant helices, Eikonal Darboux helices and non-modified Eikonal Darboux helices in 3-dimensional Riemannian manifold M3. We show that every Eikonal slant helix is also an Eikonal Darboux helix. Furthermore, we obtain that if the curve α is a non-modified Eikonal Darboux helix, then α is an Eikonal slant helix if and only if κ2 + τ2 = constant, where κ and τ are curvature and torsion of α, respectively.

Dual Darboux Frame of a Spacelike Ruled Surface and Darboux Approach to Mannheim Offsets of Spacelike Ruled Surfaces

In this paper, we define dual geodesic trihedron(dual Darboux frame) of a spacelike ruled surface. Then, we study Mannheim offsets of spacelike ruled surfaces in dual Lorentzian space by considering the E. Study Mapping. We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim offsets of the spacelike ruled surfaces by means of dual Darboux frame. We obtain relationships between the invariants of Mannheim spacelike offset surfaces and offset angle, offset distance. Furthermore, we give conditions for these surface offsets to be developable.

On the kinematic interpretation of timelike ruled surfaces

Timelike ruled surfaces are studied in dual Lorentzian space by considering E. Study Mapping and Blaschke frame. A reference timelike ruled surface is considered and associated surfaces are defined. First, it is shown that the surface generated by the instantaneous screw axis (ISA) is a Mannheim offset of reference surface. Later, the kinematic interpretations between these surfaces are introduced by means of Blaschke invariants.

Construction of a surface pencil with a common special surface curve

In this study, we introduce a new type of surface curves called D-type curve. This curve is defined by the property that the unit Darboux vector W0 of a space curve r(s) and unit surface normal n along the curve r(s) satisfy the condition =constant. We point out that a D-type curve is a geodesic curve or an asymptotic curve in some special cases. Then, by using the Frenet vectors and parametric representation of a surface pencil as a linear combination of the Frenet vectors, we investigate necessary and sufficient condition for a curve to be a D-type curve on a surface pencil. Moreover, we introduce some corollaries by considering D-type curve as a helix, a Salkowski curve or a planar curve. Finally, we give some examples for obtained results and plot the surfaces by using Mapple.

CHARACTERIZATIONS OF SPACE CURVES WITH 1-TYPE DARBOUX INSTANTANEOUS ROTATION VECTOR

Abstract. In this study, by using Laplace and normal Laplace operators,we give some characterizations for the Darboux instantaneous rotationvector field of the curves in the Euclidean 3-space E 3 . Further, we givenecessary and sufficient conditions for unit speed space curves to have1-type Darboux vectors. Moreover, we obtain some characterizations ofhelices according to Darboux vector. 1. IntroductionOne of the most important problems of local differential geometry is toobtain the relations characterizingspecial curves with respect to their curvatureand torsion. The well-known types of such special curves are constant slopecurves or general helices which are defined by the property that the tangentvectors of curves make a constant angle with fixed directions. A necessaryand sufficient condition for a curve to be a general helix in the Euclidean 3-space E 3 is that the ratio of curvature to torsion is constant [11]. So, manymathematicians have focused their studies on these special curves in differentspaces such as Euclidean space and Minkowski space [3, 4, 5, 10].Furthermore, Chen and Ishikawa [1] classified biharmonic curves, the curvesfor which ∆H~ = 0 holds in semi-Euclidean space E