31 research outputs found
An approach for minimal surface family passing a curve
We investigate minimal surfaces passing a given curve in . Using the
Frenet frame of a given curve and isothermal parameter, we derive the necessary
and sufficient condition for minimal surface. Also we derive the parametric
representation of two minimal surface families passing a circle and a helix as
examples
Ruled Surfaces according to Rotation Minimizing Frame
In this paper, we investigate the ruled surfaces generated by a straight line
according to rotation minimizing frame (RMF). Using this frame of a straight
line, we obtained the necessary and sufficient conditions when the ruled
surface is developable. Also, we give some new results and theorems related to
be the asymptotic curve, the geodesic curve and the line of curvature of the
base curve on the ruled surface.Comment: 10 pages, 2 Figure
Intrinsic Equations For a Relaxed Elastic Line of Second Kind on an Oriented Surface
Let {\alpha}(s) be an arc on a connected oriented surface S in E3,
parameterized by arc length s, with torsion {\tau} and length l. The total
square torsion F of {\alpha} is defined by T=\int_{0}^{l}\tau ^{2}ds\ $. . The
arc {\alpha} is called a relaxed elastic line of second kind if it is an
extremal for the variational problem of minimizing the value of F within the
family of all arcs of length l on S having the same initial point and initial
direction as {\alpha}. In this study, we obtain differential equation and
boundary conditions for a relaxed elastic line of second kind on an oriented
surface.Comment: 8 page
Surfaces Family With Common Null Asymptotic
We analyzed the problem of finding a surfaces family through an asymptotic
curve with Cartan frame. We obtain the parametric representation for surfaces
family whose members have the same as an asymptotic curve. By using the Cartan
frame of the given null curve, we present the surface as a linear combination
of this frame and analysed the necessary and sufficient condition for that
curve to satisfy the asymptotic requirement. We illustrate the method by giving
some examples.Comment: 10 pages and 5 figure
Surfaces with a common asymptotic curve in Minkowski 3-space
In this paper, we express surfaces parametrically through a given spacelike
(timelike) asymptotic curve using the Frenet frame of the curve in Minkowski
3-space. Necessary and sufficient conditions for the coefficients of the Frenet
frame to satisfy both parametric and asymptotic requirements are derived. We
also present some interesting examples to show the validity of this study.Comment: 10 pages, 7 figure
Surface family with a common natural asymptotic lift of a timelike curve in Minkowski 3-space
In the present paper, we find a surface family possessing the natural lift of
a given timelike curve as a asymptotic in Minkowski 3-space. We express
necessary and sufficient conditions for the given curve such that its natural
lift is a asymptotic on any member of the surface family. Finally, we
illustrate the method with some examples
Structure and characterization of ruled surfaces in Minkowski 3-space
In this paper, we consider non developable ruled surface with spacelike
ruling, timelike ruling, respectively. We give the relations between the
structure functions with the curvature and torsion of the striction line of the
timelike and spacelike non developable ruled surfaces. Also, we have calculated
the gaussian and mean curvatures of timelike and spacelike non developable
ruled surfaces using the structure functions.Comment: 9 page
Parametric Representation of a Hypersurface Family With a Common Spatial Geodesic
In this paper, we study the problem of finding a hypersurface family from a
given spatial geodesic curve in R4. We obtain the parametric representation for
a hypersurface family whose members have the same curve as a given geodesic
curve. Using the Frenet frame of the given geodesic curve, we present the
hypersurface as a linear combination of this frame and analyse the necessary
and sufficient condition for that curve to be geodesic. We illustrate this
method by presenting some examples.Comment: 17 pages, 3 figure
An approach for designing a surface pencil through a given asymptotic curve
Surfaces and curves play an important role in geometric design. In recent
years, problem of finding a surface passing through a given curve has attracted
much interest. In the present paper, we propose a new method to construct a
surface interpolating a given curve as the asymptotic curve of it. Also, we
analyze the conditions when the resulting surface is a ruled surface.
Furthermore, we prove that there exists no developable surface possessing a
given curve as an asymptotic curve except plane. Finally, we illustrate this
method by presenting some examples.Comment: 13 pages, 7 figures. arXiv admin note: text overlap with
arXiv:1305.3381, arXiv:1406.0618 by other author
An approach for designing a surface pencil through a given geodesic curve
Surfaces and curves play an important role in geometric design. In recent
years, problem of finding a surface passing through a given curve have
attracted much interest. In the present paper, we propose a new method to
construct a surface interpolating a given curve as the geodesic curve of it.
Also, we analyze the conditions when the resulting surface is a ruled surface.
In addition, developablity along the common geodesic of the members of surface
family are discussed. Finally, we illustrate this method by presenting some
examples.Comment: 13 pages, 5 figure