Received 18 June 2015; accepted 24 July 2015; published 27 July 2015

ABSTRACT

In this paper we consider the homothetic motion of Lorentzian circle by studying the scalar curvature for the corresponding cyclic surface locally. We prove that if the scalar curvature is constant, then. We describe the equations that govern such surfaces.

Keywords:

Minkowski Space, Cyclic Surfaces, Homothetic Motion, Scalar Curvature

1. Introduction

Homothetic motion is general form of Euclidean motion. It is crucial that homothetic motions are regular motions. These motions have been studied in kinematic and differential geometry in recent years. An equiform transformation in the n-dimensional Euclidean space is an affine transformation whose linear part is composed from an orthogonal transformation and a homothetical transformation add see [1] -[3] . Such an equiform transformation maps points according to

(1)

The number s is called the scaling factor. A homothetic motion is defined if the parameters of (1), including s, are given as functions of a time parameter t. Then a smooth one-parameter equiform motion moves a point x via. The kinematic corresponding to this transformation group is called similarity kinematic. See [4] . Recently, the similarity kinematic geometry has been used in computer vision and reverse engineering of geometric models such as the problem of reconstruction of a computer model from an existing object which is known (a large number of) data points on the surface of the technical object [5] [6] . Abdel-All and Hamdoon studied a cyclic surface in. In this sense, they proved that such surface in is in general contained in a canal hypersurface [7] . Solouma ( [8] - [10] ) studied locally some geometric problems on surfaces obtained by the equiform motion up to the first order. In Minkowski (semi-Euclidean) space, hyperbolas (Lorentzian circles) play role in Euclidean space [11] .

In this work we consider the homothetic motion of the hyperbolas(Lorentzian circles). Let and be two copies of Euclidean space. Under a one-parameter homothetic motion of moving space with respect to fixed space, we consider which is moved according homothetic motion. The point paths of the Lorentzian circle generate a cyclic surface X, containing the position of the starting Lorentzian circle. At any moment, the infinitesimal transformations of the motion will map the points of the Lorentzian circle into the velocity vectors whose end points will form an affine image of that will be, in general, a Lorentzian circle in the moving space. Both curves are planar and therefore, they span a subspace W of, with. This is the reason because we restrict our considerations to dimension.

Let be a parametrization of and the resultant surface by the homothetic motion. We consider a certain position of the moving space, given by, and we would like to obtain information about the motion at least during a certain period around if we know its characteristics for one instant. Then we restrict our study to the properties of the motion for the limit case. A first choice is then approximate by the first derivative of the trajectories. The purpose of this paper is to describe the cyclic surfaces obtained by the homothetic motion of the Lorentzian circle and whose scalar curvature is constant.

The proof of our results involves explicit computations of the scalar curvature of the surface. As we shall see, equation reduces to an expression that can be written as a linear combination of the hyperbolic functions and, , namely, and and are functions on the variable t. In particular, the coefficients must vanish. The work then is to compute explicitly these coefficients and by successive manipulations. The authors were able to obtain the results using the symbolic program Mathematica to check their work. The computer was used in each calculation several times, giving understandable expressions of the coefficients and.

This paper is organized as follows: In Section 2, we obtain the expression of the scalar curvature for the cyclic surfaces obtained by homothetic motion of Lorentzian circle. In successive Sections 3 and 4, we distinguish the cases and, respectively. Finally, in Section 5 explicit examples of surfaces with and are given.

2. Scalar Curvature of Cyclic Surfaces

In two copies, of semi-Euclidean 5-space, we consider a unit Lorentzian circle in the - plane of centered at the origin and represented by

Under a one-parameter homothetic motion of in the moving space with respect to fixed space. The position of a point at “time” t may be represented in the fixed system as

(2)

where describes the position of the origin of at the time t,

, is a semi orthogonal matrix and provides the scaling factor of the moving

system. For varying t and fixed, gives a parametric representation of the path (or trajectory) of. Moreover we assume that all involved functions are of class. Using the Taylor’s expansion up to the first order, the representation of the cyclic surface is

where denotes the differentiation with respect to t.

As homothetic motion has an invariant point, we can assume without loss of generality that the moving frame and the fixed frame coincide at the zero position. Then we have

Thus

where, is a semi skew-symmetric matrix. In this paper all values of and their derivatives are computed at and for simplicity, we write and instead of and respectively. In these frames, the representation of is given by

or in the equivalent form

(3)

For any fixed t in the above expression (3), we generally get an ellipse centered at the point. The latter ellipse reduce to a Lorentzian circle subject to the following conditions

(4)

where. We now compute the scalar curvature of the cyclic surface. The tangent vectors to the parametric curves of are

A straightforward computation leads to the coefficients of the first fundamental form defined by, ,. The scalar product in the above equation in Lorentzian metric. According to the inner product this equation tends to, , where

is the sign matrix. Then we get

Under the conditions (4) a computation yields

(5)

and

(6)

The Christoffel symbols of the second kind are defined by

where, are indices that take the value 1 or 2 and is the inverse matrix of. From here, the scalar curvature of is defined by

Although the explicit computation of the scalar curvature can be obtained, for example, by using the Mathematica programme, its expression is some cumbersome. However, the key in our proofs lies that one can write as

(7)

The assumption of the constancy of the scalar curvature implies that (7) converts into

(8)

Equation (8) means that if we write it as a linear combination of the functions namely,

, the corresponding coefficients must vanish. From here, we will be able to

describe all cyclic surfaces with constant scalar curvature obtained by the homothetic motion of the Lorentzian circle. As we will see, it is not necessary to give the (long) expression of but only the coefficients of higher order for the hyperbolic functions.

We distinguish the cases and.

3. Cyclic Surfaces with K = 0

In this section we assume that on the surface. From (7), we have

(9)

We distinguish different cases that fill all possible cases (Note that we have all solutions by using the symbolic program Mathematica under the condition).

3.1. Case

At and, the coefficients for and the coefficients for. Also, since implies that. But if and only if. That’s means gives contradiction with Equation (9), so we have. We then conclude the following theorem.

Theorem 3.1. Let be a cyclic surfaces obtained by the homothetic motion of Lorentzian circle c₀ and given by (3) under condition (4). Assume, then on the surface if and only if the following conditions hold:

1)

2)

In particular, if for, then circles generating the cyclic surfaces are coaxial.

3.2. Case, But either or Is Not Zero

We have two possibilities:

1) If and, then we have, , the coefficients for and the

coefficients that’s means the equation. From expression (6), we have two

conditions

2) If and , then we have, , the coefficients for and

the coefficients that’s means the equation. From expression (6), we have

Theorem 3.2. Let be a cyclic surfaces obtained by the homothetic motion of Lorentzian circle c₀ and given by (3) under condition (4) hold:

1) Assume and, then on the surface if and only if the following conditions

2) Assume and, then on the surface if and only if the following conditions

3.3. Case

If, then we have, then coefficients for, and for that’s means the equation (8) hold (i.e.,). From expression (6), we have the two conditions

Theorem 3.3. Let be a cyclic surfaces obtained by the homothetic motion of Lorentzian circle c₀ and given by (3) under condition (4). Assume, then on the surface if and only if the following conditions hold:

1)

2)

4. Cyclic Surfaces with K ¹ 0

In this section we assume that the scalar curvature of the cyclic surface obtained by the homothetic motion of Lorentzian circle and given by (3) under condition (4) is a non-zero constant. The identity (8) writes then as

(10)

Following the same scheme as in the case studied in Section 3, we begin to compute the coefficients and. Let us put.

1) CASE. The coefficients, and are

If, we distinguish different possibilities:

1., , , we conclude that

2., and, we have the same result as in the above case.

3., and, we have the same result as in cases from (1) and (2).

From (1), (2) and (3) we have, , under the following conditions

4., ,. The coefficients and are

If, we have the following conditions

2) CASE, but either or is not zero. We have two possibilities:

1. If and, then the coefficient, implies that: contradiction

2. If and, then the coefficient, implies that which gives a contradiction also.

3) CASE. The computations of implies that, contradiction. As conclusion of the above reasoning, we conclude the following theorem.

Theorem 4.1. Let be a cyclic surfaces obtained by the homothetic motion of Lorentzian circle

and given by (3) under condition (4). Assume that, then the scalar curvature or

on the surface if and only if the following conditions hold:

5. Examples of a Cyclic Surfaces with K = 0 and K ¹ 0

In this section, we construct two examples of a cyclic surfaces with constant scalar curvature and. The first example corresponds with the case. In the second example, we assume and.

Example 1. Case. Let now the semi orthogonal matrix

(11)

We assume and, then

Theorem 3.3 says that. In Figure 1, we display a piece of of Example 1 in axonometric view- point. For this, the unit vectors and are mapped onto the vectors and respectively [2] . Then

and

and both and parametrize domains of the -plane.

Example 2. Case. Consider the semi orthogonal matrix

(12)

Let and, then

Theorem 4.1 says that or. In Figure 2, we display a piece of of Example 2 in

axonometric viewpoint. Then

and

and both and parametrize domains of the -plane.

Cite this paper

M. M.Wageeda,E. M.Solouma, (2015) Local Study of Scalar Curvature of Cyclic Surfaces Obtained by Homothetic Motion of Lorentzian Circle. Applied Mathematics,06,1344-1352. doi: 10.4236/am.2015.68127

References