Local Study of Scalar Curvature of Cyclic Surfaces Obtained by Homothetic Motion of Lorentzian Circle ()
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.