Studying Scalar Curvature of Two Dimensional Kinematic Surfaces Obtained by Using Similarity Kinematic of a Deltoid ()
1. Introduction
The last name of tricuspid is deltoid. The deltoid has no real discoverer because of its relation to the cycloid. The deltoid is a special case of a cycloid, and it is also called a three-cusped hypocycloid or a tricuspid. It was named the deltoid because of its resemblance to the Greek letter Delta. Despite this, Leonhard Euler was the first to claim credit for investigating the deltoid in 1754. Though, Jakob Steiner was the first to actually study the deltoid in depth in 1856. From this, the deltoid is often known as Steiner’s Hypocycloid.
To understand the deltoid, aka the tricuspid hypocycloid, we must first look to the hypocycloid, A hypocy- cloid is the trace of a point on a small circle drawn inside of a large circle, the small circle rolls along inside the circumference of the larger circle, and the trace of a point in the small circle will form the shape of the hypocy- cloid, The ratio of the radius of the inner circle to that of the outer circle
is what makes each Hypocy-
cloid unique, curved are an engineering point replace the circumference of a circle with a radius of a roll within a radius 3a, Where a is the radius of the large fixed circle and b is the radius of the small rolling circle [1] .
From the view of differential geometry, deltoid is a geometric curve with non vanishing constant curvature K [2] . Similarity kinematic transformation in the n-dimensional an Euclidean space
is an affine transfor- mation whose linear part is composed by an orthogonal transformation and a homothetical transformation [3] - [7] . Such similarity kinematic transformation maps points
according to the rule
(1)
The number s is called the scaling factor. Similarity kinematic motion is defined if the parameters of (1), including s, are given as functions of a time parameter t. Then a smooth one-parameter similarity kinematic motion moves a point x via
. The kinematic corresponding to this transformation group is called equiform kinematic. See [8] [9] . Consider hypersurfaces in space forms generated by one- parameter family of spheres and having constant curvature [10] -[13] .
In this work, we consider the similarity kinematic motion of the deltoid
. Let
and
be two copies of Euclidean space
. Under a one-parameter similarity kinematic motion of moving space
with respect to fixed space
, we consider
which is moved according similarity kinematic motion. The point paths of the deltoid generate a kinematic surface X, containing the position of the starting tricuspid. At any moment, the infinitesimal transformations of the motion will map the points of the deltoid
into the velocity vectors whose end points will form an affine image of
that will be, in general, a deltoid in the moving space
. Both curves are planar and therefore, they span a subspace W of
, with
. This is the reason why we restrict our considerations to dimension
.
2. Locally Representation of the Motion
In two copies
,
of Euclidean 5-space
, we consider a unit deltoid
in the
-plane of
with its centered at the origin and represented by
![]()
Under a one-parameter similarity kinematic 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 an 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 kinematic surface is
![]()
where
denotes the differentiation with respect to t.
As similarity kinematic 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 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 a deltoid with its centered at the point
subject to the following condition
(4)
3. Scalar Curvature of Two-Dimensional Kinematic Surfaces
In this section we compute the scalar curvature of the two-dimensional kinematic surface
. The tangent vectors to the parametric curves of
are
![]()
A straightforward computation leads to the coefficients of the first fundamental form defined by
,
,
:
![]()
Under the conditions (5) a computation yields
(5)
where
(6)
The scalar curvature of
is defined by
![]()
where
be the Christoffel symbols of the second kind are
![]()
where
,
are indices that take the value 1 or 2 and
is the inverse matrix of
see [14] . 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.
4. Kinematic Surfaces with K = 0
In this section we assume that
on the surface
. From (7), we have
![]()
Then the work consists in the explicit computations of the coefficients
and
.
We distinguish different cases that fill all possible cases. The coefficients
are trivially zero and the coefficient
is
![]()
We have two possibilities.
1) If
. From expression (6), we have
which yields to a contradiction.
2) If
. Then most coefficients are trivially zero and the coefficients
and
are
![]()
![]()
If
, we have
or
. Then if
we have all coefficients are trivially zero. Now if
, from expression (6), we have
for
. We then conclude:
Theorem 4.1 Let
be a two dimensional kinematic surfaces obtained by similarity kinematic motion of deltoid s0 and given by (3) under condition (4). Assume
. Then
on the surface if
and one of the following conditions are satisfies
1)
,
2)
.
In particular, if
, the deltoid generating the two dimensional kinematic surfaces are coaxial.
5. Kinematic Surfaces with K ¹ 0
Assume in this section that the scalar curvature
of the kinematic surfaces
given in (3) is a non- zero constant. The identity (8) writes then as
(9)
Following the same scheme as in the case
studied in Section 4, we begin to compute the coefficients
and
. Let us put
.
The coefficient
is
![]()
We have to two possibilities:
1) If
. The coefficient
and
are
![]()
![]()
It follows that
and
or
. If
and
, then coefficient
is
![]()
Then
implies that
which give a contradiction. Now if
, then the coefficient
is
. Then
leads to
which gives a contradiction also.
2). If
. Then the coefficient
is
![]()
Then
implies that
contradiction. As conclusion of the above reasoning, we conclude:
Theorem 5.1 There are not two dimensional kinematic surfaces obtained by similarity kinematic motion of a deltoid
and given by (3) under condition(4) whose scalar curvature
is a non-zero constant.
6. Examples of Two Dimensional Kinematic Surfaces with Vanishing Scalar Curvature
In this section, we construct two examples of a kinematic surface
with constant scalar curvature
. The first example corresponds with the case
. In the second example, we assume
.
Example 1 Case
.
Consider the following orthogonal matrix.
(10)
We assume that the factor
and
. Here we have
,
,
for
,
,
and
, for
. Then Theorem 4.1 says us that the corresponding surface
has
. In Figure 1, we display a piece of
of Example 1 in axonometric viewpoint
. For this, the unit vectors
and
are mapped onto the vectors
and
respectively (5). Then
![]()
and
![]()
Example 2 Case
. Let now the orthogonal matrix
(11)
We assume
and
. Then
.
Theorem 4.1 says that
. In Figure 2, we display a piece of
of Example 2 in axonometric viewpoint
. For this, the unit vectors
and
are mapped onto the vectors
and
respectively (5). Then
![]()
and
![]()
7. A Local Isometry between Two Dimensional Surfaces
In this section, we shall study the existence of a local isometry between a two dimensional surface in
represented by
in (3) with constant scalar curvature and a two dimensional surface in Euclidean three-space
. For more details see [6] [15] .
Now, we construct a two dimensional surface
in
locally isometric
determined by (3). Where
and
defined in the same domain U such that
and
in U. Then the map
is a local isometry.
For this, we assume that the initial deltoid
is the same that in
. Then
writes as
(12)
The computation of the first fundamental form of
leads to
(13)
And
(14)
As in the case studied
, we have assumed that the original two axis of the deltoid are orthogonal. This means
. On the other hand, the first fundamental form of
was calculated in (5). From X and
, we have equations on the trigonometric functions
and
.
The identities
imply
,
,
,
and
,
,
,
,
.
Thus
![]()
Then
. We impose that the scalar curvature k is constant. We know that
Or when
In particular ,
or
. We conclude:
Theorem 7.1 Consider a two dimensional kinematic surface in
given by the parametrization
in (3) under condition (4) and with constant scalar curvature. Let
be a two dimensional kinematic surface in
defined by (12). If the following equations hold:
![]()
Then both surfaces
and
are locally isometric. The Gaussian curvature of the surface
in Euclidean space
must vanish.