Studying Scalar Curvature of Two Dimensional Kinematic Surfaces Obtained by Using Similarity Kinematic of a Deltoid

We consider a similarity kinematic of a deltoid by studying locally the scalar curvature for the corresponding two dimensional kinematic surfaces in the Euclidean space  . We prove that there is no two dimensional kinematic surfaces with scalar curvature K is non-zero constant. We describe the equations that govern such the surfaces.


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 hypocycloid 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 hypocycloid, The ratio of the radius of the inner circle to that of the outer circle ( ) a b 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 n  is an affine transformation whose linear part is composed by an orthogonal transformation and a homothetical transformation [3]- [7].Such similarity kinematic transformation maps points 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 oneparameter family of spheres and having constant curvature [10]- [13].
In this work, we consider the similarity kinematic motion of the deltoid 0 s .Let 0 Σ and Σ be two copies of Euclidean space n  .Under a one-parameter similarity kinematic motion of moving space 0 Σ with respect to fixed space Σ , we consider 0 0 s ⊂ Σ 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 0 s into the velocity vectors whose end points will form an affine image of 0 s that will be, in general, a deltoid in the moving space Σ .
Both curves are planar and therefore, they span a subspace W of n R , with ( ) 5 dim W ≤ .This is the reason why we restrict our considerations to dimension 5 n = .

Locally Representation of the Motion
In two copies 0 Σ , Σ of Euclidean 5-space 5  , we consider a unit deltoid 0 s in the 1 2 x x -plane of 0 Σ with its centered at the origin and represented by ( ) ( ) T 2 cos cos 2 , 2sin sin 2 , 0, 0, 0 , , .
Under a one-parameter similarity kinematic motion of 0 s in the moving space 0 Σ with respect to fixed space Σ .The position of a point ( ) 0 x φ ∈ Σ at "time" t may be represented in the fixed system as where ( ) describes the position of the origin of 0 Σ at the time t, ≤ is an orthogonal matrix and ( ) s t provides the scaling factor of the moving system.For varying t and fixed ( ) x φ , ( ) , X t φ gives a parametric representation of the path (or trajectory) of ( ) x φ .Moreover, we assume that all involved functions are of class 1 C .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 0 Σ and the fixed frame Σ coincide at the zero position 0 t = .Then we have ( ) ( ) ( ) 0 , 0 1 and 0 0.
where ( ) ( ) is a skew-symmetric matrix.In this paper all values of , i s b and their derivatives are computed at 0 t = and for simplicity, we write s′ and i b′ instead of ( ) In these frames, the representation of ( ) , X t φ is given by ( ) For any fixed t in the above expression (3), we generally get a deltoid with its centered at the point ( ) ω ω ω ω ω ω ω ω ω ω ω ω

Scalar Curvature of Two-Dimensional Kinematic Surfaces
In this section we compute the scalar curvature of the two-dimensional kinematic surface ( ) , X t φ .The tangent vectors to the parametric curves of ( ) A straightforward computation leads to the coefficients of the first fundamental form defined by where ( ) ( ) The scalar curvature of ( ) , , .
where k ij Γ be the Christoffel symbols of the second kind are i j k are indices that take the value 1 or 2 and ( ) lm g is the inverse matrix of ( ) ij g see [14].Although the explicit computation of the scalar curvature K 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 K as The assumption of the constancy of the scalar curvature K implies that (7) converts into ( ) ( ) Equation ( 8) means that if we write it as a linear combination of the functions { } cos ,sin n n φ φ namely, ( ) , the corresponding coefficients must vanish.

Kinematic Surfaces with K = 0
In this section we assume that 0 = K on the surface ( ) , X t φ .From (7), we have A is ( ) We have two possibilities.1) If 0 η = .From expression (6), we have 0 s′ = which yields to a contradiction.

( )
, X t φ be a two dimensional kinematic surfaces obtained by similarity kinematic motion of deltoid s 0 and given by (3) under condition (4).

Kinematic Surfaces with K ≠ 0
Assume in this section that the scalar curvature K of the kinematic surfaces ( ) 3) is a non- zero constant.The identity (8)  ) ) . Then 9 0 F = leads to 0 s′ = which gives a contradiction also.

2). If
( ) s and given by (3) under condition(4) whose scalar curvature K is a non-zero constant.

A Local Isometry between Two Dimensional Surfaces
In this section, we shall study the existence of local isometry between a two dimensional surface in 5  represented by ( ) , X t φ in (3) with constant scalar curvature and a two dimensional surface in Euclidean three-space 3   .For more details see [6] [15].Now, we construct a two dimensional surface ( ) The computation of the first fundamental form of ( ) And As in the case studied 5   , we have assumed that the original two axis of the deltoid are orthogonal.This means 2 3 0 ω ω = .On the other hand, the first fundamental form of ( ) , X t φ was calculated in (5).From X and X , we have equations on the trigonometric functions ω ω ω ω ω = = + + .We impose that the scalar curvature k is constant.We know that 0,2 7.  given by the parametrization ( )   must vanish.

∑
Then the work consists in the explicit computations of the coefficients n A and n B .We distinguish different cases that fill all possible cases.The coefficients9 6 3   , , B B B are trivially zero and the coefficient 9 coefficients are trivially zero and the coefficients 3 A and 6 A are ( ) Assume the deltoid generating the two dimensional kinematic surfaces are coaxial.

F 5 . 1
= implies that 0 s′ = contradiction.As conclusion of the above reasoning, we conclude: Theorem There are not two dimensional kinematic surfaces obtained by similarity kinematic motion of a deltoid 0 X t φ with constant scalar curvature 0 = K .The first example corresponds with the case 1 2 0 b b ′ ′ ≠ .In the second example, we assume

Figure 1 .
Figure 1.In (a), we have a piece of the two dimensional kinematic surface in axonometric view ( ) , Y t φ with zero scalar

Figure 2 .
Figure 2. In (a), we have a piece of the two dimensional kinematic surface in axonometric view ( ) , Y t φ with zero scalar

1
Consider a two dimensional kinematic surface in5 are locally isometric.The Gaussian curvature of the surface ( ) , X t φ in Euclidean space3 ).