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This paper presents building one-parameter motion by complex numbers on a time scale. Firstly, we assumed that **E** and **E**′ were moving in a fixed time scale complex plane and {0, e_{1},e_{2}} and {0', e'_{1},e'_{2}} were their orthonormal frames, respectively. By using complex numbers, we investigated the delta calculus equations of the motion on T. Secondly, we gave the velocities and their union rule on the time scale. Finally, by using the delta-derivative, we got interesting results and theorems for the instantaneous rotation pole and the pole curves (trajectory). In kinematics, investigating one-parameter motion by complex numbers is important for simplifying motion calculation. In this study, our aim is to obtain an equation of motion by using complex numbers on the time scale.

The calculus on time scales was initiated by B. Aulbach and S. Hilger in order to create a theory that can unify discrete and continuous analysis, [

In this study, some properties of motion in references [

A time scale is an arbitrary nonempty closed subset of the real numbers.

Definition 2.1. Let

and the backward jump operator

In this definition, we put

Finally, the graininess function

If

Let us define the interior of

Definition 2.2. Assume

We call

Theorem 2.1. Assume

1) If f is differentiable at t, then f is continuous at t.

2) If f is continuous at t and t is right-scattered, then f is differentiable at t with

3) If t is right-dense, then f is differential at t if the limit

exists as a finite number. In this case a given

4) If f is differentiable at t then

Theorem 2.2. Assume

1) The sum

2) For any constant,

3) The product

4) If

5) If

In the reference [

Theorem 2.3. Assume

Theorem 2.4. Let

holds.

Theorem 2.5. Assume that

Definition 2.3. For the given time scales

where

Definition 2.4. For

and for

Definition 2.5. If

where the cylinder transformation

Theorem 2.6. If

1)

2)

3)

4)

5)

6)

7)

8)

Theorem 2.7. Assume

Theorem 2.8. If

Theorem 2.9. If

Assume that

Here, let

The translation vector

by using the definition of the time scale complex plane. The translation vector is more suitable as

for doing the formulas symmetric on the moving plane.

Thus,

For any point

By substituting

Then, we can obtain the vector

Here, assume the functions

are

Definition 3.1. A velocity vector of the point X with respect to E is called

for the moving time scale complex plane.

Definition 3.2. A velocity vector of the point X with respect to E is called

for the fixed time scale complex plane.

Definition 3.3. A velocity vector of the point X with respect to the time scale complex plane

Definition 3.4. On the planar motion

So, we obtain the

by Theorem 2.5. Also

and using Theorem 2.7, we have

Here,

with the restriction

and using Equation (3.2), we get

Theorem 3.1. A

Proof. By using Equation (3.10) and Equation (3.5), we can get the following equations:

and thus, we get the relation of the velocities:

We have

We will calculate

and

Theorem 3.2. There is only one point at which the

Proof. The points at which the

we can obtain the following complex vectors;

which are given

Definition 3.5. The point

Definition 3.6. The point

We can get the following equations from Equation (3.15) and Equation (3.16):

By eliminating

and;

Result 4.1. Two results for the

1) Since scalar product of the vector is

and the vector

2) The length of the vector

here

Theorem 4.1. On the motion

Theorem 4.2. Every point of X of the moving plane E is doing rotational movement (instantaneous rotation movement) with a

Since X is an arbitrary point of the time scale complex plane E, we can give the following theorem:

Theorem 4.3. A one-parameter motion consists of rotation with

Theorem 4.4. The velocity vectors of the instantaneous rotation pole

Theorem 4.5. On one-parameter planar motion

Result 4.2. Without being depended on time, a motion