MOTIONS ON CURVES AND SURFACES USING GEOMETRIC ALGEBRA

Geometric algebra is a useful tool to overcome some problems in kinematics. Thus, the geometric algebra has attracted the attention of many researchers. In this paper, quaternion operators on curves and surfaces in Euclidean 3-space are defined by using geometric algebra. These operators generate the curves or the surfaces from the points, curves or surfaces. Using quaternion operators, we obtain motions that have orbits along the generated curve or surface. Also, these motions are expressed as 1-parameter or 2-parameter homothetic motions.


Introduction
Kinematics is a research field of geometry to describe the motion of points, lines and other geometric objects. Thus, kinematics is used in many fields such as physics, mechanics, robotics and neuroscience. Homothetic motion is one of the most commonly researched topic in kinematics. 1-parameter and 2-parameter homothetic motions were researched in Euclidean 3-space E 3 [1,2]. Yaylı gave homothetic motions in Euclidean 4-space with Hamilton operators [3].
Sir William Rowan Hamilton [4] interpreted the quaternions as an extension to the complex numbers in 1843. K. Shoemake defined the system of rotation in E 3 by using quaternions [5]. Quaternions are more useful than Euler angles and matrices in representing of rotations of vectors. Therefore, quaternions have been used in many fields such as computer graphics, robotics and control theory.
Some problems and difficulties have been encountered in modeling of the mathematics of 3-dimensional (3D) kinematics. These difficulties have been tried to overcome by using quaternions. Bayro-Corrochano [6] used geometric algebra for the mathematical model of 3D kinematics of eye movements. Then, Leclercq at 40 S. ASLAN, Y. YAYLI al. modeled some movements in 3D kinematics such as rotations, translations and screw movements [7]. In [8], an isomorphism was given between the algebra of split semi-quaternions and the Clifford algebra Cl 1,0,1 . Moreover, semi-Euclidean planar motion was defined by using the algebra of split semi-quaternions. Some surfaces were obtained by quaternions or homothetic motions in [9][10][11][12][13][14][15]. Some results have been achieved about these surfaces using quaternions. Also, using quaternions in the shape operator expressed by Darboux frame, we defined the quaternionic shape operator [16]. Moreover, we used the quaternionic shape operator in researching of the differential properties of surfaces.
In this study, we define quaternion operators using curves and surfaces in E 3 . These operators have allowed us to obtain a quaternionic or a homothetic motion on each curve and surface in E 3 . These motions have orbits along curves or surfaces. Quaternion operator with curve orbit converts a point to a curve or a curve to a curve. This operator is expressed as 1-parameter homothetic motion. Similarly, quaternion operator with surface orbit converts a point to a surface, a curve to a surface, or a surface to a surface. Moreover, quaternion operator with surface orbit is expressed as 2-parameter homothetic motion. Finally, we give some applications of the quaternion operators.

Preliminaries
In this section, definitions and some algebraic properties of the concepts real quaternions, homothetic motions and geometric algebra will be given to provide a background.
The set H = {q = a 0 + a 1 i + a 2 j + a 3 k : a 0 , a 1 , a 2 , a 3 ∈ R} of real quaternions is equal to the 4-dimensional vector space R 4 . Quaternions have a basis {1, i, j, k} shortly given with some properties as The set of real quaternion is associative and not commutative algebra. 1 is identity element of H. Scalar and vector component of q are S(q) = a 0 ∈ R and V (q) = a 1 i + a 2 j + a 3 k ∈ E 3 , respectively. We can write quaternion q as q = S(q) + V (q). If S(q) = 0, q is called pure quaternion. Quaternion product * of q = S(q) + V (q) and p = S(p) + V (p) is defined as Conjugate, norm, modulus and inverse of q is respectively. If N q = 1, q is called unit quaternion. A unit quaternions can be written in the trigonometric form as q = cos θ + sin θv, where v ∈ E 3 and ∥v∥ = 1.
Let v 1 and v 2 be unit vectors in E 3 (i.e., pure quaternions), and θ = arccos (v 1 · v 2 ), Thus, the unit quaternion q can be given as where . ∥ ∥ is the modulus in E 3 . Unit quaternion q = cos θ + sin θv rotates the vector v 1 to the vector v 2 around the axis vector v, see Figure 1. For further information about real quaternions, see [3][4][5]17]. Let p = a 0 + a 1 i + a 2 j + a 3 k be a unit quaternion and w be a pure quaternion (i.e., vector in E 3 ). Linear mapping ϕ can be defined as Matrix corresponding to the linear mapping ϕ can be given as where R is orthogonal since RR T = I and det R = 1. Thus, ϕ represents a rotation in E 3 . If unit quaternion p is in the form then ϕ(w) rotates the vector w by 2θ [5]. 1-parameter homothetic motion in E 3 can be given as where y and x are the position vectors of the same point in the fixed space R ı and the moving space R, respectively. h, A and c are homothetic scalar, orthogonal matrix and translation vector, respectively. And "t" is homothetic parameter [1,2].
Similarly, 2-parameter homothetic motion in E 3 can be given as where y and x are the position vectors of the same point in the fixed space R ı and the moving space R, respectively. h, A and c are homothetic scalar, orthogonal matrix and translation vector, respectively. And "t and s" are homothetic parameters [1,2]. The geometric product of two unit vectors a and b is written as a * b and can be expressed as a sum of its symmetric and antisymmetric parts where the inner product a · b and the outer product a × b are defined by The inner product of two vectors is the standard scalar or dot product which results in a scalar. The outer or wedge product of two vectors is a new quantity we call a bivector. We think of a bivector as a directed area in the plane containing a and b, formed by sweeping a along b [6].

Quaternion Operators
In this part, we have defined quaternion operators by geometric algebra. By using this operator, we have obtained some results on the curves and surfaces. Definition 1. Let a and b be vectors in E 3 . By using the inner product a · b and the vectorial product a × b, quaternion operator can be defined as The quaternion operator Q converts the vector a to the vector b around the axis vector a × b in the plane formed by a and b as where a and b are pure quaternion. Using a · b = ∥a∥∥b∥ cos θ and ∥a × b∥ = ∥a∥∥b∥ sin θ in Eq. (10), we get where q = cos θ + sin θv, h = ∥b∥ ∥a∥ and v = a × b ∥a × b∥ . Thus, quaternion operator Q can be given as Q = hq. Hence Eq. (11) can be expressed as Q * a = hq * a can be given in Figure 2. Theorem 1. Let α(t) and P be a curve and a point in E 3 , respectively. Quaternion operator can be given as Q(t) generates the curve α(t) from the point P as where α(t) is the orbit of Q(t) * P and P , α(t) are pure quaternions.
Proof. If we take the unit quaternion p = cos θ + sin θv in Eq. (3) as q 1 (t) = cos θ(t) 2 + sin θ(t) 2 v(t), we get the orthogonal matrix corresponding to the mapping ϕ as ). In this case, matrix R(t) performs a rotation by angle 2 θ(t) 2 = θ(t) of the vector P around the axis v(t). Thus, we can give the equalities q(t) * P = ϕ(P ) = R(t)P.
Using these equations and h(t) = ∥α(t)∥ ∥P ∥ , we get It means that Q(t) * P can be expressed as 1-parameter homothetic motion Q(t) * P = h(t)R(t)P in E 3 . If we take the point P on the curve α(t) as P = α(t 0 ), then Q(t) * α(t 0 ) = h(t)R(t)α(t 0 ) can be given in Figure 3.  M (t, s). Thus, these operators can allow us to obtain a 1-parameter motion on every surface in E 3 . Proposition 1. Let α(t) and β(t) be curves in E 3 . Quaternion operator Q(t) can be given as where α(t) and β(t) are position vectors in E 3 . This quaternion operator converts the curve α(t) to the curve β(t) as 46 S. ASLAN, Y. YAYLI where the curve β(t) is the orbit of Q(t) * α(t). Moreover, Q(t) * α(t) can be given by 1-parameter homothetic motion as where R(t) is the orthogonal matrix satisfying is a homothetic scalar and t is homothetic parameter.

Quaternion Operator with Surface Orbit.
Theorem 3. Let M (t, s) and P be a surface and a point in E 3 , respectively. Quaternion operator Q(t, s) can be defined as where M (t, s) and P are position vectors in E 3 . The operator Q(t, s) generates the surface M (t, s) from the point P as where M (t, s) is the orbit of the Q(t, s) * P and P , M (t, s) are pure quaternions.
Proof. The proof of this theorem is similar to the proof of Theorem 1.
where M (t, s) and α(t) are pure quaternions. The operator Q(t, s) generates the surface M (t, s) from the curve α(t) as where Q(t, s) * α(t) has the surface orbit M (t, s).
Proposition 3. Let M (t, s) and N (t, s) be surfaces in E 3 . Quaternion operator can be defined as where M (t, s) and N (t, s) are pure quaternions. The operator Q(t, s) generates the surface N (t, s) from the surface M (t, s) as where Q(t, s) * M (t, s) has the surface orbit N (t, s).
where R(t, s) is the orthogonal matrix satisfying R(t, s)M (t, s) = q(t, s) * M (t, s), is a homothetic scalar, and t, s are homothetic parameters.

Conclusions
In this paper, we define quaternion operators using geometric algebra and classify these operators according to their orbits (i.e., curves or surfaces). Quaternion operator with curve orbit generates a curve from a point or a curve. This operator is given as 1-parameter homothetic motion. Similarly, quaternion operator with surface orbit generates a surface from a point, a curve or a surface. Quaternion operator with surface orbit is also expressed as 2-parameter homothetic motion. Thus, quaternion operators can form a homothetic and a quaternionic motion on every surface and curve in E 3 . Finally, we give some examples of the quaternion operators.