Abstract

We show a new method to construct constant slope surfaces with quaternions. Moreover, we give some results and illustrate an interesting shape of constant slope surfaces by using Mathematica.

1. Introduction

Quaternions were invented by Sir William Rowan Hamilton as an extension to the complex number in 1843. Hamiltonโ€™s defining relation is most succinctly written as๐ข2=๐ฃ2=๐ค2=๐ขร—๐ฃร—๐ค=โˆ’๐Ÿ.(1.1)

Computing rotations is a common problem in both computer graphics and character animation. Shoemake [1] introduced an algorithm using quaternions, spherical linear interpolation (SLERP), and Bezier curves to solve this. Quaternions are used as a powerful tool for describing rotations about an arbitrary axis. Many physical laws in classical, relativistic, and quantum mechanics can be written nicely using them. They are also used in aerospace applications, flight simulators, computer graphics, navigation systems, visualizations, fractals, and virtual reality.

Kinematic describes the motion of a point or a point system depending on time. If a point moves with respect to one parameter, then it traces its 1-dimensional path, orbit curve. If a line segment or a rectangle moves with respect to one parameter, then they sweep their two- and three-dimensional paths, respectively [2]. So we are thinking of a curve as the path traced out by a particle moving in Euclidean 3-space. The position vector of the curve is very important to determine behaviour of the curve.

The Serret-Frenet formulae for a quaternionic curve in ๐‘3 and ๐‘4 were given by Bharathi and Nagaraj in [3].

In the last few years, the study of the geometry of surfaces in 3-dimensional spaces, in particular of product type ๐Œ2ร—๐‘, was developed by a large number of mathematicians. Recently, constant angle surfaces were studied in product spaces ๐’2ร—๐‘ in [4] and ๐‡2ร—๐‘ in [5, 6], where ๐’2 and ๐‡2 represent the unit 2-sphere and hyperbolic plane, respectively. The angle is considered between the unit normal of the surface ๐Œ and the tangent direction to ๐‘.

Boyadzhiev [7] explored three-dimensional versions of these two properties: surfaces that are equiangular and those that are self-similar. He investigated the relationships among these surfaces and gave some examples. Thereafter, Munteanu [8] defined constant slope surfaces. Such surfaces are those whose position vectors make a constant angle with the normals at each point on the surface. Munteanu showed that they can be constructed by using an arbitrary curve on the sphere ๐’2 or an equiangular spiral.

There is also a kinematic generation of these surfaces as follows. Take a logarithmic spiral and roll its plane along a general cone such that the eye of the spiral sits in the vertex of the cone. Then the spiral sweeps out a surface with the required property.

More recently, we [9] gave some characterizations of constant slope surfaces and Bertrand curves in Euclidean 3-space. We found parametrization of constant slope surfaces for the tangent indicatrix, principal normal indicatrix, binormal indicatrix, and the Darboux indicatrix of a space curve. Furthermore we investigated Bertrand curves corresponding to parameter curves of constant slope surfaces.

By the definition of surfaces of revolution, we can see that such surfaces can be obtained by rotation matrices. Similarly, in this study, we show that constant slope surfaces can be obtained by quaternion product and the matrix representations ๐‘€. Subsequently, we give some results and an example of constant slope surfaces.

2. Preliminaries

The algebra ๐ป={๐ช=๐‘Ž0๐Ÿ+๐‘Ž1๐ข+๐‘Ž2๐ฃ+๐‘Ž3๐คโˆถ๐‘Ž0,๐‘Ž1,๐‘Ž2,๐‘Ž3โˆˆ๐‘} of quaternions is defined as the four-dimensional vector space over ๐‘ having a basis {๐Ÿ,๐ข,๐ฃ,๐ค} with the following properties:๐ข2=๐ฃ2=๐ค2=๐ขร—๐ฃร—๐ค=โˆ’๐Ÿ,๐ขร—๐ฃ=โˆ’๐ฃร—๐ข=๐ค.(2.1) It is clear that ๐ป is an associative and not commutative algebra and ๐Ÿ is identity element of ๐ป.

We use the following four-tuple notation to represent a quaternion:๎€ท๐‘Ž๐ช=0,๐‘Ž1,๐‘Ž2,๐‘Ž3๎€ธ=๎€ท๐‘Ž0๎€ธ,๐ฐ=๐‘Ž0+๐‘Ž1๐ข+๐‘Ž2๐ฃ+๐‘Ž3๐ค,(2.2) where ๐‘†๐ช=๐‘Ž0 is the scalar component of ๐ช and ๐‘‰๐ช={๐‘Ž1,๐‘Ž2,๐‘Ž3} form the vector part and entire set of ๐ชโ€™s is spanned by the basis quaternions:๐Ÿ=(1,0,0,0),๐ข=(0,1,0,0),๐ฃ=(0,0,1,0),๐ค=(0,0,0,1).(2.3)

We also write ๐ช=๐‘†๐ช+๐‘‰๐ช. The conjugate of ๐ช=๐‘†๐ช+๐‘‰๐ช is then defined as ๐ช=๐‘†๐ชโˆ’๐‘‰๐ช. We call a quaternion as pure if its scalar part vanishes. Summation of two quaternions ๐ช=๐‘†๐ช+๐‘‰๐ช and ๐ฉ=๐‘†๐ฉ+๐‘‰๐ฉ is defined as ๐ช+๐ฉ=(๐‘†๐ช+๐‘†๐ฉ)+(๐‘‰๐ช+๐‘‰๐ฉ). Multiplication of a quaternion ๐ช=๐‘†๐ช+๐‘‰๐ช with a scalar ๐œ†โˆˆ๐‘ is defined as ๐œ†๐ช=๐œ†๐‘†๐ช+๐œ†๐‘‰๐ช. Quaternion product is defined in the most general form for two quaternions ๐ช=๐‘†๐ช+๐‘‰๐ช and ๐ฉ=๐‘†๐ฉ+๐‘‰๐ฉ as๐ชร—๐ฉ=๐‘†๐ช๐‘†๐ฉโˆ’โŸจ๐‘‰๐ช,๐‘‰๐ฉโŸฉ+๐‘†๐ช๐‘‰๐ฉ+๐‘†๐ฉ๐‘‰๐ช+๐‘‰๐ชโˆง๐‘‰๐ฉ,(2.4) where โŸจ๐‘‰๐ช,๐‘‰๐ฉโŸฉ and ๐‘‰๐ชโˆง๐‘‰๐ฉ denote the familiar dot and cross-products, respectively, between the three-dimensional vectors ๐‘‰๐ช and ๐‘‰๐ฉ. Quaternionic multiplication satisfies the following properties: for any two quaternions ๐ช and ๐ฉ we have ๐ชร—๐ฉ=๐ฉร—๐ชand the formula for the dot product โŸจ๐ช,๐ฉโŸฉ=(๐ชร—๐ฉ+๐ฉร—๐ช)/2. In particular, if ๐ช=๐ฉ, we obtain |๐ช|2=โŸจ๐ช,๐ชโŸฉ=๐ชร—๐ช. If |๐ช|=1, then the quaternion ๐ช is unitary. The inverse of a quaternion ๐ช is given by๐ชโˆ’1=1||๐ช||2||๐ช||๐ช,โ‰ 0,(2.5) and it satisfies the relation ๐ชร—๐ชโˆ’1=๐ชโˆ’1ร—๐ช=1 [10]. If ๐ช is a unitary quaternion, we may write ๐ช in the trigonometric form as (cos๐œƒ,sin๐œƒ๐ฏ), where |๐ฏ|=1.

The most important property of quaternions is that they can characterize rotations in a three-dimensional space. The conventional way of representing three-dimensional rotations is by using a set of Euler angles {๐œƒ,๐œ‘,๐‘ข}, which denote rotations about independent coordinate axes. Any general rotation can be obtained as the result of a sequence of rotations, as given byโŽกโŽขโŽขโŽขโŽขโŽฃ๐‘ฅ๎…ž๐‘ฆ๎…ž๐‘ง๎…žโŽคโŽฅโŽฅโŽฅโŽฅโŽฆ=โŽกโŽขโŽขโŽขโŽขโŽฃโŽคโŽฅโŽฅโŽฅโŽฅโŽฆโŽกโŽขโŽขโŽขโŽขโŽฃโŽคโŽฅโŽฅโŽฅโŽฅโŽฆโŽกโŽขโŽขโŽขโŽขโŽฃโŽคโŽฅโŽฅโŽฅโŽฅโŽฆโŽกโŽขโŽขโŽขโŽขโŽฃ๐‘ฅ๐‘ฆ๐‘งโŽคโŽฅโŽฅโŽฅโŽฅโŽฆcos๐‘ขโˆ’sin๐‘ข0sin๐‘ขcos๐‘ข0001cos๐œ‘0sin๐œ‘010โˆ’sin๐œ‘0cos๐œ‘1000cos๐œƒโˆ’sin๐œƒ0sin๐œƒcos๐œƒ.(2.6)

Let ๐‘†โŠ‚๐‘3 be the set obtained by rotating a regular plane curve ๐ถ about an axis in the plane which does not meet the curve; we shall take ๐‘ฅ๐‘ง plane as the plane of the curve and the ๐‘ง axis as the rotation axis. Let ๐‘ฅ=๐‘”(๐‘ฃ),๐‘ง=โ„Ž(๐‘ฃ),๐‘Ž<๐‘ฃ<๐‘,๐‘”(๐‘ฃ)>0, be a parametrization for ๐ถ and denote by ๐‘ข the rotation angle about the ๐‘ง axis. Thus, we obtain a mapโŽกโŽขโŽขโŽขโŽขโŽฃโŽคโŽฅโŽฅโŽฅโŽฅโŽฆโŽกโŽขโŽขโŽขโŽขโŽฃ0โŽคโŽฅโŽฅโŽฅโŽฅโŽฆ=โŽกโŽขโŽขโŽขโŽขโŽฃโŽคโŽฅโŽฅโŽฅโŽฅโŽฆ๐‘ฅ(๐‘ข,๐‘ฃ)=cos๐‘ขโˆ’sin๐‘ข0sin๐‘ขcos๐‘ข0001๐‘”(๐‘ฃ)โ„Ž(๐‘ฃ)๐‘”(๐‘ฃ)cos๐‘ข๐‘”(๐‘ฃ)sin๐‘ขโ„Ž(๐‘ฃ)(2.7) from the open set ๐‘ˆ={(๐‘ข,๐‘ฃ)โˆˆ๐‘3โˆถ0<๐‘ข<2๐œ‹,๐‘Ž<๐‘ฃ<๐‘} into ๐‘†. We can see that ๐‘ฅ satisfies the conditions for a parametrization in the definition of a regular surface. Thus ๐‘† is a regular surface which is called a surface of revolution.

A one-parameter homothetic motion of a rigid body in Euclidean 3-space is given analytically by๐ฑ๎…ž=โ„Ž๐ด๐ฑ+๐‚,(2.8) in which ๐ฑ๎…ž and ๐ฑ are the position vectors, represented by column matrices, of a point ๐‘‹ in the fixed space ๐‘๎…ž and the moving space ๐‘, respectively; ๐ด is an orthogonal 3ร—3-matrix, ๐‚ is a translation vector, and โ„Ž is the homothetic scale of the motion. Also โ„Ž,๐ด, and ๐‚ are continuously differentiable functions of a real parameter ๐‘ก [2].

The map ๐œ™ acting on a pure quaternion ๐ฐโˆถ๐œ™โˆถ๐‘3โŸถ๐‘3,๐œ™(๐ฐ)=๐ชร—๐ฐร—๐ชโˆ’๐Ÿ(2.9) is linear. Without loss of generality we choose |๐ช|=1 and if ๐ช=๐‘Ž0+๐‘Ž1๐ข+๐‘Ž2๐ฃ+๐‘Ž3๐ค then๎€ท๐‘Ž๐“(๐ข)=20+๐‘Ž21โˆ’๐‘Ž22โˆ’๐‘Ž23๎€ธ๎€ท๐ข+2๐‘Ž0๐‘Ž3+2๐‘Ž1๐‘Ž2๎€ธ๎€ท๐ฃ+2๐‘Ž1๐‘Ž3โˆ’2๐‘Ž0๐‘Ž2๎€ธ๎€ท๐ค,๐“(๐ฃ)=โˆ’2๐‘Ž0๐‘Ž3+2๐‘Ž1๐‘Ž2๎€ธ๎€ท๐‘Ž๐ข+20+๐‘Ž22โˆ’๐‘Ž21โˆ’๐‘Ž23๎€ธ๎€ท๐ฃ+2๐‘Ž0๐‘Ž1+2๐‘Ž2๐‘Ž3๎€ธ๎€ท๐ค,๐“(๐ค)=2๐‘Ž0๐‘Ž2+2๐‘Ž1๐‘Ž3๎€ธ๎€ท๐ข+2๐‘Ž2๐‘Ž3โˆ’2๐‘Ž0๐‘Ž1๎€ธ๎€ท๐‘Ž๐ฃ+20+๐‘Ž23โˆ’๐‘Ž21โˆ’๐‘Ž22๎€ธ๐ค,(2.10) so that the matrix representation of the map ๐œ™ isโŽกโŽขโŽขโŽขโŽขโŽฃ๐‘Ž๐‘€=20+๐‘Ž21โˆ’๐‘Ž22โˆ’๐‘Ž23โˆ’2๐‘Ž0๐‘Ž3+2๐‘Ž1๐‘Ž22๐‘Ž0๐‘Ž2+2๐‘Ž1๐‘Ž32๐‘Ž0๐‘Ž3+2๐‘Ž1๐‘Ž2๐‘Ž20+๐‘Ž22โˆ’๐‘Ž21โˆ’๐‘Ž232๐‘Ž2๐‘Ž3โˆ’2๐‘Ž0๐‘Ž12๐‘Ž1๐‘Ž3โˆ’2๐‘Ž0๐‘Ž22๐‘Ž0๐‘Ž1+2๐‘Ž2๐‘Ž3๐‘Ž20+๐‘Ž23โˆ’๐‘Ž21โˆ’๐‘Ž22โŽคโŽฅโŽฅโŽฅโŽฅโŽฆ.(2.11) It is not difficult to check that ๐‘€ is orthogonal: ๐‘€๐‘€๐‘‡=๐ผ and det๐‘€=1 so that the linear map ๐“(๐ฐ)=๐ชร—๐ฐร—๐ชโˆ’๐Ÿ represents a rotation in ๐‘3 [11].

Now we give the characterization of constant slope surfaces as the following theorem.

Theorem 2.1. Let ๐ซโˆถ๐‘†โ†’๐‘3 be an isometric immersion of a surface ๐‘† in the Euclidean 3-space. Then ๐‘† is a constant slope surface if and only if either it is an open part of the Euclidean 2-sphere centered in the origin, or it can be parametrized by ๐ซ๎€ท(๐‘ข,๐‘ฃ)=๐‘ขsin๐œƒcos๐œ‰๐Ÿ(๐‘ฃ)+sin๐œ‰๐Ÿ(๐‘ฃ)โˆง๐Ÿ๎…ž๎€ธ(๐‘ฃ),(2.12) where ๐œƒ is a constant (angle) different from 0,๐œ‰=๐œ‰(๐‘ข)=cot๐œƒlog๐‘ข, and ๐Ÿ is a unit speed curve on the Euclidean sphere ๐’2 [8].

3. New Approach

A quaternion function ๐(๐‘ข,๐‘ฃ)=cos๐œ‰(๐‘ข)โˆ’sin๐œ‰(๐‘ข)๐Ÿ๎…ž(๐‘ฃ) defines a 2-dimensional surface in ๐’3โŠ‚๐‘4, where ๐Ÿ๎…ž(๐‘ฃ)=(๐‘“๎…ž1(๐‘ฃ),๐‘“๎…ž2(๐‘ฃ),๐‘“๎…ž3(๐‘ฃ)) and |๐Ÿโ€ฒ|=1. Thus, for the unitary quaternion ๐(๐‘ข,๐‘ฃ), the matrix representation of the map ๐œ™โˆถ๐‘3โ†’๐‘3 is given byโŽกโŽขโŽขโŽขโŽขโŽขโŽฃ๐‘€=cos2๐œ‰+sin2๐œ‰๎‚€๐‘“โ€ฒ21โˆ’๐‘“2๎…ž2โˆ’๐‘“3๎…ž2๎‚๎€ท2sin๐œ‰cos๐œ‰๐‘“๎…ž3+sin๐œ‰๐‘“๎…ž1๐‘“๎…ž2๎€ธ๎€ท2sin๐œ‰sin๐œ‰๐‘“๎…ž1๐‘“๎…ž3โˆ’cos๐œ‰๐‘“๎…ž2๎€ธ๎€ท2sin๐œ‰sin๐œ‰๐‘“๎…ž1๐‘“๎…ž3โˆ’cos๐œ‰๐‘“๎…ž3๎€ธcos2๐œ‰+sin2๐œ‰๎‚€โˆ’๐‘“โ€ฒ21+๐‘“2๎…ž2โˆ’๐‘“3๎…ž2๎‚๎€ท2sin๐œ‰sin๐œ‰๐‘“๎…ž2๐‘“๎…ž3+cos๐œ‰๐‘“๎…ž1๎€ธ๎€ท2sin๐œ‰sin๐œ‰๐‘“๎…ž1๐‘“๎…ž3+cos๐œ‰๐‘“๎…ž2๎€ธ๎€ท2sin๐œ‰sin๐œ‰๐‘“๎…ž2๐‘“๎…ž3โˆ’cos๐œ‰๐‘“๎…ž1๎€ธcos2๐œ‰+sin2๐œ‰๎‚€โˆ’๐‘“โ€ฒ21โˆ’๐‘“2๎…ž2+๐‘“3๎…ž2๎‚โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ.(3.1)

We are now ready to show the main result of this study.

Theorem 3.1. Let ๐ซโˆถ๐‘†โ†’๐‘3 be an isometric immersion of a surface ๐‘† in the Euclidean 3-space. Then the constant slope surface ๐‘† can be reparametrized by ๐ซ(๐‘ข,๐‘ฃ)=๐(๐‘ข,๐‘ฃ)ร—๐1(๐‘ข,๐‘ฃ), where โ€œร—โ€ is the quaternion product, ๐1(๐‘ข,๐‘ฃ)=๐‘ขsin๐œƒ๐Ÿ(๐‘ฃ)โˆˆ๐‘3 is a surface and a pure quaternion.

Proof. Since ๐(๐‘ข,๐‘ฃ)=cos๐œ‰(๐‘ข)โˆ’sin๐œ‰(๐‘ข)๐Ÿโ€ฒ(๐‘ฃ) and ๐๐Ÿ(๐‘ข,๐‘ฃ)=๐‘ขsin๐œƒ๐Ÿ(๐‘ฃ), we obtain ๐(๐‘ข,๐‘ฃ)ร—๐1(๐‘ข,๐‘ฃ)=(cos๐œ‰(๐‘ข)โˆ’sin๐œ‰(๐‘ข)๐Ÿโ€ฒ(๐‘ฃ))ร—(๐‘ขsin๐œƒ๐Ÿ(๐‘ฃ))=๐‘ขsin๐œƒ(cos๐œ‰(๐‘ข)โˆ’sin๐œ‰(๐‘ข)๐Ÿโ€ฒ(๐‘ฃ))ร—๐Ÿ(๐‘ฃ)=๐‘ขsin๐œƒcos๐œ‰(๐‘ข)๐Ÿ(๐‘ฃ)โˆ’๐‘ขsin๐œƒsin๐œ‰(๐‘ข)๐Ÿโ€ฒ(๐‘ฃ)ร—๐Ÿ(๐‘ฃ).(3.2) By using (2.4), we get ๐Ÿโ€ฒ(๐‘ฃ)ร—๐Ÿ(๐‘ฃ)=๐Ÿโ€ฒ(๐‘ฃ)โˆง๐Ÿ(๐‘ฃ)=โˆ’๐Ÿ(๐‘ฃ)โˆง๐Ÿโ€ฒ(๐‘ฃ).(3.3) If we substitute this into the last equation, we have ๐(๐‘ข,๐‘ฃ)ร—๐๐Ÿ(๐‘ข,๐‘ฃ)=๐‘ขsin๐œƒ(cos๐œ‰(๐‘ข)๐Ÿ(๐‘ฃ)+sin๐œ‰(๐‘ข)๐Ÿ(๐‘ฃ)โˆง๐Ÿโ€ฒ(๐‘ฃ)).(3.4) Hence applying Theorem 2.1, we find that the constant slope surface is given by ๐(๐‘ข,๐‘ฃ)ร—๐๐Ÿ(๐‘ข,๐‘ฃ)=๐ซ(๐‘ข,๐‘ฃ).(3.5) This completes the proof.

As a consequence of this theorem, we get the following important corollary.

Corollary 3.2. Let ๐‘€ be the matrix representation of the map ๐œ™โˆถ๐‘3โ†’๐‘3 for the unitary quaternion ๐(๐‘ข,๐‘ฃ). Then, for the pure quaternion ๐๐Ÿ(๐‘ข,๐‘ฃ), we get the constant slope surface as ๐ซ(๐‘ข,๐‘ฃ)=๐‘€๐๐Ÿ(๐‘ข,๐‘ฃ).(3.6) We know that a surface of revolution can be obtained by a rotation matrix. Similarly, we view that the constant slope surface ๐ซ(๐‘ข,๐‘ฃ) can be obtained by the matrix representation ๐‘€, too.

Finally we state the following result.

Corollary 3.3. For the homothetic motion ๎‚๐(๐‘ข,๐‘ฃ)=๐‘ขsin๐œƒ๐(๐‘ข,๐‘ฃ), the constant slope surface can be written as ๎‚๐ซ(๐‘ข,๐‘ฃ)=๐(๐‘ข,๐‘ฃ)ร—๐Ÿ(๐‘ฃ). Therefore we have ๐ซ(๐‘ข,๐‘ฃ)=๐‘ขsin๐œƒ๐‘€๐Ÿ(๐‘ฃ).(3.7)

Now, we give some remarks regarding our Theorem 3.1 and Corollary 3.3.

Remark 3.4. Theorem 3.1 says that both the points and the position vectors on the surface ๐๐Ÿ(๐‘ข,๐‘ฃ) are rotated by ๐(๐‘ข,๐‘ฃ) through the angle ๐œ‰(๐‘ข) about the axis ๐‘†๐‘{๐Ÿ๎…ž(๐‘ฃ)}.

Remark 3.5. Corollary 3.3 shows that the position vector of the curve ๐Ÿ(๐‘ฃ) is rotated by ๎‚๐(๐‘ข,๐‘ฃ) through the angle ๐œ‰(๐‘ข) about the axis ๐‘†๐‘{๐Ÿ๎…ž(๐‘ฃ)} and extended through the homothetic scale ๐‘ขsin๐œƒ.

4. Example

We give an example of constant slope surfaces and draw its picture by using Mathematica.

Example 4.1. We consider the unit speed spherical curve 1๐Ÿ(๐‘ฃ)=2๎‚€โˆšcos2๐‘ฃ,๎‚3,sin2๐‘ฃ.(4.1) If the angle is taken ๐œƒ=๐œ‹/4 then we have ๐(๐‘ข,๐‘ฃ)=cos(log๐‘ข)+(sin(log๐‘ข)sin2๐‘ฃ,0,โˆ’sin(log๐‘ข)cos2๐‘ฃ),(4.2)๐๐Ÿ๎ƒฉโˆš(๐‘ข,๐‘ฃ)=24โˆš๐‘ขcos2๐‘ฃ,64โˆš๐‘ข,24๎ƒช๐‘ขsin2๐‘ฃ.(4.3) Thus, by using (3.1) and (3.6), we get the following constant slope surface: =โŽกโŽขโŽขโŽขโŽขโŽฃ๐ซ(๐‘ข,๐‘ฃ)cos2(log๐‘ข)โˆ’sin2(log๐‘ข)cos4๐‘ฃsin(2log๐‘ข)cos2๐‘ฃโˆ’sin2(log๐‘ข)sin4๐‘ฃโˆ’sin(2log๐‘ข)cos2๐‘ฃcos(2log๐‘ข)โˆ’sin(2log๐‘ข)sin2๐‘ฃโˆ’sin2(log๐‘ข)sin4๐‘ฃsin(2log๐‘ข)sin2๐‘ฃcos2(log๐‘ข)+sin2โŽคโŽฅโŽฅโŽฅโŽฅโŽฆโ‹…โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽฃโˆš(log๐‘ข)cos4๐‘ฃ24โˆš๐‘ขcos2๐‘ฃ64๐‘ขโˆš24โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽฃโˆš๐‘ขsin2๐‘ฃ24๎‚€โˆš๐‘ขcos2๐‘ฃcos(2log๐‘ข)+๎‚โˆš3sin(2log๐‘ข)64โˆš๐‘ขcos(2log๐‘ข)โˆ’24โˆš๐‘ขsin(2log๐‘ข)24๎‚€โˆš๐‘ขsin2๐‘ฃcos(2log๐‘ข)+๎‚โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ.3sin(2log๐‘ข)(4.4) Hence, we can give Mathematica code of this constant slope surface as โˆšParametricPlot3D๎ƒฌ๎ƒฏ24[]๎‚€๎€บ[๐‘ข]๎€ป+โˆš๐‘ขcos2๐‘ฃcos2log๎€บ[๐‘ข]๎€ป๎‚,โˆš3sin2log64๎€บ[๐‘ข]๎€ปโˆ’โˆš๐‘ขcos2log24๎€บ[๐‘ข]๎€ป,โˆš๐‘ขsin2log24[]๎‚€๎€บ[๐‘ข]๎€ป+โˆš๐‘ขsin2๐‘ฃcos2log๎€บ[๐‘ข]๎€ป๎‚๎ƒฐ,๎‚†3sin2log๐‘ข,0,๐‘ƒ๐‘–2๎‚‡๎ƒญ,{๐‘ฃ,0,๐‘ƒ๐‘–}(4.5) and the picture of ๐(๐‘ข,๐‘ฃ)ร—๐๐Ÿ(๐‘ข,๐‘ฃ) is drawn as follows (Figure 1).

5. Conclusion

By the definition of surfaces of revolution, we can see that such surfaces can be obtained by rotation matrices for the position vectors of given regular plane curves. Similarly, in this study, we show that constant slope surfaces can be obtained by quaternion product and the matrix representations ๐‘€. Afterwards, we give some results and illustrate an example of constant slope surfaces by using quaternions and draw its picture by using Mathematica computer program.

Acknowledgment

The authors wish to express their gratitude to Marian Ioan Munteanu for suggesting them the subject and for helpful hints given during the preparation of this work.