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ISRN Geometry
Volume 2012 (2012), Article ID 126358, 8 pages
http://dx.doi.org/10.5402/2012/126358
Research Article

A New Approach to Constant Slope Surfaces with Quaternions

1Department of Mathematics, Faculty of Arts and Sciences, Bozok University, 66100 Yozgat, Turkey
2Department of Mathematics, Faculty of Science, Ankara University, 06100 Ankara, Turkey

Received 5 November 2011; Accepted 8 December 2011

Academic Editor: M. Przanowski

Copyright © 2012 Murat Babaarslan and Yusuf Yayli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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. Hamilton’s defining relation is most succinctly written as𝐢2=𝐣2=𝐤2=𝐢×𝐣×𝐤=𝟏.(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 𝐑3 and 𝐑4 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 𝐌2×𝐑, was developed by a large number of mathematicians. Recently, constant angle surfaces were studied in product spaces 𝐒2×𝐑 in [4] and 𝐇2×𝐑 in [5, 6], where 𝐒2 and 𝐇2 represent the unit 2-sphere and hyperbolic plane, respectively. The angle is considered between the unit normal of the surface 𝐌 and the tangent direction to 𝐑.

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 𝐒2 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 𝑀. Subsequently, we give some results and an example of constant slope surfaces.

2. Preliminaries

The algebra 𝐻={𝐪=𝑎0𝟏+𝑎1𝐢+𝑎2𝐣+𝑎3𝐤𝑎0,𝑎1,𝑎2,𝑎3𝐑} of quaternions is defined as the four-dimensional vector space over 𝐑 having a basis {𝟏,𝐢,𝐣,𝐤} with the following properties:𝐢2=𝐣2=𝐤2=𝐢×𝐣×𝐤=𝟏,𝐢×𝐣=𝐣×𝐢=𝐤.(2.1) It is clear that 𝐻 is an associative and not commutative algebra and 𝟏 is identity element of 𝐻.

We use the following four-tuple notation to represent a quaternion:𝑎𝐪=0,𝑎1,𝑎2,𝑎3=𝑎0,𝐰=𝑎0+𝑎1𝐢+𝑎2𝐣+𝑎3𝐤,(2.2) where 𝑆𝐪=𝑎0 is the scalar component of 𝐪 and 𝑉𝐪={𝑎1,𝑎2,𝑎3} form the vector part and entire set of 𝐪’s is spanned by the basis quaternions:𝟏=(1,0,0,0),𝐢=(0,1,0,0),𝐣=(0,0,1,0),𝐤=(0,0,0,1).(2.3)

We also write 𝐪=𝑆𝐪+𝑉𝐪. The conjugate of 𝐪=𝑆𝐪+𝑉𝐪 is then defined as 𝐪=𝑆𝐪𝑉𝐪. We call a quaternion as pure if its scalar part vanishes. Summation of two quaternions 𝐪=𝑆𝐪+𝑉𝐪 and 𝐩=𝑆𝐩+𝑉𝐩 is defined as 𝐪+𝐩=(𝑆𝐪+𝑆𝐩)+(𝑉𝐪+𝑉𝐩). Multiplication of a quaternion 𝐪=𝑆𝐪+𝑉𝐪 with a scalar 𝜆𝐑 is defined as 𝜆𝐪=𝜆𝑆𝐪+𝜆𝑉𝐪. Quaternion product is defined in the most general form for two quaternions 𝐪=𝑆𝐪+𝑉𝐪 and 𝐩=𝑆𝐩+𝑉𝐩 as𝐪×𝐩=𝑆𝐪𝑆𝐩𝑉𝐪,𝑉𝐩+𝑆𝐪𝑉𝐩+𝑆𝐩𝑉𝐪+𝑉𝐪𝑉𝐩,(2.4) where 𝑉𝐪,𝑉𝐩 and 𝑉𝐪𝑉𝐩 denote the familiar dot and cross-products, respectively, between the three-dimensional vectors 𝑉𝐪 and 𝑉𝐩. Quaternionic multiplication satisfies the following properties: for any two quaternions 𝐪 and 𝐩 we have 𝐪×𝐩=𝐩×𝐪and the formula for the dot product 𝐪,𝐩=(𝐪×𝐩+𝐩×𝐪)/2. In particular, if 𝐪=𝐩, we obtain |𝐪|2=𝐪,𝐪=𝐪×𝐪. If |𝐪|=1, then the quaternion 𝐪 is unitary. The inverse of a quaternion 𝐪 is given by𝐪1=1||𝐪||2||𝐪||𝐪,0,(2.5) and it satisfies the relation 𝐪×𝐪1=𝐪1×𝐪=1 [10]. If 𝐪 is a unitary quaternion, we may write 𝐪 in the trigonometric form as (cos𝜃,sin𝜃𝐯), where |𝐯|=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 {𝜃,𝜑,𝑢}, which denote rotations about independent coordinate axes. Any general rotation can be obtained as the result of a sequence of rotations, as given by𝑥𝑦𝑧=𝑥𝑦𝑧cos𝑢sin𝑢0sin𝑢cos𝑢0001cos𝜑0sin𝜑010sin𝜑0cos𝜑1000cos𝜃sin𝜃0sin𝜃cos𝜃.(2.6)

Let 𝑆𝐑3 be the set obtained by rotating a regular plane curve 𝐶 about an axis in the plane which does not meet the curve; we shall take 𝑥𝑧 plane as the plane of the curve and the 𝑧 axis as the rotation axis. Let 𝑥=𝑔(𝑣),𝑧=(𝑣),𝑎<𝑣<𝑏,𝑔(𝑣)>0, be a parametrization for 𝐶 and denote by 𝑢 the rotation angle about the 𝑧 axis. Thus, we obtain a map0=𝑥(𝑢,𝑣)=cos𝑢sin𝑢0sin𝑢cos𝑢0001𝑔(𝑣)(𝑣)𝑔(𝑣)cos𝑢𝑔(𝑣)sin𝑢(𝑣)(2.7) from the open set 𝑈={(𝑢,𝑣)𝐑30<𝑢<2𝜋,𝑎<𝑣<𝑏} into 𝑆. We can see that 𝑥 satisfies the conditions for a parametrization in the definition of a regular surface. Thus 𝑆 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 by𝐱=𝐴𝐱+𝐂,(2.8) in which 𝐱 and 𝐱 are the position vectors, represented by column matrices, of a point 𝑋 in the fixed space 𝐑 and the moving space 𝐑, respectively; 𝐴 is an orthogonal 3×3-matrix, 𝐂 is a translation vector, and is the homothetic scale of the motion. Also ,𝐴, and 𝐂 are continuously differentiable functions of a real parameter 𝑡 [2].

The map 𝜙 acting on a pure quaternion 𝐰𝜙𝐑3𝐑3,𝜙(𝐰)=𝐪×𝐰×𝐪𝟏(2.9) is linear. Without loss of generality we choose |𝐪|=1 and if 𝐪=𝑎0+𝑎1𝐢+𝑎2𝐣+𝑎3𝐤 then𝑎𝝓(𝐢)=20+𝑎21𝑎22𝑎23𝐢+2𝑎0𝑎3+2𝑎1𝑎2𝐣+2𝑎1𝑎32𝑎0𝑎2𝐤,𝝓(𝐣)=2𝑎0𝑎3+2𝑎1𝑎2𝑎𝐢+20+𝑎22𝑎21𝑎23𝐣+2𝑎0𝑎1+2𝑎2𝑎3𝐤,𝝓(𝐤)=2𝑎0𝑎2+2𝑎1𝑎3𝐢+2𝑎2𝑎32𝑎0𝑎1𝑎𝐣+20+𝑎23𝑎21𝑎22𝐤,(2.10) so that the matrix representation of the map 𝜙 is𝑎𝑀=20+𝑎21𝑎22𝑎232𝑎0𝑎3+2𝑎1𝑎22𝑎0𝑎2+2𝑎1𝑎32𝑎0𝑎3+2𝑎1𝑎2𝑎20+𝑎22𝑎21𝑎232𝑎2𝑎32𝑎0𝑎12𝑎1𝑎32𝑎0𝑎22𝑎0𝑎1+2𝑎2𝑎3𝑎20+𝑎23𝑎21𝑎22.(2.11) It is not difficult to check that 𝑀 is orthogonal: 𝑀𝑀𝑇=𝐼 and det𝑀=1 so that the linear map 𝝓(𝐰)=𝐪×𝐰×𝐪𝟏 represents a rotation in 𝐑3 [11].

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

Theorem 2.1. Let 𝐫𝑆𝐑3 be an isometric immersion of a surface 𝑆 in the Euclidean 3-space. Then 𝑆 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 𝐫(𝑢,𝑣)=𝑢sin𝜃cos𝜉𝐟(𝑣)+sin𝜉𝐟(𝑣)𝐟(𝑣),(2.12) where 𝜃 is a constant (angle) different from 0,𝜉=𝜉(𝑢)=cot𝜃log𝑢, and 𝐟 is a unit speed curve on the Euclidean sphere 𝐒2 [8].

3. New Approach

A quaternion function 𝐐(𝑢,𝑣)=cos𝜉(𝑢)sin𝜉(𝑢)𝐟(𝑣) defines a 2-dimensional surface in 𝐒3𝐑4, where 𝐟(𝑣)=(𝑓1(𝑣),𝑓2(𝑣),𝑓3(𝑣)) and |𝐟|=1. Thus, for the unitary quaternion 𝐐(𝑢,𝑣), the matrix representation of the map 𝜙𝐑3𝐑3 is given by𝑀=cos2𝜉+sin2𝜉𝑓21𝑓22𝑓322sin𝜉cos𝜉𝑓3+sin𝜉𝑓1𝑓22sin𝜉sin𝜉𝑓1𝑓3cos𝜉𝑓22sin𝜉sin𝜉𝑓1𝑓3cos𝜉𝑓3cos2𝜉+sin2𝜉𝑓21+𝑓22𝑓322sin𝜉sin𝜉𝑓2𝑓3+cos𝜉𝑓12sin𝜉sin𝜉𝑓1𝑓3+cos𝜉𝑓22sin𝜉sin𝜉𝑓2𝑓3cos𝜉𝑓1cos2𝜉+sin2𝜉𝑓21𝑓22+𝑓32.(3.1)

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

Theorem 3.1. Let 𝐫𝑆𝐑3 be an isometric immersion of a surface 𝑆 in the Euclidean 3-space. Then the constant slope surface 𝑆 can be reparametrized by 𝐫(𝑢,𝑣)=𝐐(𝑢,𝑣)×𝐐1(𝑢,𝑣), where “×” is the quaternion product, 𝐐1(𝑢,𝑣)=𝑢sin𝜃𝐟(𝑣)𝐑3 is a surface and a pure quaternion.

Proof. Since 𝐐(𝑢,𝑣)=cos𝜉(𝑢)sin𝜉(𝑢)𝐟(𝑣) and 𝐐𝟏(𝑢,𝑣)=𝑢sin𝜃𝐟(𝑣), we obtain 𝐐(𝑢,𝑣)×𝐐1(𝑢,𝑣)=(cos𝜉(𝑢)sin𝜉(𝑢)𝐟(𝑣))×(𝑢sin𝜃𝐟(𝑣))=𝑢sin𝜃(cos𝜉(𝑢)sin𝜉(𝑢)𝐟(𝑣))×𝐟(𝑣)=𝑢sin𝜃cos𝜉(𝑢)𝐟(𝑣)𝑢sin𝜃sin𝜉(𝑢)𝐟(𝑣)×𝐟(𝑣).(3.2) By using (2.4), we get 𝐟(𝑣)×𝐟(𝑣)=𝐟(𝑣)𝐟(𝑣)=𝐟(𝑣)𝐟(𝑣).(3.3) If we substitute this into the last equation, we have 𝐐(𝑢,𝑣)×𝐐𝟏(𝑢,𝑣)=𝑢sin𝜃(cos𝜉(𝑢)𝐟(𝑣)+sin𝜉(𝑢)𝐟(𝑣)𝐟(𝑣)).(3.4) Hence applying Theorem 2.1, we find that the constant slope surface is given by 𝐐(𝑢,𝑣)×𝐐𝟏(𝑢,𝑣)=𝐫(𝑢,𝑣).(3.5) This completes the proof.

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

Corollary 3.2. Let 𝑀 be the matrix representation of the map 𝜙𝐑3𝐑3 for the unitary quaternion 𝐐(𝑢,𝑣). Then, for the pure quaternion 𝐐𝟏(𝑢,𝑣), we get the constant slope surface as 𝐫(𝑢,𝑣)=𝑀𝐐𝟏(𝑢,𝑣).(3.6) We know that a surface of revolution can be obtained by a rotation matrix. Similarly, we view that the constant slope surface 𝐫(𝑢,𝑣) can be obtained by the matrix representation 𝑀, too.

Finally we state the following result.

Corollary 3.3. For the homothetic motion 𝐐(𝑢,𝑣)=𝑢sin𝜃𝐐(𝑢,𝑣), the constant slope surface can be written as 𝐫(𝑢,𝑣)=𝐐(𝑢,𝑣)×𝐟(𝑣). Therefore we have 𝐫(𝑢,𝑣)=𝑢sin𝜃𝑀𝐟(𝑣).(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 𝐐𝟏(𝑢,𝑣) are rotated by 𝐐(𝑢,𝑣) through the angle 𝜉(𝑢) about the axis 𝑆𝑝{𝐟(𝑣)}.

Remark 3.5. Corollary 3.3 shows that the position vector of the curve 𝐟(𝑣) is rotated by 𝐐(𝑢,𝑣) through the angle 𝜉(𝑢) about the axis 𝑆𝑝{𝐟(𝑣)} and extended through the homothetic scale 𝑢sin𝜃.

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 1𝐟(𝑣)=2cos2𝑣,3,sin2𝑣.(4.1) If the angle is taken 𝜃=𝜋/4 then we have 𝐐(𝑢,𝑣)=cos(log𝑢)+(sin(log𝑢)sin2𝑣,0,sin(log𝑢)cos2𝑣),(4.2)𝐐𝟏(𝑢,𝑣)=24𝑢cos2𝑣,64𝑢,24𝑢sin2𝑣.(4.3) Thus, by using (3.1) and (3.6), we get the following constant slope surface: =𝐫(𝑢,𝑣)cos2(log𝑢)sin2(log𝑢)cos4𝑣sin(2log𝑢)cos2𝑣sin2(log𝑢)sin4𝑣sin(2log𝑢)cos2𝑣cos(2log𝑢)sin(2log𝑢)sin2𝑣sin2(log𝑢)sin4𝑣sin(2log𝑢)sin2𝑣cos2(log𝑢)+sin2(log𝑢)cos4𝑣24𝑢cos2𝑣64𝑢24=𝑢sin2𝑣24𝑢cos2𝑣cos(2log𝑢)+3sin(2log𝑢)64𝑢cos(2log𝑢)24𝑢sin(2log𝑢)24𝑢sin2𝑣cos(2log𝑢)+.3sin(2log𝑢)(4.4) Hence, we can give Mathematica code of this constant slope surface as ParametricPlot3D24[][𝑢]+𝑢cos2𝑣cos2log[𝑢],3sin2log64[𝑢]𝑢cos2log24[𝑢],𝑢sin2log24[][𝑢]+𝑢sin2𝑣cos2log[𝑢],3sin2log𝑢,0,𝑃𝑖2,{𝑣,0,𝑃𝑖}(4.5) and the picture of 𝐐(𝑢,𝑣)×𝐐𝟏(𝑢,𝑣) is drawn as follows (Figure 1).

126358.fig.001
Figure 1: Constant slope surface 𝐐(𝐮,𝐯)×𝐐𝟏(𝐮,𝐯),𝐟(𝐯)=(𝟏/𝟐)(𝐜𝐨𝐬𝟐𝐯,𝟑,𝐬𝐢𝐧𝟐𝐯),𝜽=𝝅/𝟒.

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 𝑀. 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.

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