Cumhuriyet Sci. J., Vol.39-1 (2018) 71-79 - DergiPark

Cumhuriyet Science Journal CSJ e-ISSN: 2587-246X ISSN: 2587-2680

Cumhuriyet Sci. J., Vol.39-1(2018) 71-79

Some Geometric Properties of The Spacelike Bezier Curve with a Timelike Principal Normal in Minkowski 3-Space Hatice KUSAK SAMANCI Bitlis Eren Univ. Faculty of Art and Science, Bitlis, TURKEY Received: 14.10.2017; Accepted: 10.12.2017

http://dx.doi.org/10.17776/csj.344353

Abstract: The aim of present paper is to introduce and investigate the spacelike Bezier curve with a timelike principal normal in Minkowski 3-space. The Serret-Frenet frame, curvatures and the derivation formulas of the curve at the starting and ending points are studied. Keywords: Bezier curve, causal character, curvatures, Minkowski 3-space

Zamanımsı Asal Normalli Uzayımsı Bezier Eğrisinin Bazı Geometrik Özellikleri Özet: Bu çalışmanın amacı Minkowski 3-uzayında zamanımsı asal normalli uzayımsı Bezier eğrisini tantmak ve incelemektir. Eğrinin başlangıç ve bitiş noktasındaki Serret-Frenet çatısı, eğrilikleri ve türev formülleri çalışılmıştır. Anahtar Kelimeler: Bezier eğrisi, causal karakterler, eğrilikler, Minkowski 3-uzayı

1.

INTRODUCTION

Minkowski space is founded by German mathematician Hermann Minkowski, [1]. Let

u   u1 , u2 , u3  , v   v1 , v2 , v3  be vectors in Minkowski 3-space. If the metric g  ,  is given by

g  u , v   u1v1  u2v2  u3v3 , then the space

R  3 1

 R , g , 3

is called the Minkowski 3-

space where the metric g  ,  is called the Lorentzian or Minkowskian metric. Let  be a curve in Minkowski 3-space and T be the tangent

vector for all points of the curve. The curve

 is

called a spacelike curve if g T , T   0 or

T  0, a timelike curve if g T , T   0 and a lightlike (null) curve if g T , T   0 and T  0 . Furthermore, there are three possibilities depending on the causal character of T . If the vector T '

N s =

is timelike, then the equations

T s and B  s  = T  s   N  s  are  s '

provided where the N and B are called the principal normal and the binormal vectors

* Corresponding author. Email address: [email protected] http://dergipark.gov.tr/csj ©2016 Faculty of Science, Cumhuriyet University

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Kusak Samanci / Cumhuriyet Sci. J., Vol.39-1 (2018) 71-79

respectively. The curvature and torsion of



 is

defined by   s  = T  s  and  =  g N , B '

'



. Then the Serret- Frenet equations are given with

T ' =  N , N ' = T   B , B' =  B for the spacelike curve with a timelike principal normal, [2]. If the spacelike vectors u and v satisfy the condition | g (u , v) |< u is

a

timelike

| g (u, v) |= u uv

L

= u

L

vector

were given in the paper [6]. In the [7,8] the authors considered a spacelike quadratic and

and

equations

cubic Bezier curves in Minkowski 2-plane R 12 .

and

Furthermore, similarly a spacelike curve with a spacelike principle normal in Minkowski 3-space were studied in [9].

the

v L sin  are used where  is

L

the spacelike angle between, u and v spacelike vectors. If the spacelike vectors u and v ensure the condition | g (u , v) |> u

L

v L , then u  v

vector is a timelike vector, and the equations

g (u , v) =  u uv

L

= u

L

v L cosh 

L

and

v L sinh  are satisfied where 

is the hyperbolic angle between the u and v spacelike vectors. If the vectors u and v are the spacelike vectors provided the equation

| g (u, v) |L = u

L

v L , then u  v is a lightlike

Our main intention of this paper to investigate some differential geometric properties of the spacelike Bezier curve with a timelike principal normal, e.g. curvature, torsion, Serret-Frenet equations and the derivativation formula of Serret-Frenet equations. 2.

MAIN RESULTS

Let b0 , b1 ,..., bn  R13 be the control points, and

b n  t  be a spacelike Bezier curve, which does n not have unit speed. If the Bezier curve b  t  is

vector, [1]. n On the other hand, the curve b (t ) 

n

 bi Bin (t ) i 0

is called a Bezier curve given with the control points b0 , b1 ,..., bn for each t [0,1] where

n Bin (t )    t i (1  t ) n i i

curves at the beginning point, i.e   0  and   0 

v L , then u  v vector

v L cos 

L

interactive, in production because of the easiness in usage [3,4,5]. Spacelike Bezier curves in the three dimensional Minkowski 3-space was firstly introduced by G.H. Georgiev in 2008. The curvature and torsion of the spacelike Bezier

are called Bernstein

polynomials. Bezier curves were first developed by a French engineer (1958-1960), Pierre Bezier (1910-1999) and a French mathematician Paul de Faget de Casteljau (1930) independently with different mathematical approaches. Bezier curves provide easiness in design processes because they are controllable in such a way that Bezier curves can be used almost in every design of any of the products used now. The Bezier curves have a wide variety of usage in design because of being

spacelike curve, then the tangent T must be spacelike vector. However, in case T ' i.e. the n principle normal is timelike b  t  spacelike

Bezier curve is called as "the spacelike Bézier curve with timelike principle normal". The orthonormal frame

T , N , B t=0

of the bn (t)

spacelike Bezier curve in the t = 0 starting point is include the T spacelike vector, N timelike vector and B spacelike vector. So the conditions

g T , T  = 1, g  N , N  = 1, g  B, B  = 1, g T , N  = 0, g T , B  = 0, g  N , B  = 0

are

satisfied. If the Bezier curve bn (t ) is spacelike,

 db n (t ) db n (t )  ,  > 0 . Therefore, the dt   dt

then g 


length of the speed vector and the arc parameter

b t  n

of

v=

db n (t ) dt t1

L

are

 db n (t ) db n (t )  = g ,  and dt   dt

b0  L b1 . b0 L . b1 L .sin 

Proof: Since bn (t ) is a spacelike Bezier curve, the tangent vector must also be spacelike. For this

 db (t ) db (t )  s =  g , dt , recpectively. If dt   dt t0 n

B t =0 =

73

n

the orthonormal frame vectors are T spacelike, N timelike, B spacelike , then the vectoral products are satisfy the equations T L N = B,

N L B = T , and B L T = N .

reason

b0

L

= g  b0 , b0  . The tangent

vector in the starting point is given with the following formula:

T t =0

Let b0 , b1 ,..., bn  R13 be the control points of the

db n (t ) dt = = n db (t ) dt L

b0

g  b0 , b0 

.

spacelike Bezier curve with timelike principal normal in Minkowski 3-space. The convex polygon vectors bi are spacelike vectors which

Then the binormal vector is calculated by

are existed in the same spacelike cone. If the vectors b0 and b1 are spacelike, the vectors

B t =0

have three conditions for using inner and exterior product as following: Condition (1). g  b0 , b1  < b0

= L

. b1

Condition (2). g  b0 , b1  > b0

. b1 L

Condition (3). g  b0 , b1   b0

L

. b1

L

In our paper, we will deal with Condition (1) and (2). Theorem 2.1. If b0 and b1 vectors satisfy the Condition (1), then the Serret-Frenet frame

T , N , B t =0 obtained by T

N t =0 =

in the starting point t = 0 is t =0

=

b0

g  b0 , b0 

b0 b1 cot   cos ec b0 L b1 L

t =0

n.b0  L  n.(n  1) b1  b0  n.b0  L  n.(n  1) b1  b0 

=

b0  L b1  b0  L b0 b0  L b1  b0  L b0 L

=

b0  L b1 . b0 L b1 L sin 

L

L

db n (t ) d 2b n (t ) L dt dt 2 = db n (t ) d 2b n (t ) L dt dt 2 L

L

Since T spacelike, N timelike and B spacelike, the principal normal vector N is provided by

N t =0 = B t =0  L T =

t =0

b0  L b1 b0 L b0  L b1 L b0 L

  g  b0 , b0  .b1  g  b1 , b0  .b0  =   b0 L . b1 L sin  . b0 L  

74


=

b0 b1 cot   cos ec b0 L b1 L

norm is given by b0

N t =0 = B t =0  L T

between the b0 and b1 spacelike vector.

=

Theorem 2.2. If b0 and b1 vectors satisfy the Condition (2), then the Serret-Frenet frame in the starting point t = 0 is

provided by

T t =0 =

N t =0 = 

B t =0

b0

g  b0 , b0 

b0  L b1 b0 L b0  L b1 L b0 L

  g  b0 , b0  .b1  g  b1 , b0  .b0  =    b .  b sinh  .  b 0 1 0  L L L  b0 b1 = coth   csc h . b0 L b1 L

control points. If the Condition (1) is satisfied, then the curvature and torsion of the spacelike Bezier curve bn (t ) with timelike principal normal at the starting point t = 0 are

b0  L b1 . = b0 L . b1 L .sinh 

 |t =0 =

n  1 b1 .sin  n b0 2

Proof: The proof of the tangent vector T |t =0 is the same as the Theo.2.1. The binormal vector

B t =0 is found by

B t =0

=

=

t =0

Theorem 2.3. Let b0 , b1 ,..., bn be the spacelike

b0 b1 coth   csc h b0 L b1 L

db n (t ) d 2b n (t ) L dt dt 2 = db n (t ) d 2b n (t ) L dt dt 2 L

= g  b0 , b0  . Thus the

principal normal is

where  =  b0 , b1  is the spacelike angle

T , N , B t =0

2

 |t =0 = 

n  2 det  b0 , b1 , b2  . 2 n b0  L b1

Proof: b0 and b1 vectors that ensure the Condition (1), the curvature at the starting point for spacelike vectors is calculated with: t =0

n.b0  L  n.(n  1) b1  b0  n.b0  L  n.(n  1) b1  b0 

 t =0 = L

b0  L b1 b0  L b1 = b0  L b1 L b0 L b1 L sinh 

=

where  =  b0 , b1  is called a hiperbolic angle. Since T spacelike, N timelike, B spacelike, the equation of the principal normal N will be taken by N = B L T . Since b0 is spacelike, the

db n  t  d 2b n  t  L dt dt 2 db  t  dt n

L

t =0

n  1 b0  L  b1  b0  3 n b0 L

=

L

3

n  1 b0  L b1 3 n b0 L

L

L


=

n  1 b1 n b0

.sin 

L 2

=

L

=

 db  t  d b t  d b t   g L ,  dt dt 2 dt 3    |t =0 = 2 db n  t  d 2b n  t  L dt dt 2 L n

2 n

3 n

t =0

n  2 g  b0  IL b1 , b2  2b1  b0  2 n b0  L  b1  b0 

n  2 g  b0  L b1 , b2  n b0  L  b1  b0  2 L

=

n  1 b1 n b0

Theorem 2.5. For the spacelike vectors provided the Condition (1), the Serret-Frenet frame derivation formula of the at the t = 0 of the curve

bn (t ) is

Theorem 2.4. Let b0 , b1 ,..., bn be the spacelike Condition (2), the curvature and torsion of the spacelike Bezier curve with timelike principal normal at the starting point are

 |t =0 = 

L

b1

L

b0

L

.sin  .T det  b0 , b1 , b2  L

b0  L b1

Proof: The curvature at the starting point is

b0 L b1

2

.N

L

where v1 = n b1  b0 = n b0 . Hence we get

T  =  v1.N L

3

L

.B

L

det(b0 , b1 , b2 ) L

2 n

db n  t  dt

2

 T '   0  v1 0  T        N '     v1 0  v1  N   B'   0 v  0  1     B 

L

 t =0 =

L

.sin  .N

Proof: The Frenet derivation formula for spacelike curve with timelike principal normal is

.sinh 

db  t  d b t  L dt dt 2

L

b0

B =   n  2  b0

n  2 det  b0 , b1 , b2  . 2 n b0  L b1

n

b1

  n  2  b0

control points. If b0 and b1 vectors satify the

L 2

.sinh  .

L

N  =  n  1

L

n  1 b1 n b0

L 2

T  =  n  1

n  2 det  b0 , b1 ,b2  2 n b0  L b1

 |t =0 =

L

Here, the torsion equation for the Condition (2) can be proven similar method with Theo.2.3.

L

=

n  1 b0  L  b1  b0  3 n b0 L

Now, let us find the torsion at the starting point.

=

75

=

 n  1

b1

n

b0

t =0

= (n  1)

sin  .n b0

L 2 L

b1

L

b0

L

sin  .N

L

.N

76


N  =  v1T   v1B = (n  1)

N  |t =0 =  v1T   v1B

b1

L

b0

L

sin  .T

=

det(b0 , b1 , b2 )

 (n  2) b0

L

b0  L b1

2

.B

det(b0 , b1 , b2 ) b0 L b1

L

2

.N

L

derivation formula of the Serret-Frenet frame is yield by

N  =  n  1

L

b0

L

.sinh  .N

b1

L

b0

L

  n  2  b0

B =   n  2  b0

2 L

b0 L b1

2

.N .

L

Proof: Following calculations give us the derivation formula of Serret-Frenet frame:

T  |t =0 =  v1.N =

n  1 b1 n b0

=  n  1

L 2

.sinh  .n b0

L

b1

L

b0

L

.sinh  .N

b1

L

b0

L

 (n  2) b0

.sinh .T det  b0 , b1 , b2  L

b0  b1

.B

2 L

B |t =0 =  v1 N =

 n  2   b0 , b1 , b2  .n b0  L b1

n

2

b0

L

.N

L

det(b0 , b1 , b2 ) L

b0  b1

2

.N .

L

Bezier curve with principal normal.

.B

det(b0 , b1 , b2 ) L

n  2 det  b0 , b1 , b2  n b0 L .B 2 n b0  L b1

Let bi  R 13 be the control points of the spacelike

det  b0 , b1 , b2  b0  L b1



.sinh .n b0 L T

L

=   n  2  b0

.sinh  .T

L

b0

=  n  1

b1 vectors that satisfy the Condition (2), the

b1

n

L 2

L

Theorem 2.6. For the spacelike vectors b0 and

T  =  n  1

b1

L

B |t =0 =  v1 N =   n  2  b0

 n  1

L

N

Theorem 2.7. If the bn1 and bn2 vectors are satisfy Condition (1), the Serret-Frenet frame

T , N , B t =1 at the ending point t = 1 is obtained by

bn 1

T t =1 =

N t =1 =

g  bn 1 , bn 1  bn 2 bn 1 cos ec  cot  bn 2 L bn 1 L

B t =1 = 

bn 1  L bn 2 bn 1 L . bn  2 L sin 

Proof: The proof is similar with Theo. 2.1.


Theorem 2.8. If the bn1 and bn2 vectors satisfy Condition (2), the Serret-Frenet frame

T , N , B t =1 at the ending point t = 1 is obtained by

the derivation formula of Serret-Frenet frame at the ending point t = 1 is calculated by

T  = (n  1)

bn 1

T t =1 =

g  bn 1 , bn 1 

N |t =1 =

N  = (n  1)

bn  2 bn 1 csc h  coth  bn  2 L bn 1 L

B |t =1 = 

bn 1  L bn  2 . bn 1 L . bn  2 L sinh 

77

bn  2

L

bn 1

L

bn  2

L

bn 1

L

(n  2) bn 1

B =  n  2 

sin  .N

sin  .T

det(bn 1 , bn  2 , bn 3 ) bn 1  L bn  2

L

det(bn1 , bn2 , bn3 ) bn1  bn2

2

2

.B

L

bn1 L .N .

L

Proof: The proof is similar with Theo. 2.5. Proof: The proof is similar with Theo. 2.2. Theorem 2.9. For the spacelike vectors that provide the Condition (1), the curvature and torsion of the curve at the ending point are obtained by

n  1 bn 2  |t =1 = n bn 1

L 2

Theorem 2.12. If the spacelike vectors bn1 and

bn2 vectors that ensure the Condition (2), then the derivation formula of Serret-Frenet frame at the ending point t = 1 is calculated by

T  = (n  1)

.sin 

L

n  2 det  bn 1 , bn  2 , bn 3   |t =1 = . 2 n bn 1  L bn  2 L

N  = (n  1)

bn  2

L

bn 1

L

bn  2

L

bn 1

L

 (n  2) bn 1

Proof: The proof is similar with Theo. 2.3. Theorem 2.10. If the spacelike vectors satisfy the Condition (2), the curvature and torsion are given by

 |t =1 =  |t =1 =

n  1 bn 2 n bn 1

L 2

B =  n  2 

sinh  .N sinh  .T det(bn 1 , bn  2 , bn 3 ) L

bn 1  L bn  2

det(bn1 , bn2 , bn3 ) bn1  bn2

2

2

.B

L

bn1 L .N .

L

Proof: The proof is similar with Theo.2.6. 3. NUMERIC EXAMPLE

.sinh 

L

n  2 det  bn 1 , bn  2 , bn 3  . 2 n bn 1  L bn  2 L

Proof: The proof is similar with Theo. 2.4. Theorem 2.11. If the spacelike vectors bn1 and

bn2 vectors that ensure the Condition (1), then

n Consider a cubic Bezier curve b  t  with

spacelike

control

points

b1 =  4, 4,1 , b2 =  5, 7,3  , Minkowski 3-space.

b0 =  3,3, 2  , b3 =  6,3,5  in

78


g  b0 , b1  =  b0 uv

L

= u

L

L

. b1

L

cosh  ,

v L sinh 

will be used in this example. The Serret-Frenet frame formula is obtained by

T t =0 = 1,1, 1

Nt =0 =

1  5,3, 8 30

B t =0 =

1  5,3, 2  . 30

The curvature and torsion of the curve are

 t =0 =

Figure 1. The cubic Bezier curve.

2 30 3

and  |t =0 =

7 , respectively. 30

Consequently, the derivation matrix of SerretFrenet is founded by following matrix Then the spacelike convex hull is found by the vectors

b0 = 1,1, 1 ,

b1 = 1,3, 2  ,

2 30 0  T  T'   0     ' 0 7 10   N   N  =  2 30   B '   0 7 10 0   B     

b2 = 1, 4, 2  . The first, second and third derivations at t = 0 are

dbn  t  =  3,3, 3 dt t =0

dbn  t  where v = dt

d 2b n  t  =  0,12,18  dt 2 t =0

4.

d 3b n  t  = (0, 54, 18) . dt 3 t =0 The norms b0 calculated.

L

= 1 and b1

Because

g  b0 , b1   6 , g  b0 , b1  > b0 Hence the equations

of

L

the

the L

. b1

L

= 14 are equation inequality

is

satisfied.

= 3. L

CONCLUSION

Bezier curves are a type of curve commonly used for ease of use in design geometry. In this paper, we studied on the spacelike Bezier curves with timelike principal normal in Minkowski 3-space. We think that this work will be a guide to research that can be done on this subject in the future. 5.

ACKNOWLEDGEMENT

The author is grateful to referees for their careful reading of the paper which improved it greatly.


REFERENCES

[1]. Ratcliffe J.G., Foundations of hyperbolic manifolds, Graduate texts in mathematics, vol. 149, 2nd edition, 2006. [2]. Lopez R., Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int. Electron. J. Geom., 7-1 (2014) 44-107. [3]. Farin G., Curves and Surfaces for ComputerAided Geometric Design, Academic Press, 1996. [4]. Marsh D., Applied Geometry for Computer Graphics and CAD, Springer Science and Business Media, 2006. [5]. İncesu M., Gursoy, O., Bezier Eğrilerinde Esas Formlar ve Eğrilikler, XVII Ulusal Matematik Sempozyumu, Bildiriler, Abant İzzet Baysal Üniversitesi, (2004) 146-157.

[6]. Georgiev G.H. Spacelike Bezier Curves in the Three-dimensional Minkowski space, Proceedings of AIP Conference, 1067-1 (2008) . [7]. Chalmoviansky P., Pokorna B., Quadratic Spacelike Bezier Curves in the three dimensional Minkowski Space, Proceeding of Symposium on Computer Geometry, 20 (2011) 104-110. [8]. Pokorna B., Chalmovianski P., Planar Cubic Spacelike Bezier Curves in Three Dimensional Minkowski Space, Proceeding of Syposium on Computer Geometry, SCG, 23 (2012).93-98. [9]. Bükcü B., Karacan M., Bishop Frame of the Spacelike Curve with a Spacelike Principal Normal in Minkowski 3-space , Commun. Fac. Sci. Univ. Ank. Series, A1, 57-1 (2008) 13-22.

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