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 uv
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 uv
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
Kusak Samanci / Cumhuriyet Sci. J., Vol.39-1 (2018) 71-79
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
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Kusak Samanci / Cumhuriyet Sci. J., Vol.39-1 (2018) 71-79
=
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
Kusak Samanci / Cumhuriyet Sci. J., Vol.39-1 (2018) 71-79
=
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 2b1 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
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Kusak Samanci / Cumhuriyet Sci. J., Vol.39-1 (2018) 71-79
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 bn1 and bn2 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.
Kusak Samanci / Cumhuriyet Sci. J., Vol.39-1 (2018) 71-79
Theorem 2.8. If the bn1 and bn2 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(bn1 , bn2 , bn3 ) bn1 bn2
2
2
.B
L
bn1 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 bn1 and
bn2 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(bn1 , bn2 , bn3 ) bn1 bn2
2
2
.B
L
bn1 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 bn1 and
bn2 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
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Kusak Samanci / Cumhuriyet Sci. J., Vol.39-1 (2018) 71-79
g b0 , b1 = b0 uv
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.
Kusak Samanci / Cumhuriyet Sci. J., Vol.39-1 (2018) 71-79
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[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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