Journal of Logic, Mathematics and Linguistics in Applied Sciences
Volume: 1 Issue:1, 2016
Recieved: 13. 02. 2016
Accepted: 22. 03. 2016
A METHOD OF DETERMINING DUAL UNIT SPHERICAL BÉZIER CURVES
Ferhat Taş
Istanbul University, Faculty of Science, Department of Mathematics, 34134, Istanbul, Turkey
tasf@istanbul.edu.tr
Abstract: In this paper, we defined the spherical Bézier curves with a new method on the real
unit sphere(RUS). Then similarly, dual unit spherical bézier curves are obtained with the help
of a spherical interpolation between the points on the dual unit sphere(DUS). So we have
found a new way to the design of a ruled surface.
Key Words: Kinematics, Bézier curves, E. Study’s map, Spherical interpolation.
Classification: 53A17-53A25-65D17-65D18-68U07
1.
Introduction
Closed dual curves on the DUS have an important role in kinematics. With the another
purpose, Shoemake constructed the Spherical Bézier Curves using the spherical interpolation
which is called by Slerp between unit quaternions [3]. This interpolation accepts for two
quaternions. That is a Bézier curve to get on the RUS, more than two quaternions had to be
interpolated and the control points not on the RUS supposed to use. P.Crouch, G.Kun, F.Silva
Leite and Tomasz Popiel, Lyle Noakes have studied on this subject. But the methods used in
all are based on the result of generalization of the de Casteljau algorithm.
There are many studies about ruled surfaces to apply in CAGD. As known, design and
control problems have an important role in the theory of mechanisms. In this area, Bézier
curves and surfaces provide very useful results. The main purpose of Bézier technic is to
present the algorithmic basis to applicable computer design. So the aim of this study is to
generate any dual closed Bézier curve and the corresponding ruled surfaces. Unlike Shoemake
and the others, in our method we take the control points on the DUS namely, every control
points is a unit dual vector. So we can apply E.Study's transformation principle between the
unit dual vectors and the oriented lines in 3-dimensional Euclidean space [5]. Moreover, this
method represents a flexible design since control points provide flexibility in practise i.e.
mechanisms.
For this aim, we give a special spherical linear interpolation between the unit dual control
vectors to define a dual unit spherical Bézier curve. In this case, a dual unit spherical Bézier
curve can be obtained by using control points on the DUS and can be represented in a matrix
form and this is very useful for design and manufacturing.
2.
Basic Concepts
2.1
Kinematics in 3-dimensional space
In E³, let a moving orthonormal trihedron {v₁,v₂,v₃} make a closed spatial motion along a
closed differentiable curve c=c(t). Then v₁-axis of the orthonormal trihedron generates a
closed ruled surface
x(t,u)=c(t)+uv₁(t),
x(t,u)=x(t+2π,u),
t,u∈ℝ.
(1)
which is represented by v₁(t)-c.r.s.
The derivative equations of the closed spatial motion described above are
𝑗
𝑗
d𝐯𝑖 = ∑3𝑗=1 𝑤𝑖 𝐯𝑗 , 𝑤𝑖 (𝑡) = −𝑤𝑗𝑖 (𝑡), t∈ℝ, i,j=1,2,3
(2)
d𝐜 = ∑3𝑗=1 𝑎𝑖 𝐯𝑗
(3)
and
𝑗
where 𝑤𝑖 and 𝑎𝑖 are the Pfaff differential forms of the motion [2].
2.2 Dual Numbers and Dual Vectors
The set of dual numbers is defined in [5] and [7] as
𝐼𝐷 = {𝐴 = 𝑎 + 𝜀𝑎̅ |𝑎, 𝑎̅ ∈ ℝ, 𝜀 2 = 0 }.
(4)
Dual number A also can be written by an ordered pair as 𝐴 = (𝑎, 𝑎̅ ) and dual unit ε
represented by (0,1).
The set of dual numbers forms an associative ring over real numbers with the following
operations,
𝐴 + 𝐵 = 𝑎 + 𝜀𝑎̅ + 𝑏 + 𝜀𝑏̅ = 𝑎 + 𝑏 + 𝜀(𝑎̅ + 𝑏̅)
(5)
𝐴𝐵 = (𝑎 + 𝜀𝑎̅)(𝑏 + 𝜀𝑏̅ ) = 𝑎𝑏 + 𝜀(𝑎̅𝑏 + 𝑎𝑏̅).
(6)
and
Also a dual number 𝐴 = (𝑎, 𝑎̅ ) can be represented by a 2×2 matrix as
𝐴 = (𝑎, 𝑎̅ ) = [
𝑎
0
𝑎̅
].
𝑎
(7)
Let F be a differentiable function with dual variable 𝑋 = (𝑥, 𝑥̅ ). Then by using the Taylor
series of f, it can be shown that,
F(X)=F(𝑥 + 𝜀𝑥̅ )=𝑓(𝑥) + 𝜀𝑥̅ 𝑓′(𝑥),
where 𝑓′(𝑥) is derivative of 𝑓(𝑥).
(8)
The set of dual vectors is defined [5], [7] as
𝐼𝐷3 = {𝐕 = 𝐯 + 𝜀𝐯̅ |𝐯, 𝐯̅ ∈ ℝ3 , 𝜀 2 = 0 }.
(9)
The dual vectors also have the 6×2 matrix representation [5] as
𝐯
𝐕 = 𝐯 + 𝜀𝐯̅ = [𝟎 3
ℝ
𝐯̅
𝐯 ].
(10)
Let K be moving DUS generated by a dual orthonormal system
𝐕 ′
{𝐕1 , 𝐕2 = ‖𝐕1 ′‖ , 𝐕3 = 𝐕1 ∧ 𝐕2 }
1
(11)
and K´ be a fixed DUS generated by the fixed dual system {E₁, E₂, E₃} with the same centre.
Then the derivative equations of the dual spherical closed motion of K with respect to K′ are
𝑗
d𝐕𝑖 = ∑3𝑗=1 Ω𝑖 𝐕𝑗
(12)
𝑗
𝑗
̅̅̅̅𝑗 (𝑡), t∈ℝ, Ω𝑗 = −Ω𝑖 , i,j=1,2,3.
where Ω𝑖 (𝑡) = 𝑤𝑖 (𝑡) + 𝜀𝑤
𝑗
𝑖
𝑖
According to Study' s transformation, the dual equations (12) correspond to the real
equations (2) and (3) of the closed spatial motion. Thus, during the dual spherical motion,
V₁(t)-closed dual curve, drawn by V₁-axis, corresponds to (1) i.e. v₁(t)-closed ruled surface in
ℝ³ [2].
2.3 Bézier Curves
A Bézier curve of degree n is defined [1] by
𝑛
𝐛(𝑡) = ∑𝑛𝑖=0 𝐛𝑖 𝐵𝑖𝑛 (𝑡) = ∑𝑛𝑖=0 𝐛𝑖 ( ) (1 − 𝑡)𝑛−𝑖 𝑡 𝑖 , t∈[0,1] (13)
𝑖
𝑛
where 𝐛𝑖 are the control points of the curve in E³ and ( ) denotes the binomial coefficient of
𝑖
the Bézier function.
Furthermore, since the barycentric combination of the control points, it is easily seen that;
the first control point (when t=0) and the last control point (when t=1) are on the same curve
segment. Besides if we take the first and last control points same, then curve has a closed
shape. But it is not a sufficient condition to ensure the C¹ continuity of the curve. That is why,
𝐛0 = 𝐛𝑛 , 𝐛1 , 𝐛𝑛−1 have to be collinear to smooth curve [6].
2.4 Quaternions
A quaternion can be defined as ordered four numbers written with 1,i,j,k units like
q=a1+bi+cj+dk where a,b,c,d are real numbers and i²=j²=k²=-1, ijk=-1, ij=-ji=k, jk=-kj=i,
ki=-ik=j. q can be represented by 4-tuples of numbers (a,b,c,d). This is an extended complex
number and represents a rotation in 3-dimensional space. Another representation of a
quaternion is q=(a,v), where v=(b,c,d) is a 3-dimensional vector. So it has scalar and vectorial
part.
Quaternions construct an algebra with addition and multiplication operations. Addition and
multiplication of quaternions q₁=(a₁,v₁) and q₂=(a₂,v₂) are defined by
q₁+q₂ = (a₁+a₂,v₁+v₂)
(14)
q₁q₂ = (a₁a₂-v₁v₂,a₁v₂+a₂v₁+v₁×v₂)
(15)
respectively. As seen in equation (15) q₁q₂ is a quaternion. Quaternion multiplication is not
commutative because of v₁×v₂=-v₂×v₁.
̃ = (𝑎, −𝐯) represent the conjugate of quaternion q and q𝐪
̃=a²+‖v‖². So the norm of q can
𝐪
̃ = √𝑎2 + ‖𝐯‖2 . Also, q has an inverse and it is denoted by q⁻¹ and
be defined by ‖𝐪‖ = √𝐪𝐪
̃ /(‖𝐪‖²)). If ‖𝐪‖=1 then we can say that q is a unit quaternion. Besides a
defined by q⁻¹=(𝐪
unit quaternion can be represented by q=(cosθ,vsinθ), where 0≤θ≤π and ‖v‖=1.
Let q=(cos(θ/2),vsin(θ/2)) be a unit quaternion and r=(a,u) is any quaternion then
q⁻¹rq=(cos(θ/2),-vsin(θ/2))(a,u)(cos(θ/2),vsin(θ/2))
(16)
is the process of rotation of quaternion r with an angle θ about the v-axis. Specifically, as we
shall see in the next section, if quaternion r is a pure vector (i.e. r=(0,v)) then we just turned
the vector u with an angle θ about v-axis [1], [3], [4], [6].
3. Spherical Bézier Curves
In 3-dimensional Euclidean space, let p and q be two points on the RUS. The spherical
interpolation of these points is given with
v(t)=p((sin((1-t)θ))/(sinθ))+q((sin(tθ))/(sinθ)),
t∈[0,1] (17)
where θ=arccos(<u,v>), [3].
Similarly, if we want to interpolate more points on the RUS we can apply equation (17) to
them in twain. So if we continue this process, we will get a spherical Bézier curve on the
RUS. Here our main purpose is getting a spherical Bézier curve by interpolating the points on
the RUS continuously. For generalization here we take θ is constant. Then we must choose
these points equally spaced on the RUS. But this does not give a guarantee that the curve has
unit magnitude since θ is not invariant in any interpolation step. We just choose it constant for
generalization.
Thus, given (n+1)-spherical control points p₀,p₁,...,pn (n≥1) the spherical Bézier curve of
degree n is defined as
∑𝑛 𝐩 𝑠𝑛 (𝑡)
𝑖 𝑖
𝐬(𝑡) = ‖∑𝑖=0
𝑛 𝐩 𝑠𝑛 (𝑡)‖
𝑖=0 𝑖 𝑖
(18)
𝑛
1
where 𝑠𝑖𝑛 (𝑡) = (sin𝜃)𝑛 ( ) (sin(𝜃 − 𝑡𝜃))𝑛−𝑖 sin(𝑡𝜃)𝑖 , 𝐩𝑖 ∈ E 3 , ‖𝐩𝑖 ‖ = 1, t∈[0,1], θ∈(0,2π).
𝑖
3.1 Closed Spherical Bézier Curves
We have mentioned that closed Bézier curve which has tangent continuity in 2.3. Similarly,
we want to construct a closed Bézier curve on the RUS. Choosing the control point, before
the last one, is important. Because the curve must be regular in the neighborhood of the last
point p₀=pn. For this aim, by using the quaternion rotation, we can find the control point pn-1
as:
𝜃
𝜃
𝜃
𝜃
𝐩𝑛−1 = (cos ( 2) , −sin (2) 𝐩0 ∧ 𝐩1 ) (0, 𝐩0 ) (cos (2) , sin ( 2) 𝐩0 ∧ 𝐩1 ) (19)
where θ=arccos(<𝐩0 , 𝐩1 >). The right-hand side of the above equation is a quaternion rotation
around unit vector p₀∧p₁ with angle θ. Thus the three consecutive points pn-1, p₀, p₁ lie on the
same geodesic of the RUS.
4. Dual Spherical Bézier Curves
̅𝑖 , i=0,1,...,n, are dual unit vectors in ID-module and 𝛩 = 𝜃 + 𝜀𝜃̅ be the
Let 𝐏𝑖 = 𝐩𝑖 + 𝜀𝐩
constant dual angle between the consecutive dual vectors. Then the dual spherical Bézier
curve can be given as
∑𝑛
𝐏 𝑆 𝑛 (𝑡)
𝑖 𝑖
𝐒(𝑡) = ‖∑𝑖=0
𝑛 𝐏 𝑆 𝑛 (𝑡)‖
(20)
𝑖=0 𝑖 𝑖
𝑛
1
where 𝑆𝑖𝑛 (𝑡) = (sin𝛩)𝑛 ( ) (sin(𝛩 − 𝑡𝛩))𝑛−𝑖 sin(𝑡𝛩)𝑖 , 𝐏𝑖 ∈ DUS, t∈[0,1],
𝑖
𝛩=arccos(<𝐩0 , 𝐩1 >).
We know from equation (10) that a dual unit vector can be given in a matrix form so to write
B(t) in matrix form, we can take its components in matrix form as:
𝑥𝑖
𝑦𝑖
𝑧𝑖
𝐏𝑖 = 0
0
[0
𝑚𝑥𝑖
𝑚𝑦𝑖
𝑚𝑧𝑖
𝑛
𝑛
𝑥𝑖 , ( 𝑖 ) ≡ ( 𝑖 ) . [𝐼]6×6
𝑦𝑖
𝑧𝑖 ]
(21)
and
1
(sin𝛩)𝑛
𝑎11
(sin(𝛩 − 𝑡𝛩))𝑛−𝑖 sin(𝑡𝛩) = [𝑎
21
𝑎12
𝑎22 ]
(22)
where [𝑥𝑖 , 𝑦𝑖, 𝑧𝑖 ] and [𝑚𝑥𝑖 , 𝑚𝑦𝑖, 𝑚𝑧𝑖 ] are real part and dual(moment) part of dual vector,
respectively. Expanding dual trigonometric functions as in (8) we have
𝑛
1
𝑎11 = 𝑎22 = (sin𝜃)𝑛 ( ) (sin(𝜃 − 𝑡𝜃))𝑛−𝑖 sin(𝑡𝜃)𝑖
𝑖
𝑎21 = 0
̅
̅
𝑎12 = (𝑛 − 𝑖)(𝜃 − 𝑡𝜃)(𝑐𝑜𝑠𝜃)𝑛−𝑖 (𝑠𝑖𝑛𝜃)−1−𝑖 (sin(𝑡𝜃))𝑖
+𝑛𝑡𝜃̅ cos(𝑡𝜃) (𝑠𝑖𝑛𝜃)−1 (sin(𝜃 − 𝑡𝜃))𝑛−𝑖
−𝑛𝜃̅𝑐𝑜𝑠𝜃(𝑠𝑖𝑛𝜃)−𝑛−1 (sin(𝑡𝜃))𝑖 (𝑠𝑖𝑛(𝜃 − 𝑡𝜃))𝑛−𝑖
(23)
Then the equation (20) can be given in matrix form as with the equations (21),(22),(23)
𝐒(𝑡) =
𝑥𝑖 𝑎11
𝑦𝑖 𝑎11
𝑛 𝑧𝑖 𝑎11
∑𝑛
𝑖=0( 𝑖 )
0
0
[ 0
𝑥𝑖 𝑎11
𝑦𝑖 𝑎11
‖ 𝑛 𝑛 𝑧𝑖 𝑎11
∑𝑖=0( )
𝑖
0
‖
0
[ 0
𝑥𝑖 𝑎12 +𝑚𝑥𝑖 𝑎22
𝑦𝑖 𝑎12 +𝑚𝑦𝑖 𝑎22
𝑧𝑖 𝑎12 +𝑚𝑧𝑖 𝑎22
𝑥𝑖 𝑎22
𝑦𝑖 𝑎22
𝑧𝑖 𝑎22
]
𝑥𝑖 𝑎12 +𝑚𝑥𝑖 𝑎22 .
𝑦𝑖 𝑎12 +𝑚𝑦𝑖 𝑎22
𝑧𝑖 𝑎12 +𝑚𝑧𝑖 𝑎22 ‖
𝑥𝑖 𝑎22
‖
𝑦𝑖 𝑎22
𝑧𝑖 𝑎22
]
(24)
So the dual spherical Bézier curve B(t) corresponds to the ruled surface with respect to
E.Study's transference principle.
5. Conclusion
In modern technology, motion design is very useful in manufacture. Geometry has also an
important role in this area. In parallel studies in this direction, we examined dual spherical
Bézier curves, which its control points are dual vectors, and so according to the principle of
transformation of E.Study, a trajectory surface design was constructed. Bézier curves are both
convex as well as allowing flexible work led us to these definitions. Closed motion, which
holds an important place in kinematic, and was obtained by Bézier spherical interpolation
method and so this study is suitable for application on computer systems. In addition, matrix
representation of dual spherical Bézier curve as covered by the subject in CAGD will give
direction and convenience to the future studies.
References
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Number 3 (1985).
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[5]
Veldkamp, G. R. On The Use of Dual Numbers, Vectors and Matrices in
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