On the Design and Invariants of a Ruled Surface
Ferhat Taş Ph.D., Research Assistant. Department of Mathematics, Faculty of Science, Istanbul University, Vezneciler, 34134, Istanbul,Turkey. [email protected]
ABSTRACT
This paper deals with a kind of design of a ruled surface. It combines concepts from the fields of computer aided geometric design and kinematics. A dual unit spherical Bézier- like curve on the dual unit sphere (DUS) is obtained with respect the control points by a new method. So, with the aid of Study [1] transference principle, a dual unit spherical
Bézier-like curve corresponds to a ruled surface. Furthermore, closed ruled surfaces are determined via control points and integral invariants of these surfaces are investigated.
The results are illustrated by examples.
Key Words: Kinematics, Bézier curves, E. Study’s map, Spherical interpolation. Classification: 53A17-53A25
1 Introduction
Many scientists like mechanical engineers, computer scientists and mathematicians are
interested in ruled surfaces because the surfaces are widely using in mechanics, robotics
and some other industrial areas. Some of them have investigated its differential
geometric properties and the others, its representation mathematically for the design of
these surfaces. Study [1] used dual numbers and dual vectors in his research on the geometry of lines and kinematics in the 3-dimensional space. He showed that there
1 exists a one-to-one correspondence between the position vectors of DUS and the
directed lines of space 3. So, a one-parameter motion of a point on DUS corresponds
to a ruled surface in 3-dimensionalℝ real space.
Hoschek [2] found integral invariants for characterizing the closed ruled surfaces.
Furthermore, Gürsoy [3] studied some relationship finding between the dual integral invariant and the real integral invariants of a closed ruled surface. He concerned the motion of a point on DUS which draws a closed trajectory. However, these studies still remain theoretical, except for some examples.
The purpose of investigation of ruled surfaces is to find its computational properties since it is used in design and manufacturing. So, we have found a useful method to the representation of a ruled surfaces to designing it. For this, we need a designable curve on DUS.
Spherical Bézier curves were introduced by Shoemake [4] using the spherical interpolation which is called by Slerp. This interpolation method is valid for two quaternions. He developed a method to get a spherical Bézier curve via combining the spherical curve segments continuously. P. Crouch, G. Kun, F. Silva Leite [5] have studied on this subject. Q.J. Ge and B. Ravani [6, 7] developed a computational geometric structure by using dual quaternion interpolation for continuous Bézier motion of the rigid body. Y. Zhou, J. Schulze and S. Schäffler [8] studied about optimization on DUS and defined the dual spherical spline for the design of a blade surface as ruled surface to cutting and shaping the shoes. In [9], In this study, the dual unit spherical Bezier curve was created by the global interpolation method of equally spaced points on the dual
2 sphere. Of course, this has led to some restrictions. Thus, this curve thereby forming a
negative in terms of ability to flexibly design the ruled surface.
It is well-known that dual curves on DUS have an important role in kinematics, design and control problems. In the theory of mechanisms, 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 differentiable dual unit spherical Bézier curve and investigate the properties of corresponding ruled surfaces in terms of control points.
Moreover, our method represents a flexible design since control points provide flexibility in practice i.e. mechanisms. For this aim, we use a map from the domain of the real unit sphere (RUS) to its surface to define a dual unit spherical Bézier-like curve. In this case, a dual unit spherical Bézier-like curve can be obtained by using the control points on the RUS.
2 Basic Concepts
This section is about fundamentals of the kinematics that generate a ruled surface and its representation via dual unit vectors.
2.1 Dual Vector Representation of a Ruled Surface
Clifford [10] defined dual numbers as a tool for his geometrical studies. The set of dual numbers is = { = + | , , = 0}. 2 Here , namely𝔻𝔻 , the𝐴𝐴 dual𝑎𝑎 unit𝜀𝜀𝑎𝑎� can𝑎𝑎 𝑎𝑎� be∈ representedℝ 𝜀𝜀 by an ordered pair = (0, 1). This set
forms𝜀𝜀 an associative ring over real numbers with the following operations,𝜀𝜀
+ = + + + = + + ( + )
𝐴𝐴 𝐵𝐵 𝑎𝑎 𝜀𝜀𝑎𝑎� 𝑏𝑏 𝜀𝜀𝑏𝑏� 𝑎𝑎 𝑏𝑏 𝜀𝜀 𝑎𝑎� 𝑏𝑏�
3 and
= ( + ) + = + + . � � Let𝐴𝐴𝐴𝐴 be𝑎𝑎 a differentiable𝜀𝜀𝑎𝑎� �𝑏𝑏 𝜀𝜀𝑏𝑏� function𝑎𝑎𝑎𝑎 𝜀𝜀with�𝑎𝑎�𝑏𝑏 respect𝑎𝑎𝑏𝑏� to the dual variable = + . Then, by
using𝑓𝑓 the Taylor series of , we can write the following equation [1]𝑋𝑋, 𝑥𝑥 𝜀𝜀𝑥𝑥̅
𝑓𝑓 ( ) ( ) = ( + ) = ( ) + . 𝑑𝑑𝑑𝑑 𝑥𝑥 𝑓𝑓 𝑋𝑋 𝑓𝑓 𝑥𝑥 𝜀𝜀𝑥𝑥̅ 𝑓𝑓 𝑥𝑥 𝜀𝜀𝑥𝑥̅ We can give some examples: 𝑑𝑑𝑑𝑑
sin( + ) = sin( )+ cos( ),
cos(𝑥𝑥 +𝜀𝜀𝜀𝜀 ) = cos(𝑥𝑥 )- 𝜀𝜀𝜀𝜀 sin(𝑥𝑥),
exp(𝑥𝑥 +𝜀𝜀𝜀𝜀) = exp(𝑥𝑥 )+𝜀𝜀𝜀𝜀 exp𝑥𝑥( ).
After𝑥𝑥 Clifford𝜀𝜀𝜀𝜀 ’ s works𝑥𝑥 , 𝜀𝜀𝜀𝜀Study used𝑥𝑥 the dual numbers and the dual vectors in his research
on the geometry of lines and kinematics. The set of dual vectors is defined as
= { = + | , , = 0}. 3 3 2 𝔻𝔻In 3, a𝐗𝐗 –𝐱𝐱 valued𝜀𝜀𝐱𝐱� 𝐱𝐱 bilinear𝐱𝐱� ∈ ℝ form𝜀𝜀 is defined by
𝔻𝔻, =𝔻𝔻 , + ( , + , )
where〈𝐔𝐔 𝐕𝐕〉 〈=𝐮𝐮 𝐯𝐯〉+ ε and〈𝐮𝐮 𝐯𝐯 �V〉= 〈𝐮𝐮�+𝐯𝐯〉 . Then we can define a dual unit vector as follow
=𝐔𝐔 ,𝐮𝐮 =ε𝐮𝐮� + 2𝐯𝐯 , ε𝐯𝐯� = 1. 2 2 ‖So,𝐔𝐔‖ is a〈 𝐔𝐔unit𝐔𝐔 〉vector‖𝐮𝐮 ‖iff ε=〈𝐮𝐮1𝐮𝐮�, 〉 , = 0.
We 𝐔𝐔can explain what the‖ 𝐮𝐮geometric‖ 〈𝐮𝐮 meaning𝐮𝐮�〉 of scalar product of two dual vector and
is: Let U and V two lines in 3-dimensional real space. Then the real and dual parts𝐔𝐔 of
the𝐕𝐕 scalar product which is defined above measure the angle and the shortest distance
between two lines, respectively. The direction vector of the common perpendicular line
is found by w = u×v.
4 Definition 1 The dual angle between the dual vectors and can be represented by
( , ) = ( , ) + d( , ). 𝐔𝐔 𝐕𝐕 where∢� 𝐔𝐔 𝐕𝐕 the dual∢ 𝐮𝐮 angle𝐯𝐯 𝜀𝜀between𝑈𝑈 𝑉𝑉 two lines, the angle between direction vectors and the shortest distance between two lines are denoted by ( , ), ( , ) and d( , ),
respectively. ∢� 𝐔𝐔 𝐕𝐕 ∢ 𝐮𝐮 𝐯𝐯 𝑈𝑈 𝑉𝑉
Let ( ) = ( ) + (t) be a one-parameter dual unit curve. Then its correspondence in
w.r.t.𝐗𝐗 𝑡𝑡 Study’s𝐱𝐱 𝑡𝑡 transference𝜀𝜀𝐱𝐱� principle is given by 3 ℝ( , ) = ( ) + ( )
where𝐫𝐫 𝑡𝑡 𝑢𝑢 ,𝐜𝐜 𝑡𝑡 . 𝑢𝑢 𝐱𝐱=𝑡𝑡 × and x are directrix curve and generator vector of the ruled
surface, 𝑡𝑡respectively.𝑢𝑢 ∈ ℝ 𝐜𝐜 𝐱𝐱 𝐱𝐱�
2.2 Kinematic Generation of a Ruled Surface
In this section, following Blaschke [11] we give some differential geometric properties of
a ruled surface which is drawn by a one-parameter motion in 3-dimensional real space.
Let L(t) a one-parameter family of lines in 3-dimesional real space and defined by:
c, x: D ⊂ → , r: D× → , 3 3 ( , ) =𝐸𝐸 ( 𝐸𝐸) + ( )ℝ 𝐸𝐸 (1) where𝐫𝐫 𝑡𝑡 𝑤𝑤 c( )𝐜𝐜 ve𝑡𝑡 ( )𝑤𝑤 𝐱𝐱are𝑡𝑡 base and generator curve, respectively. In addition, we assume
= 1. The𝑡𝑡 n 𝐱𝐱the𝑡𝑡 mapping x(D) be the spherical image of L. Since vector c is the position vector‖𝐱𝐱‖ and x is the direction vector at a point on the line L, then the dual vector
0 representation of a line L is 𝑡𝑡
( ) = ( ) + ( ) × (t) = ( ) + ( ). (2)
𝐗𝐗 𝑡𝑡 𝐱𝐱 𝑡𝑡 ε𝐜𝐜 𝑡𝑡 𝐱𝐱 𝐱𝐱 𝑡𝑡 ε𝐱𝐱� 𝑡𝑡 as shown Fig. 1.
5 Let
= , = , = × (3) 𝐗𝐗′𝟏𝟏 �𝐗𝐗𝟏𝟏 𝐗𝐗 𝐗𝐗𝟐𝟐 𝐗𝐗𝟑𝟑 𝐗𝐗𝟏𝟏 𝐗𝐗𝟐𝟐� ‖𝐗𝐗′𝟏𝟏‖ be a dual orthonormal frame. While generate the ruled surface, we get the derivative
𝟏𝟏 equations of this motion like this; 𝐗𝐗
0 0 = 0 𝟏𝟏 𝟏𝟏 𝐗𝐗′ 0 𝜅𝜅̂ 0 𝐗𝐗 �𝐗𝐗′𝟐𝟐� �−𝜅𝜅̂ 𝜏𝜏̂� �𝐗𝐗𝟐𝟐� 𝟑𝟑 𝐗𝐗′𝟑𝟑 −𝜏𝜏̂ 𝐗𝐗 [ , , ] ( ) ( ) = = , = = ′ ′′ 𝐗𝐗1 𝐗𝐗1 𝐗𝐗1 𝜅𝜅̂ 𝜅𝜅̂ 𝑡𝑡 ‖𝐗𝐗′1‖ 𝜏𝜏̂ 𝜏𝜏̂ 𝑡𝑡 ′ 𝟐𝟐 ‖𝐗𝐗 1‖ where and are differentiable dual invariants and called by dual curvature and dual torsion𝜅𝜅 of̂ (𝜏𝜏̂ )-ruled surface, respectively. Here, if we expand this functions to the real
1 and dual parts,𝐗𝐗 𝑡𝑡 we get
= + , = + ,
𝜅𝜅̂ 𝜅𝜅 𝜀𝜀𝜅𝜅̅ 𝜏𝜏̂ 𝜏𝜏 𝜀𝜀𝜏𝜏̅ and then derivative equations become
0 0 ′ = 0 (4) 1 1 𝐱𝐱′ 0 𝜅𝜅 0 𝐱𝐱 2 �𝐱𝐱2� �−𝜅𝜅 𝜏𝜏� �𝐱𝐱 � ′ 3 𝐱𝐱3 −𝜏𝜏 𝐱𝐱 = + , = + + , = . (5) ′ ′ ′ 𝐱𝐱�1 𝜅𝜅̅𝐱𝐱2 𝜅𝜅𝐱𝐱�2 𝐱𝐱�2 −𝜅𝜅̅𝐱𝐱1 𝜏𝜏̅𝐱𝐱3 − 𝜅𝜅𝐱𝐱�1 𝜏𝜏𝐱𝐱�2 𝐱𝐱�3 −𝜏𝜏̅𝐱𝐱2 − 𝜏𝜏𝐱𝐱�2 Since
6 = = +
�‖𝐗𝐗′1‖𝑑𝑑𝑑𝑑 � 𝜅𝜅̂𝑑𝑑𝑑𝑑 � 𝜅𝜅𝜅𝜅𝜅𝜅 𝜀𝜀 � 𝜅𝜅̅𝑑𝑑𝑑𝑑 dual arc-length of ( ) dual spherical curve is an invariant of the ruled surface then
1 and are𝐗𝐗 integral𝑡𝑡 invariants of this surface. Analogously,
∫ 𝜅𝜅𝜅𝜅𝜅𝜅 ∫ 𝜅𝜅̅𝑑𝑑𝑑𝑑 = = +
�‖𝐗𝐗′3‖𝑑𝑑𝑑𝑑 � 𝜏𝜏̂𝑑𝑑𝑑𝑑 � 𝜏𝜏𝜏𝜏𝜏𝜏 𝜀𝜀 � 𝜏𝜏̅𝑑𝑑𝑑𝑑 is dual arc-length of ( ) dual spherical curve thus, and are integral
3 invariants of this surface𝐗𝐗. So𝑡𝑡, we get real and dual parts of these∫ 𝜏𝜏𝜏𝜏𝜏𝜏 functions∫ 𝜏𝜏̅𝑑𝑑 𝑑𝑑like that:
= , ′ 𝜅𝜅 [‖𝐱𝐱 ,1‖ , ] = ′ ′′ 𝐱𝐱1 𝐱𝐱1 𝐱𝐱1 𝜏𝜏 ′ 𝟐𝟐 ‖𝐱𝐱 1‖ , = ′ ′ 〈𝐱𝐱 1 𝐱𝐱� 1〉 𝜅𝜅̅ [ ,𝜅𝜅 , ] + [ , , ] + [ , , ] 2 = . (6) ′ ′′ ′ ′′ ′ ′′ 𝐱𝐱�1 𝐱𝐱1 𝐱𝐱1 𝐱𝐱1 𝐱𝐱�1 𝐱𝐱1 𝐱𝐱1 𝐱𝐱1 𝐱𝐱�1 𝜏𝜏𝜅𝜅̅ 𝜏𝜏̅ 2 − 𝜅𝜅 𝜅𝜅 The distribution parameter of ( ) ruled surface is given by
𝐗𝐗1 𝑡𝑡 − 1 , = = = , ( 0). ′ 〈𝐱𝐱1 𝐱𝐱�1′〉 𝜅𝜅̅ 𝛿𝛿 ′2 𝜅𝜅 ≠ 𝑑𝑑 𝐱𝐱1 𝜅𝜅 If = 0 then ( ) denotes a cylindrical surface.
𝜅𝜅 𝐗𝐗1 𝑡𝑡 We may give a base curve of ( ) ruled surface like that,
𝐗𝐗1 𝑡𝑡 − ( ) = ( ) × ( ).
𝐜𝐜 𝑡𝑡 𝐱𝐱𝟏𝟏 𝑡𝑡 𝐱𝐱�𝟏𝟏 𝑡𝑡
7 Then ( ) ruled surface can be given in vector form:
𝐗𝐗1 𝑡𝑡 − ( , ) = ( ) × ( ) + ( ).
𝐫𝐫 𝑡𝑡 𝑤𝑤 𝐱𝐱1 𝑡𝑡 𝐱𝐱�𝟏𝟏 𝑡𝑡 𝑤𝑤𝐱𝐱1 𝑡𝑡 Let striction curve be m(t). Since m(t) is on the ruled surface and is surface normal
2 along striction curve then we can write the tangent vector of the striction𝐱𝐱 curve of ruled surface like 𝐫𝐫 d = + , , . d 𝐦𝐦 𝑎𝑎𝐱𝐱1 𝑏𝑏𝐱𝐱3 𝑎𝑎 𝑏𝑏 ∈ ℝ 𝑡𝑡 Besides, m(t) is the intersection point of , , then,
𝐱𝐱1 𝐱𝐱2 𝐱𝐱3 × = , × =
𝐦𝐦 𝐱𝐱1 𝐱𝐱�1 𝐦𝐦 𝐱𝐱3 𝐱𝐱�3 Differentiating these equations and with the help of the equation (5), we find
d = + . (7) d 𝐦𝐦 𝜏𝜏̅𝐱𝐱1 𝜅𝜅̅𝐱𝐱3 𝑡𝑡 Then, the arc-length function ( ) of the striction curve is found by
𝑠𝑠 𝑡𝑡 ( ) = + . 2 2 𝑠𝑠 𝑡𝑡 � �𝜏𝜏̅ 𝜅𝜅̅ 𝑑𝑑𝑑𝑑 The equation of the striction curve is given as follow
( ), ( ) ( ) = ( ) ( ). ′ ( )′ 〈𝐱𝐱𝟏𝟏 𝑡𝑡 𝐜𝐜 𝑡𝑡 〉 𝐦𝐦 𝑡𝑡 𝐜𝐜 𝑡𝑡 − ′ 𝟐𝟐 𝐱𝐱𝟏𝟏 𝑡𝑡 ‖𝐱𝐱𝟏𝟏 𝑡𝑡 ‖ 2.3 Striction of a Ruled Surface
8 From the fundamental theorem of space curves, we can determine a likely shape of a
space curve from its curvature and torsion. If we want to determine the shape of a ruled surface (non-cylindrical) we have to know its Kruppa invariants; curvature, torsion and striction.
Let be arc-length of striction curve then,
𝑠𝑠 d = cos ( ) ( ) + sin ( ) ( ), (0 ( ) 2 ) (8) ds 𝐦𝐦 𝜎𝜎 𝑠𝑠 𝐱𝐱1 𝑠𝑠 𝜎𝜎 𝑠𝑠 𝐱𝐱3 𝑠𝑠 ≤ 𝜎𝜎 𝑠𝑠 ≤ 𝜋𝜋 can be written where ( ) is striction of the ruled surface.
𝜎𝜎 𝑠𝑠 Theorem 1 (G. Sannia, 1925)
Let and be continuously differentiable functions and be continuous on an interval
I 𝜅𝜅. Then,𝜎𝜎 these functions determine a twice continuously𝜏𝜏 differentiable ruled surface,
[⊂12ℝ].
Theorem 2 Let ( ), ( ) are -valued differentiable functions. Then these functions
determine a dual𝜅𝜅 curvê 𝑡𝑡 𝜏𝜏̂ 𝑡𝑡 : 𝔻𝔻 on an interval I , [13]. 3 𝐗𝐗𝟏𝟏 𝐼𝐼 → 𝔻𝔻 ⊂ ℝ Corollary 1 Considering Study’ s transference principle, this theorem corresponds to that
if we know -valued differentiable functions ( ), ( ) then we get only one ruled
surface in 3-dimensional𝔻𝔻 Euclidean space. 𝜅𝜅̂ 𝑡𝑡 𝜏𝜏̂ 𝑡𝑡
2.4 Differential Geometric Properties of a Ruled Surface
9 The Differential geometric properties of a ruled surface can be expressed in terms of
, , , functions.
Let𝜅𝜅 𝜏𝜏 the𝜅𝜅̅ 𝜏𝜏 ruled̅ surface be parametrized by (1). Then partial derivatives of the ruled surface
= + + ,
𝐫𝐫𝑡𝑡 =𝜏𝜏̅𝐱𝐱1, 𝑤𝑤𝑤𝑤𝐱𝐱=2 0, 𝜅𝜅 ̅ 𝐱𝐱 3 (9)
𝑤𝑤 𝟏𝟏 𝑤𝑤𝑤𝑤 𝐫𝐫 =𝐱𝐱 𝐫𝐫
𝐫𝐫𝑡𝑡𝑡𝑡 𝜅𝜅𝐱𝐱2 may determine the unit normal of the ruled surface as follow
× ( + + ) × ( ) = = = . × ( + + ) × ( ) ( ) + 𝐫𝐫𝑡𝑡 𝐫𝐫𝑤𝑤 𝜏𝜏̅𝐱𝐱1 𝑤𝑤𝑤𝑤𝐱𝐱2 𝜅𝜅̅𝐱𝐱3 𝐱𝐱𝟏𝟏 𝜅𝜅̅𝐱𝐱2 − 𝑤𝑤𝑤𝑤𝐱𝐱3 𝐧𝐧 𝑡𝑡 𝑤𝑤 1 2 3 𝟏𝟏 2 2 ‖𝐫𝐫 𝐫𝐫 ‖ ‖ 𝜏𝜏̅𝐱𝐱 𝑤𝑤𝑤𝑤𝐱𝐱 𝜅𝜅̅𝐱𝐱 𝐱𝐱 ‖ � 𝑤𝑤𝑤𝑤 𝜅𝜅̅ Furthermore, we can achieve coefficients of metric.
g = , = + ( ) + 2 2 2 g11 = 〈𝐫𝐫𝑡𝑡, 𝐫𝐫𝑡𝑡〉 =𝜏𝜏̅ 𝑤𝑤 𝑤𝑤 𝜅𝜅 ̅
12 𝑡𝑡 𝑤𝑤 g = 〈𝐫𝐫 ,𝐫𝐫 〉 =𝜏𝜏1̅
22 〈𝐫𝐫𝑤𝑤 𝐫𝐫𝑤𝑤〉 are coefficients of the first fundamental form [I] and
( + ) ( + ) h = , = ′ ( ) + 11 𝑡𝑡𝑡𝑡 𝜅𝜅̅ 𝜏𝜏̅ 𝑤𝑤𝜅𝜅 − 𝜏𝜏 − 𝑤𝑤𝑤𝑤 𝑤𝑤𝑤𝑤𝑤𝑤 𝜅𝜅̅′ 〈𝐫𝐫 𝐧𝐧〉 2 2 � 𝑤𝑤𝑤𝑤 𝜅𝜅̅ h = , = ( ) + 12 𝑡𝑡𝑡𝑡 𝜅𝜅̅𝜅𝜅 〈𝐫𝐫 𝐧𝐧〉 2 2 h = , =�0 𝑤𝑤𝑤𝑤 𝜅𝜅̅
22 〈𝐫𝐫𝑤𝑤𝑤𝑤 𝐧𝐧〉
10 are coefficients of the second fundamental form [II]. After some calculations, we get
Gauss and mean curvatures of the ruled surface:
h h h K ( , ) = = , g g g 2 (( )2 +2 ) 11 22 − 12 𝜅𝜅̅ 𝜅𝜅 G 𝑡𝑡 𝑤𝑤 2 − 2 2 2 11 22 − 12 𝑤𝑤𝑤𝑤 𝜅𝜅̅ h g + g h 2h g K ( , ) = 2(g g g ) 11 22 11 22 − 12 12 M 𝑡𝑡 𝑤𝑤 2 ( + 11 22 −) 12 ( + ) 2 = . ′ 2(( ) + ) / ′ 𝜅𝜅̅ 𝜏𝜏̅ 𝑤𝑤𝜅𝜅 − 𝜏𝜏 − 𝑤𝑤𝑤𝑤 𝑤𝑤𝑤𝑤𝑤𝑤 𝜅𝜅̅ − 𝜅𝜅𝜅𝜅̅𝜏𝜏̅ 2 2 3 2 𝑤𝑤𝑤𝑤 𝜅𝜅̅ 2.5 Integral Invariants of a Closed Ruled Surface
In addition to definition of a one-parameter motion ( ), if it is periodic, i.e.
1 ( + ) = ( ), then this motion to be closed on𝐗𝐗 DUS.𝑡𝑡 So, corresponding ruled
1 1 surface𝐗𝐗 𝑡𝑡 𝑃𝑃is also𝐗𝐗 closed.𝑡𝑡 Hoschek defined integral invariants to this type surfaces: First integral invariant is pitch which is described like that: If ( ) generate a ruled surface
1 and the motion is closed then, after one period, this vector𝐱𝐱 𝑡𝑡is on the same line and
generally different point. Length between these two points is called pitch of the ruled
surface and it is formulated by
= , . (10) ′ 𝑙𝑙𝐗𝐗1 − �〈𝐜𝐜 𝐱𝐱〉𝑑𝑑𝑑𝑑 Second integral invariant is angle of pitch which is described like that: Let be a unit
vector on the plane ( , ) and generate a developable surface during this𝛏𝛏 motion:
2 3 = cos( ) + sin (𝐱𝐱 ) 𝐱𝐱
2 3 𝛏𝛏Then, the𝜙𝜙 measure𝐱𝐱 of𝜙𝜙 total𝐱𝐱 variation of the angle during this motion is called by angle of pitch of the ruled surface and formulated as follow𝜙𝜙 :
11 = = , . (11) ′ 𝜆𝜆𝐗𝐗1 � 𝑑𝑑𝑑𝑑 �〈𝐱𝐱2 𝐱𝐱3〉𝑑𝑑𝑑𝑑 3. Construction of a Bézier-Like Curve on DUS
DUS can be defined, similar to the real one, by
= { = + | = 1, , } 2 3 𝔻𝔻 𝑆𝑆Theoretically,𝐗𝐗 𝐱𝐱 a dual𝜀𝜀𝐱𝐱� ‖ parametric𝐗𝐗‖ 𝐱𝐱 𝐱𝐱�representation∈ ℝ of a DUS can be given in vector form
[12]:
( , ) = (cos sin , sin sin , cos ), (12)
where𝐗𝐗 𝑢𝑢� 𝑣𝑣� = +𝑢𝑢� , 𝑣𝑣�= +𝑢𝑢� , 𝑣𝑣� , , 𝑣𝑣�, . Using Taylor expansion of dual- parameter𝑢𝑢� function𝑢𝑢 𝜀𝜀𝑢𝑢� s𝑣𝑣�, equation𝑣𝑣 𝜀𝜀𝑣𝑣 ̅(12𝑢𝑢) 𝑣𝑣becomes𝑢𝑢� 𝑣𝑣̅ ∈ ℝ
( , ) = ( , ) + [ . ( , ) + . ( , )]
𝑢𝑢 𝑣𝑣 𝐗𝐗or 𝑢𝑢equivalently� 𝑣𝑣� 𝐱𝐱 𝑢𝑢 𝑣𝑣 𝜀𝜀 𝑢𝑢� 𝐱𝐱 𝑢𝑢 𝑣𝑣 𝑣𝑣̅ 𝐱𝐱 𝑢𝑢 𝑣𝑣
( , ) = (cos sin , sin sin , ) + [ ( sin sin , cos sin , 0) +
𝐗𝐗 𝑢𝑢 � 𝑣𝑣� (cos𝑢𝑢 cos𝑣𝑣 , sin𝑢𝑢 cos𝑣𝑣 𝑐𝑐𝑐𝑐, 𝑐𝑐𝑐𝑐 )𝜀𝜀].𝑢𝑢� − 𝑢𝑢 𝑣𝑣 𝑢𝑢 𝑣𝑣 (13)
Equation (13𝑣𝑣̅ ) represents𝑢𝑢 𝑣𝑣 four𝑢𝑢-parameter𝑣𝑣 −𝑠𝑠𝑠𝑠𝑠𝑠 subspace𝑠𝑠 in real space and this is so-called a
family of line complexes. The real part of the equation (13) depends on the , parameters. Therefore, for the convenience of all parameters in the general𝑢𝑢 equation𝑣𝑣
(13), we take , : , = ( , ) and = ( , ). Then, the number of 2 parameters descends𝑢𝑢� 𝑣𝑣̅ ℝ from↦ ℝ four𝑢𝑢� to𝑢𝑢� two𝑢𝑢 𝑣𝑣. Thus,𝑣𝑣 ̅ indicates𝑣𝑣̅ 𝑢𝑢 𝑣𝑣 the two-parameter motion on
DUS and so, we know this motion corresponds𝐗𝐗 to a line-congruence in . Now, if we 3 return to the main purpose of this paper, we will construct a one-parameterℝ devisable
motion on DUS via Bézier technic:
12 Let = { ( , ) | 0 , 0 2 } be a domain of RUS and 2 { ,ℬ , … , 𝑢𝑢 𝑣𝑣} is∈ theℝ set of≤ the𝑢𝑢 ≤ control𝜋𝜋 ≤ points𝑣𝑣 ≤ of𝜋𝜋 degree n+1 Bernstein-Bézier curve on
0 1 𝑛𝑛 this𝐩𝐩 domain𝐩𝐩 𝐩𝐩 where ( , ). Bernstein-Bézier curve is then given by
𝐩𝐩𝑖𝑖 ∈ 𝑢𝑢 𝑣𝑣 ( ) = (1 ) , . (14) 𝑛𝑛 𝑛𝑛 𝑛𝑛−𝑖𝑖 𝑖𝑖 𝐛𝐛 𝑡𝑡 ∑𝑖𝑖=0 � � − 𝑡𝑡 𝑡𝑡 𝐩𝐩𝑖𝑖 𝑡𝑡 ∈ ℝ It is known that𝑖𝑖 if = and { , , } are collinear then the curve has closed
0 𝑛𝑛 𝑛𝑛−1 0 1 shape and C1 continuity𝐩𝐩 ,𝐩𝐩 respectively𝐩𝐩 [13𝐩𝐩, 14𝐩𝐩]. Using these conditions, a closed Bézier curve b(t) can be obtained.
Mapping b(t) via , we get a closed spherical Bézier-like curve on RUS:
𝐗𝐗 = ( ) = (1 ) , = ( ) = (1 ) (15) 𝑛𝑛 𝑛𝑛−𝑖𝑖 𝑖𝑖 𝑛𝑛 𝑛𝑛−𝑖𝑖 𝑖𝑖 𝑛𝑛 1 𝑛𝑛 2 𝑢𝑢 𝑢𝑢 𝑡𝑡 ∑𝑖𝑖=0 � � − 𝑡𝑡 𝑡𝑡 𝐩𝐩𝑖𝑖 𝑣𝑣 𝑣𝑣 𝑡𝑡 ∑𝑖𝑖=0 � � − 𝑡𝑡 𝑡𝑡 𝐩𝐩𝑖𝑖 where (j=1,2) are𝑖𝑖 the jth coordinate of point . Since w𝑖𝑖e already know functions
𝐩𝐩𝑖𝑖𝑗𝑗 𝐩𝐩𝑖𝑖 and , we get a closed dual spherical Bézier-like curve on DUS:
𝑢𝑢�( ) =𝑣𝑣̅ ( ), ( ) = cos ( ) sin ( ) , sin ( ) sin ( ) , ( ) +
𝐗𝐗 𝑡𝑡 𝐗𝐗 � 𝑢𝑢 𝑡𝑡 𝑣𝑣 𝑡𝑡 � �[ ( )𝑢𝑢( 𝑡𝑡sin 𝑣𝑣( )𝑡𝑡sin (𝑢𝑢)𝑡𝑡, cos 𝑣𝑣( 𝑡𝑡) sin𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐( )𝑡𝑡, 0�) +
𝜀𝜀 𝑢𝑢�(𝑡𝑡). (−cos 𝑢𝑢( 𝑡𝑡) cos 𝑣𝑣(𝑡𝑡) , sin 𝑢𝑢( 𝑡𝑡) cos 𝑣𝑣(𝑡𝑡) , ( ))]. (16)
Then one-parameter closed𝑣𝑣̅ motion𝑡𝑡 𝑢𝑢(𝑡𝑡) correspond𝑣𝑣 𝑡𝑡 to𝑢𝑢 a𝑡𝑡 closed𝑣𝑣 ruled𝑡𝑡 − surface𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣 𝑡𝑡 in . 3 The most advantageous side of this method𝐗𝐗 𝑡𝑡 , we will determine the control pointsℝ for
the design is the absence of any constraint, except for the condition of being closed. So,
we can achieve any closed ruled surface via this technic controlled by the user.
Corollary 2
( ) represent a ruled surface in 3-dimensional real space and its shape changes with
respect𝐗𝐗 𝑡𝑡 to its control points.
4 On the Invariants of Ruled Surfaces
13 In this section, properties of ruled surfaces are expressed via coordinate functions. Let
take , : , = ( , ), = ( , ), = ( ), = ( ) in the equation (13), 2 then 𝑢𝑢� (𝑣𝑣̅)ℝ represents↦ ℝ 𝑢𝑢� the𝑢𝑢� one𝑢𝑢 𝑣𝑣-parameter𝑣𝑣̅ 𝑣𝑣̅ 𝑢𝑢 motion𝑣𝑣 𝑢𝑢 on𝑢𝑢 DUS:𝑡𝑡 𝑣𝑣 𝑣𝑣 𝑡𝑡
( ) 𝐗𝐗= 𝑡𝑡cos ( ) sin ( ) , sin ( ) sin ( ) , cos ( )
𝐗𝐗 𝑡𝑡 � 𝑢𝑢 𝑡𝑡 ( )𝑣𝑣( 𝑡𝑡sin (𝑢𝑢) sin𝑡𝑡 (𝑣𝑣) ,𝑡𝑡cos (𝑣𝑣)𝑡𝑡sin� ( ) , 0) + (17) + ( ). (cos ( ) cos ( ) , sin ( ) cos ( ) , sin ( )) 𝑢𝑢� 𝑡𝑡 − 𝑢𝑢 𝑡𝑡 𝑣𝑣 𝑡𝑡 𝑢𝑢 𝑡𝑡 𝑣𝑣 𝑡𝑡 𝜀𝜀 � � Assume that 𝑣𝑣(̅ )𝑡𝑡 = (𝑢𝑢),𝑡𝑡 then we𝑣𝑣 𝑡𝑡 have 𝑢𝑢 𝑡𝑡 𝑣𝑣 𝑡𝑡 − 𝑣𝑣 𝑡𝑡
𝟏𝟏 ( ( ), ( )𝐗𝐗) =𝑡𝑡 (𝐗𝐗 ) +𝑡𝑡 ( ).
𝟏𝟏 𝟏𝟏 𝟏𝟏 So𝐗𝐗 equation𝑢𝑢� 𝑡𝑡 𝑣𝑣� 𝑡𝑡 (5) become𝐗𝐗 𝑡𝑡 ε𝐗𝐗� 𝑡𝑡
′ = + . �1 3 1 1 𝐗𝐗′ −𝜅𝜅𝛼𝛼 𝜅𝜅̅ 𝜅𝜅𝛼𝛼 𝐗𝐗 �2 𝜏𝜏𝛼𝛼2 − 𝜅𝜅̅ −𝜅𝜅𝛼𝛼3 − 𝜏𝜏𝛼𝛼1 𝜏𝜏̅ 𝜅𝜅𝜅𝜅2 2 �𝐗𝐗′ � � � �𝐗𝐗 � 3 3 1 3 Here𝐗𝐗� , 𝜏𝜏 and𝛼𝛼 can be−𝜏𝜏 found̅ by −following𝜏𝜏𝛼𝛼 𝐗𝐗 equations;
1 2 3 = 𝛼𝛼 𝛼𝛼 𝛼𝛼, = , = .
1 3 2 2 3 2 1 3 3 1 3 2 1 1 2 Corollary𝐗𝐗� 𝛼𝛼 𝐗𝐗 −3 𝛼𝛼 𝐗𝐗 𝐗𝐗� 𝛼𝛼 𝐗𝐗 − 𝛼𝛼 𝐗𝐗 𝐗𝐗� 𝛼𝛼 𝐗𝐗 − 𝛼𝛼 𝐗𝐗
From the equation (17), the expressions of the dual curvature function and dual torsion
function can be written by coordinate functions:
= sin + ′2 2 ′2 𝜅𝜅 �𝑢𝑢sin ( 𝑣𝑣cos𝑣𝑣 + sin ) + ( + sin ) + = ′2 ′ ′ 2 ′2 𝑢𝑢 𝑣𝑣 𝑣𝑣̅ 𝑣𝑣 𝑢𝑢�𝑢𝑢 𝑣𝑣sin 𝑢𝑢 𝑣𝑣+ 𝑣𝑣𝑢𝑢̅ 𝑢𝑢�𝑣𝑣 𝑣𝑣 𝑣𝑣 𝑣𝑣𝑣𝑣̅ 𝜅𝜅̅ ′2 2 ′2 cos sin + 2 �𝑢𝑢 + sin𝑣𝑣 (𝑣𝑣 + ) = ′3 2 si′ n′2 + ′ ′′ ′′ ′ 𝑣𝑣�𝑢𝑢 𝑣𝑣 𝑢𝑢 𝑣𝑣 � 𝑣𝑣 𝑢𝑢 𝑣𝑣 𝑢𝑢 𝑣𝑣 𝜏𝜏 ′2 2 ′2 = sin ( )𝑢𝑢 𝑣𝑣 𝑣𝑣 (18) ′ where𝜏𝜏̅ the𝑣𝑣 𝑢𝑢� subscript𝑣𝑣 − 𝑣𝑣̅𝑢𝑢′s indicate the partial derivative of the coordinate functions.
14 The distribution parameter function of the ( ) ruled surface can be found by
1 coordinate functions 𝐗𝐗 𝑡𝑡
sin ( cos + sin ) + ( + sin ) + = (19) ′2 sin +′ ′ 2 2 𝑢𝑢 𝑣𝑣 𝑣𝑣̅ 𝑣𝑣 𝑢𝑢�𝑢𝑢 𝑣𝑣 𝑢𝑢 𝑣𝑣 𝑣𝑣𝑢𝑢̅ 𝑢𝑢�𝑣𝑣 𝑣𝑣 𝑣𝑣′ 𝑣𝑣𝑣𝑣̅ 𝛿𝛿 ′2 2 ′2 The directrix curve of the 𝑢𝑢 ( ) ruled𝑣𝑣 𝑣𝑣 surface can be found as follow
1 ( ) = ( cos sin cos𝐗𝐗 𝑡𝑡 sin , sin sin cos + cos , sin ). 2 Then𝐚𝐚 𝑡𝑡 one− can𝑢𝑢� write𝑢𝑢 𝑣𝑣 𝑣𝑣 − 𝑣𝑣̅ 𝑢𝑢 −𝑢𝑢� 𝑢𝑢 𝑣𝑣 𝑣𝑣 𝑣𝑣̅ 𝑢𝑢 𝑢𝑢� 𝑣𝑣
( , ) = ( ) + ( ), (20)
1 or𝐑𝐑 𝑡𝑡explicitly𝑤𝑤 𝐚𝐚 𝑡𝑡 𝑤𝑤𝐗𝐗 𝑡𝑡 𝑤𝑤 ∈ ℝ
( , ) = ( cos sin cos sin , sin sin cos + cos , sin ) 2 𝐑𝐑 𝑡𝑡 𝑤𝑤 −+𝑢𝑢� (cos𝑢𝑢 sin𝑣𝑣 , sin𝑣𝑣 − 𝑣𝑣sin̅ 𝑢𝑢, cos−𝑢𝑢�). 𝑢𝑢 𝑣𝑣 𝑣𝑣 𝑣𝑣̅ 𝑢𝑢 𝑢𝑢� 𝑣𝑣
From equations𝑤𝑤 (7) and𝑢𝑢 (8)𝑣𝑣 we get𝑢𝑢 𝑣𝑣 𝑣𝑣 d d = + // = cos ( ) ( ) + sin ( ) ( ). (21) d ds 𝐦𝐦 𝐦𝐦 𝜏𝜏̅𝐗𝐗1 𝜅𝜅̅𝐗𝐗3 𝜎𝜎 𝑠𝑠 𝐗𝐗1 𝑠𝑠 𝜎𝜎 𝑠𝑠 𝐗𝐗3 𝑠𝑠 Lemma𝑡𝑡 1
From equation (21), we get a relation between the dual sections of dual curvature and dual torsion and the striction
sin ( ) + cot = = (22) sin ( cos + ′sin ) + ′2( 2+ sin′2 ) + 𝜏𝜏̅ 𝑣𝑣 𝑢𝑢�𝑣𝑣 − 𝑣𝑣̅𝑢𝑢′ �𝑢𝑢 𝑠𝑠𝑠𝑠𝑛𝑛 𝑣𝑣 𝑣𝑣 ′2 ′ ′ 2 ′2 𝜎𝜎 𝑢𝑢 𝑢𝑢 𝑣𝑣 𝑣𝑣 Example𝜅𝜅 ̅1 𝑢𝑢 𝑣𝑣 𝑣𝑣̅ 𝑣𝑣 𝑢𝑢� 𝑣𝑣 𝑢𝑢 𝑣𝑣 𝑣𝑣̅ 𝑢𝑢� 𝑣𝑣 𝑣𝑣 𝑣𝑣̅
In example 3.1.1. let ( ) = ( ) = be. Then we have ( ) = 0, ( ) = 2 . Then from
the equation (18), we𝑢𝑢 can𝑡𝑡 calculate𝑣𝑣 𝑡𝑡 the𝑡𝑡 invariants of the ruled𝑢𝑢� 𝑡𝑡 surface𝑣𝑣̅ 𝑡𝑡 that correspond𝑡𝑡
to these values:
15 = sin + 1 2 𝜅𝜅 �sin2 𝑡𝑡+ 3 = sin + 1 𝑡𝑡 𝑡𝑡 𝜅𝜅̅ 2 cos√ (si𝑡𝑡n + 2) = sin 2+ 1 𝑡𝑡 𝑡𝑡 𝜏𝜏 2 = sin . 𝑡𝑡
The𝜏𝜏̅ striction𝑡𝑡 𝑡𝑡 and distribution parameter of the ruled surface can be found by equations
(22) and (19), respectively:
sin + 1 cot = cos +22sin −𝑡𝑡√ 𝑡𝑡 σ sin𝑡𝑡(2 cos𝑡𝑡 + sin𝑡𝑡 ) + cos + 1 = . sin + 1 2 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝛿𝛿 2 4.1. Calculation of the𝑡𝑡 Integral Invariants of a Closed Ruled Surface
From equation (18) we can reformulate the integral invariants of -closed ruled
1 surface in terms of coordinate functions: 𝐗𝐗
= sin ( )d (24) ′ 𝑙𝑙𝐗𝐗1 � 𝑣𝑣 𝑢𝑢�𝑣𝑣 − 𝑣𝑣̅𝑢𝑢′ 𝑡𝑡 cos sin + 2 + sin ( + ) = d (25) ′3 2 si′ n′2 + ′ ′′ ′′ ′ 1 𝑣𝑣�𝑢𝑢 𝑣𝑣 𝑢𝑢 𝑣𝑣 � 𝑣𝑣 𝑢𝑢 𝑣𝑣 𝑢𝑢 𝑣𝑣 𝜆𝜆𝐗𝐗 − � ′2 2 ′2 𝑡𝑡 4.2. Design of a Ruled Surface𝑢𝑢 via Dual𝑣𝑣 Spherical𝑣𝑣 Bézier-Like Curves
To indicate that the method, in section 3.2, is appropriate for calculation, we can apply
it to the computer applications. From equations (15) we have the following equations:
1 ( ) = 𝑛𝑛−1 (1 ) ( ) , ′ 𝑛𝑛−𝑖𝑖−1 𝑖𝑖 𝑛𝑛 − 1 1 𝑢𝑢 𝑡𝑡 𝑛𝑛 � � � − 𝑡𝑡 𝑡𝑡 �𝐩𝐩 𝑖𝑖+1 − 𝐩𝐩𝑖𝑖 � 𝑖𝑖=0 𝑖𝑖
16 1 ( ) = 𝑛𝑛−1 (1 ) ( ) , ′ 𝑛𝑛−𝑖𝑖−1 𝑖𝑖 𝑛𝑛 − 2 2 𝑣𝑣 𝑡𝑡 𝑛𝑛 � � � − 𝑡𝑡 𝑡𝑡 �𝐩𝐩 𝑖𝑖+1 − 𝐩𝐩𝑖𝑖 � 𝑖𝑖=0 𝑖𝑖 2 ( ) = ( 1) 𝑛𝑛−2 (1 ) ( ) 2 ( ) + , ′′ 𝑛𝑛−𝑖𝑖−2 𝑖𝑖 𝑛𝑛 − 1 1 1 𝑢𝑢 𝑡𝑡 𝑛𝑛 𝑛𝑛 − � � � − 𝑡𝑡 𝑡𝑡 �𝐩𝐩 𝑖𝑖+2 − 𝐩𝐩 𝑖𝑖+1 𝐩𝐩𝑖𝑖 � 𝑖𝑖=0 𝑖𝑖 2 ( ) = ( 1) 𝑛𝑛−2 (1 ) ( ) 2 ( ) + . 𝑛𝑛−𝑖𝑖−2 𝑖𝑖 𝑛𝑛 − 2 2 2 𝑣𝑣′′ 𝑡𝑡 𝑛𝑛 𝑛𝑛 − � � � − 𝑡𝑡 𝑡𝑡 �𝐩𝐩 𝑖𝑖+2 − 𝐩𝐩 𝑖𝑖+1 𝐩𝐩𝑖𝑖 � 𝑖𝑖=0 𝑖𝑖 Thus, if we put these values into former equations of the ruled surface then they belong
to the control points. We can calculate the these values for some points specifically
since these calculations do not fit in this paper. For example, let the parameter value be
zero, i.e. t = 0, then we have following values:
(0) = , (0) = , ′ ′ 11 01 12 02 𝑢𝑢 (0) =𝑛𝑛�(𝐩𝐩 −1)𝐩𝐩 � 𝑣𝑣2 + 𝑛𝑛�𝐩𝐩 , − 𝐩𝐩 � ′′ 21 11 01 𝑢𝑢 (0) = 𝑛𝑛(𝑛𝑛 − 1)�𝐩𝐩 − 2𝐩𝐩 + 𝐩𝐩 �. ′′ 22 12 02 𝑣𝑣So we can𝑛𝑛 calculate𝑛𝑛 − � 𝐩𝐩the −curvature,𝐩𝐩 𝐩𝐩 torsion� , distribution parameter and the striction at this point.
(0) = sin + 2 2 2 𝜅𝜅 𝑛𝑛��𝐩𝐩11 − 𝐩𝐩01� �𝐩𝐩02� �𝐩𝐩12 − 𝐩𝐩02� n sin cos + sin (0) = 2 11 01 02 02 𝑢𝑢 02 ��𝐩𝐩 − 𝐩𝐩 � �𝐩𝐩sin� �𝑣𝑣̅ �+𝐩𝐩 � 𝑢𝑢� �𝐩𝐩 ��� 𝜅𝜅̅ 2 2 2 ��𝐩𝐩11 − 𝐩𝐩01� �𝐩𝐩02� �𝐩𝐩12 − 𝐩𝐩02� n + sin + + 2 2 11 01 12 02 𝑢𝑢 𝑣𝑣 02 12 02 𝑣𝑣 ��𝐩𝐩 − 𝐩𝐩 ��𝐩𝐩 − 𝐩𝐩 � �sin𝑣𝑣̅ 𝑢𝑢� + �𝐩𝐩 �� �𝐩𝐩 − 𝐩𝐩 � 𝑣𝑣̅ � 2 2 2 ��𝐩𝐩11 − 𝐩𝐩01� �𝐩𝐩02� �𝐩𝐩12 − 𝐩𝐩02�
17 n cos sin + 2 (0) = 2 2 2 � �𝐩𝐩02��𝐩𝐩11 − 𝐩𝐩01� ��𝐩𝐩sin11 − 𝐩𝐩01� + �𝐩𝐩02� �𝐩𝐩12 − 𝐩𝐩02� �� 2 2 𝜏𝜏 2 �𝐩𝐩11 − 𝐩𝐩01� �𝐩𝐩02� �𝐩𝐩12 − 𝐩𝐩02� (n 1)sin 2 + + 2 + + − �𝐩𝐩02� ��𝐩𝐩11 − 𝐩𝐩01��𝐩𝐩22 −sin𝐩𝐩12 𝐩𝐩0+2� �𝐩𝐩12 − 𝐩𝐩02��𝐩𝐩21 − 𝐩𝐩11 𝐩𝐩01�� 2 2 2 �𝐩𝐩11 − 𝐩𝐩01� �𝐩𝐩02� �𝐩𝐩12 − 𝐩𝐩02� (0) = nsin
𝜏𝜏̅ �𝐩𝐩02� �𝑢𝑢��𝐩𝐩12 − 𝐩𝐩02� − 𝑣𝑣̅�𝐩𝐩11 − 𝐩𝐩01�� n sin cos + sin (0) = 2 ��𝐩𝐩11 − 𝐩𝐩01� �𝐩𝐩sin02� �𝑣𝑣̅ +�𝐩𝐩02� 𝑢𝑢�𝑢𝑢 �𝐩𝐩02��� 2 2 𝛿𝛿 2 �𝐩𝐩11 − 𝐩𝐩01� �𝐩𝐩02� �𝐩𝐩12 − 𝐩𝐩02� sin + 2 2 cot ( (0)) = 2 �𝑢𝑢��𝐩𝐩12 − 𝐩𝐩02� − 𝑣𝑣̅�𝐩𝐩11 − 𝐩𝐩01��cos��𝐩𝐩11 − 𝐩𝐩+01� sin �𝐩𝐩02� �𝐩𝐩12 − 𝐩𝐩02� 𝜎𝜎 2 �𝐩𝐩11 − 𝐩𝐩01� �𝑣𝑣̅ �𝐩𝐩02� 𝑢𝑢�𝑢𝑢 �𝐩𝐩02�� For t = 1, it can be calculated in the same way.
We can show the design of a ruled surface in an example:
Example 2
Let the domain of RUS be ={( , ): 0 , 0 2 } and choose the control
points on this domain as follows:ℬ 𝑢𝑢 𝑣𝑣 ≤ 𝑢𝑢 ≤ 𝜋𝜋 ≤ 𝑣𝑣 ≤ 𝜋𝜋
p(0) = [ /8, /4], p(1) = [ /8,3 /8], p(2) = [3 /8,3 /8], p(3) = [3 /8, /4],
p(4) = [𝜋𝜋3 /8𝜋𝜋, /8], p(5) = 𝜋𝜋 [ /8,𝜋𝜋 /8], p(6) = p(0)𝜋𝜋 . 𝜋𝜋 𝜋𝜋 𝜋𝜋
Then the 𝜋𝜋Bézier𝜋𝜋 curve on this𝜋𝜋 domain𝜋𝜋 can be expressed as:
6 ( ) = 6 (1 ) 𝑛𝑛−𝑖𝑖 𝑖𝑖 𝐛𝐛 𝑡𝑡 � � � − 𝑡𝑡 𝑡𝑡 𝐩𝐩𝑖𝑖 𝑖𝑖=0 𝑖𝑖 (Fig 4.3).
So we can map this curve to the RUS by
( ) = cos ( ) sin ( ) , sin ( ) sin ( ) , cos ( )
𝐗𝐗 𝑡𝑡 � 𝑢𝑢 𝑡𝑡 𝑣𝑣 𝑡𝑡 𝑢𝑢 𝑡𝑡 𝑣𝑣 𝑡𝑡 𝑣𝑣 𝑡𝑡 � 18 parametrization (Fig 4.4).
In addition to describing the dual closed Bézier-like curve we can take = ve =
+ and if we put the control points in equation (15) we get a dual closed𝑢𝑢� 𝑢𝑢 Bézier− 𝑣𝑣 -like𝑣𝑣̅
𝑢𝑢curve𝑣𝑣 on DUS. Corresponding ruled surface is shown in figure (Fig 4.5). Furthermore, integral invariants of this ruled surface can be calculated as,
( ) = 1.419793061
𝜆𝜆𝐗𝐗 𝑡𝑡 ( ) = 0.3381414433.
𝐗𝐗 𝑡𝑡 𝑙𝑙5 Conclusion and Future Work
Several scientists investigated spherical Bézier curves by using deCasteljau-type algorithms. Those algorithms run a large number of inputs since they have to count the control points and also the angles among position vectors of the control points. This situation causes challenges in view of computation. Technically, several mathematical software can not compute the spherical Bézier curve. These methods are not suitable for us because one of our main purposes is to calculate the integral invariants of the closed ruled surface. We have developed a new method that is more efficient computable than the others in order to construct spherical Bézier-like curves.
In this paper, we have designed spherical Bézier-like curves via mapping the planar Bézier curves to the surface of RUS by unit sphere mapping. Naturally, coordinate functions emerge since we obtain ruled surfaces from a synectic congruence. Therefore, we obtained devisable ruled surfaces with respect to the control points. The control points can be chosen either on the plane(domain) or on the surface of RUS since the unit sphere mapping is one-to-one. The user can design the ruled surfaces with this method. Besides, the method is more suitable and faster for computation than the former methods.
Obviously, we can construct the dual spherical B-spline curves via constructing B-spline curves in the domain and mapping to the RUS. Thus, the design of the corresponding ruled surface belongs to the control points and knots of the spline curve.
Plane kinematics can be studied by this method and therefore results of spherical kinematics can be viewed. For example, Holditch-type theorems(plane and spherical) can be investigated in terms of control points.
In addition to Euclidean space, the ruled surfaces in hyperbolic space and Lorentz space can be designed by control points. So, timelike and spacelike ruled surfaces can be investigated by control points.
19
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22 Figure Captions List
Fig. 1 Dual representation of a line
Fig. 2 Closed Bézier curve on the domain
Fig. 3 Closed spherical Bézier curve on RUS
Fig. 4 The closed ruled surface with its striction curve
23
Fig. 1 Dual representation of a line
24
Fig. 2 Closed Bézier curve on the domain
25
Fig. 3 Closed spherical Bézier curve on RUS
26
Fig. 4 The closed ruled surface with its striction curve
27