Parametric Interval Bezier Curve Interpolation
O. Ismail, Senior Member, IEEE
Abstract— Bezier curves are a method of designing polynomial curve segments when you want to control their shape in an easy way. Bezier curves make sense for any degree. In Computer Aided Geometric Design (CAGD), the Bezier curve and surface have been widely used for geometric modeling. This paper presents a simple matrix form for parametric interval Bezier curve interpolation. The four fixed Kharitonov's polynomials (four fixed Bezier curves) 𝑷𝒏𝒋(𝒖) for (𝒋 = 𝟏, 𝟐, 𝟑, 𝟒) associated
with the original interval Bezier curve 𝑷𝒏𝑰(𝒖) are obtained. The fixed control points 𝜶𝒊,𝒏𝒋 for (𝒊 = 𝟎, 𝟏 , ⋯ , 𝒏) and (𝒋 = 𝟏, 𝟐, 𝟑, 𝟒) of the four fixed Kharitonov's polynomials (four fixed Bezier curves) that cause the four fixed Kharitonov's polynomials (four fixed Bezier curves) to pass through 𝜶𝒊,𝒏𝒋 for (𝒊 = 𝟎, 𝟏 , ⋯ , 𝒏) and (𝒋 = 𝟏, 𝟐, 𝟑, 𝟒) are computed. The desired interval control points {[𝒒𝒊−, 𝒒𝒊+]}𝒊=𝟎𝒏 of the desired parametric interval Bezier curve 𝑸𝒏𝑰(𝒖) that cause the parametric interval
Bezier curve to pass through the regions determined by {[𝒒𝒊−, 𝒒𝒊+]}𝒊=𝟎𝒏 are obtained from the fixed control points of the
four fixed Kharitonov's polynomials (four fixed Bezier curves) 𝜶𝒊,𝒏𝒋 for (𝒊 = 𝟎, 𝟏 , ⋯ , 𝒏) and (𝒋 = 𝟏, 𝟐, 𝟑, 𝟒). An illustrative example is included in order to demonstrate the effectiveness of the proposed method.
Index Term— CAGD, computer graphics, interpolation, interval Bezier curves.
I. INTRODUCTION
The construction of curves and surfaces is a key issue in computer aided geometric design (CAGD). The CAGD method arise from the need of the efficient computer representation of practical curves and surfaces, which is very broadly used in engineering design. In CAGD, the parametric curves is the combination of basis functions and control points. There are various expressions for describing a shape, but the number of practical ones is limited. A polynomial expression in parametric form with control points is most useful in most applications, for it is not only mathematically simple, but also it has favorable properties for engineering design; its expression does not depend on coordinate systems, geometrical properties are easily grasped and manipulated by control points, unwanted waviness usually does not appear in construction and modification of shapes, a Bezier curve of degree 𝑛has 𝑛 + 1 control points. Curves of high degree are not often used since there is only a weak relationship between the shape of the curve and the shape of the control polygon. Further, operations such as the evaluation of points require a large number of arithmetical operations, and so there is an increased risk of computational errors. In contrast, curves of low degree have few control points, and therefore yield a
The author is with Department of Computer Engineering, Faculty of Electrical and Electronic Engineering, University of Aleppo, Aleppo, (e-mail: oismail@ieee.org).
limited range of curve shapes. To widen the range of shapes without increasing the degree of the curve, a number of Bezier curves can be joined end to end to form a single continuous curve called a piecewise Bezier curve. In practice, the joins of the curves are required to be smooth.
In computer aided design, Bezier models act important role for free-form curve/surface modeling. They are used in many fields, such as path generation [1], image processing [2], data compression [3], rail transit [4], fluid computation [5] and virtual reality [6]. So their problems, interpolation [7], approximation [8] and parameterization [9], have been studied widely and deeply. Among these models, low-degree (rational) Bezier curve is simple and easy to use. Therefore, many researches consider its good modeling properties, for example, the optimization [10] and geometric mean [11] of its parameter.
A parametric interval Bezier curve [12-17] is a Bezier curve whose control points are rectangles (the sides of which are parallel to coordinate axis) in a plane.
The use of interpolation as curve design has become popular [18]. For interpolating curves, shape parameters offer a good way to vary curve’s shape.
In this paper, an algorithmic approach for parametric interval Bezier curve interpolation based on simple matrix form is proposed.
This paper is organized as follows. Section II contains the interval Bezier curves, and section III includes the basic results whereas section IV presents a numerical example, and the final section offers conclusions.
II. INTERVALBEZIERCURVES
An interval polynomial is a polynomial whose coefficients are interval. We shall denote such polynomials in the form 𝑃𝐼(𝑢) to distinguish them from ordinary (single-valued) polynomials. In general we express an interval polynomial of degree 𝑛 in the form:
𝑃𝐼(𝑢) = ∑[𝑎
𝑘 −, 𝑎
𝑘+] 𝑛
𝑘=0
𝐵𝑘𝑛(𝑢) , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢 ∈ [0,1]
(1) in terms of the Bernstein polynomial basis:
𝐵𝑘𝑛(𝑢) = (𝑛𝑘) (1 − 𝑢)(𝑛−𝑘)𝑢𝑘
𝑘 = 0, 1, ⋯ , 𝑛
on [0,1]. Usual interval arithmetic can be applied to the interval polynomials [19].
We will define a vector-valued interval 𝑃𝐼in the most general terms as any compact set of points (𝑥, 𝑦) in two dimensions. The addition of such sets is given by the Minkowski sum:
𝑃1𝐼+ 𝑃2𝐼= {(𝑥1+ 𝑥2, 𝑦1+ 𝑦2)|(𝑥1, 𝑦1) ∈ 𝑃1𝐼; (𝑥2, 𝑦2) ∈ 𝑃2𝐼}
(3) It is prudent to restrict our attention to the vector-valued intervals that are just the tensor products of scalar intervals:
𝑃𝐼= [𝑎−, 𝑎+] × [𝑏−, 𝑏+] = {(𝑥, 𝑦)| 𝑥 ∈ [𝑎−, 𝑎+] 𝑎𝑛𝑑 𝑦 ∈ [𝑏−, 𝑏+]}
(4)
We occasionally use the notation ([𝑎−, 𝑎+], [𝑏−, 𝑏+]) instead of ([𝑎−, 𝑎+] × [𝑏−, 𝑏+]) for 𝑃𝐼. Such vector-valued intervals are rectangular regions in the plane, and their addition a trivial matter:
𝑃1𝐼+ 𝑃2𝐼= [𝑎−+ 𝑐−, 𝑎++ 𝑐+] × [𝑏−+ 𝑑−, 𝑏++ 𝑑+]
(5) where, 𝑃1𝐼 = [𝑎−, 𝑎+] × [𝑏−, 𝑏+] and 𝑃2𝐼 = [𝑐−, 𝑐+] ×
[𝑑−, 𝑑+]. The extension of these ideas to vector-valued
intervals in spaces of higher dimension is straightforward.
III.THEBASICRESULTS
Let {[𝑝𝑖−, 𝑝𝑖+]}𝑖=0𝑛 be a given set of interval control points which defines the interval Bezier curve:
𝑃𝑛𝐼(𝑢) = ∑[𝑝𝑖−, 𝑝𝑖+] 𝑛
𝑖=0
𝐵𝑖𝑛(𝑢)
= ∑([𝑥𝑖−, 𝑥𝑖+], [𝑦𝑖−, 𝑦𝑖+]) 𝑛
𝑖=0
𝐵𝑖𝑛(𝑢)
0 ≤ 𝑢 ≤ 1
(6) of degree 𝑛 where,
𝐵𝑘𝑗(𝑢) = (𝑗
𝑘) (1 − 𝑢)(𝑗−𝑘)𝑢𝑘
(𝑘 = 0, 1, ⋯ , 𝑗)
(7) where, [𝑝𝑘−, 𝑝𝑘+] for (𝑘 = 0, 1, ⋯ , 𝑛) are interval control points (rectangular intervals of the form (4). For each 𝑢 ∈
[0,1], the value 𝑃𝐼(𝑢) of the interval Bezier curve (6) is a
vector interval that has the following significance: For any fixed Bezier curve 𝑃(𝑢) whose control points satisfy 𝑝𝑘 ∈ [𝑝𝑘−, 𝑝𝑘+] for (𝑘 = 0, 1, ⋯ , 𝑛) we have 𝑃(𝑢) ∈ 𝑃𝐼(𝑢) .
Likewise, the entire interval Bezier curve (6) defines a region in the plane that contains all Bezier curves whose control points satisfy 𝑝𝑘 ∈ [𝑝𝑘−, 𝑝𝑘+] for (𝑘 = 0, 1, ⋯ , 𝑛).
Parametric interval Bezier curve in general does not pass through any of the regions determined by control points except the first (𝑢 = 0) and last (𝑢 = 1) interval control points [𝑝0−, 𝑝0+] and [𝑝𝑛−, 𝑝𝑛+] . However, it is easy to
determine a parametric interval Bezier curve that interpolates a set of interval points using the matrix definition of an interval Bezier curve.
The problem is to find the desired parametric interval Bezier curve 𝑄𝑛𝐼(𝑢), calculated using {[𝑝𝑖−, 𝑝𝑖+]}𝑖=0𝑛 and will
pass through the regions defined by the desired interval control points {[𝑞𝑖−, 𝑞𝑖+]}𝑖=0𝑛 , that correspond to 𝑢 = 𝑐𝑖, for
(𝑖 = 0,1, ⋯ , 𝑛), such that 0 < 𝑐1< 𝑐2< ⋯ ⋯ 𝑐𝑛−1< 1 (to
ensure that the matrix with 𝑐𝑖’𝑠 for (𝑖 = 1,2, ⋯ , 𝑛 − 1) must be invertible) and 𝑐0= 0 and 𝑐𝑛= 1 (because the interval
Bezier curve passes through the regions defined by first and last interval control points) .
The four fixed Kharitonov's polynomials (four fixed Bezier curves) 𝑃𝑛𝑗(𝑢) for (𝑗 = 1,2,3,4)[20] associated with the interval Bezier curve 𝑃𝑛𝐼(𝑢) are:
𝑃𝑛1(𝑢) = 𝑝0−𝐵0𝑛(𝑢) + 𝑝1−𝐵1𝑛(𝑢) + 𝑝2+𝐵2𝑛(𝑢) + 𝑝3+𝐵3𝑛(𝑢) + 𝑝4−𝐵4𝑛(𝑢) + 𝑝5−𝐵5𝑛(𝑢) + ⋯
≡ 𝛼0,𝑛1 𝐵0𝑛(𝑢) + 𝛼1,𝑛1 𝐵1𝑛(𝑢) + 𝛼2,𝑛1 𝐵2𝑛(𝑢) + ⋯ + 𝛼𝑛,𝑛1 𝐵𝑛𝑛(𝑢)
𝑃𝑛2(𝑢) = 𝑝0−𝐵0𝑛(𝑢) + 𝑝1+𝐵1𝑛(𝑢) + 𝑝2+𝐵2𝑛(𝑢) + 𝑝3−𝐵3𝑛(𝑢) + 𝑝4−𝐵4𝑛(𝑢) + 𝑝5+𝐵5𝑛(𝑢) + ⋯
≡ 𝛼0,𝑛2 𝐵 0 𝑛(𝑢) + 𝛼
1,𝑛2 𝐵1𝑛(𝑢) + 𝛼2,𝑛2 𝐵2𝑛(𝑢) + ⋯ + 𝛼𝑛,𝑛2 𝐵𝑛𝑛(𝑢)
𝑃𝑛3(𝑢) = 𝑝0+𝐵0𝑛(𝑢) + 𝑝1+𝐵1𝑛(𝑢) + 𝑝2−𝐵2𝑛(𝑢) + 𝑝3−𝐵3𝑛(𝑢) + 𝑝4+𝐵4𝑛(𝑢) + 𝑝5+𝐵5𝑛(𝑢) + ⋯
≡ 𝛼0,𝑛3 𝐵0𝑛(𝑢) + 𝛼1,𝑛3 𝐵1𝑛(𝑢) + 𝛼2,𝑛3 𝐵2𝑛(𝑢) + ⋯ + 𝛼𝑛,𝑛3 𝐵𝑛𝑛(𝑢)
𝑃𝑛4(𝑢) = 𝑝0+𝐵0𝑛(𝑢) + 𝑝1−𝐵1𝑛(𝑢) + 𝑝2−𝐵2𝑛(𝑢) + 𝑝3+𝐵3𝑛(𝑢) + 𝑝4+𝐵4𝑛(𝑢) + 𝑝5−𝐵5𝑛(𝑢) + ⋯
≡ 𝛼0,𝑛4 𝐵0𝑛(𝑢) + 𝛼1,𝑛4 𝐵1𝑛(𝑢) + 𝛼2,𝑛4 𝐵2𝑛(𝑢) + ⋯ + 𝛼𝑛,𝑛4 𝐵𝑛𝑛(𝑢) (8)
These four fixed Kharitonov's polynomials (four fixed curves) can be written as follows:
𝑃𝑛𝑗(𝑢) = ∑ 𝛼𝑖,𝑛𝑗
𝑛
𝑖=0
𝐵𝑖𝑛(𝑢)
(𝑗 = 1,2,3,4)
(9) Now, the problem can be converted into determining the parametric four fixed Kharitonov's polynomials (four fixed Bezier curves) that interpolate a set of fixed points using the matrix definition of a fixed Bezier curve.
The above four fixed Kharitonov's polynomials (four fixed Bezier curves) can be rewritten in matrix form as follows:
𝑃𝑛𝑗(𝑢) = 𝑈𝑇. 𝑀𝐵. 𝛼𝑗
(𝑗 = 1,2,3,4)
𝑈𝑇 = [1 𝑢 𝑢2 ⋯ ⋯ 𝑢𝑛−1 𝑢𝑛]
𝛼𝑗= [𝛼
0,𝑛𝑗 𝛼1,𝑛𝑗 𝛼2,𝑛𝑗 ⋯ ⋯ 𝛼𝑛−1,𝑛𝑗 𝛼𝑛,𝑛𝑗 ]
𝑇
and
𝑀𝐵=
[
(𝑛0) (𝑛0)(−1)0 0 ⋯ ⋯ 0
(𝑛0) (𝑛1)(−1)1 (𝑛
1) (𝑛 − 10 ) (−1)0 ⋯ ⋯ 0
⋮ ⋮ ⋯ ⋯ ⋮
⋮ ⋮ ⋯ ⋯ ⋮
(𝑛0) (𝑛 − 1)𝑛 (−1)𝑛−1 (𝑛
1) (𝑛 − 1𝑛 − 2) (−1)𝑛−2 ⋯ ⋯ 0 (𝑛0) (𝑛𝑛)(−1)𝑛 (𝑛
1) (𝑛 − 1𝑛 − 1) (−1)𝑛−1 ⋯ ⋯ ( 𝑛 𝑛) (
𝑛 − 𝑛 𝑛 − 𝑛)(−1)0]
Cubic Bezier curves offer a reasonable compromise between flexibility and speed of computation. Compared to higher-order polynomials, cubic Bezier curves require less calculations and memory and they are more stable. Compared to lower-order polynomials, cubic Bezier curves are more flexible for modeling arbitrary curve shapes. Therefore, we will concentrate on cubic Bezier curves, but the techniques scale up to all degrees easily. For 𝑛 = 3 equation (10) can be written as follows:
𝑃3𝑗(𝑢) = 𝑈𝑇. 𝑀
𝐵. 𝛼𝑗
(𝑗 = 1,2,3,4)
(11) where,
𝑈𝑇 = [1 𝑢 𝑢2 𝑢3]
𝛼𝑗= [𝛼
0,3𝑗 𝛼1,3𝑗 𝛼2,3𝑗 𝛼3,3𝑗 ] 𝑇
and
𝑀𝐵= [
1 0 0 0
−3 3 0 0
3 −6 3 0
−1 3 −3 1
]
We know that 𝑃3𝑗(0) = 𝛼0,𝑛𝑗 and 𝑃3𝑗(1) = 𝛼𝑛,𝑛𝑗 for
(𝑗 = 1,2,3,4) and if we have two additional points that we
want the four fixed Kharitonov's polynomials (four fixed Bezier curves) to interpolate, say 𝑃3𝑗(𝑐1) = 𝛽1𝑗 and 𝑃3𝑗(𝑐2) =
𝛽2𝑗, then we can write:
𝑃3𝑗(𝑐1) = [1 𝑐1 𝑐12 𝑐13]. 𝑀𝐵.
[ 𝛼0,3𝑗
𝛼1,3𝑗
𝛼2,3𝑗
𝛼3,3𝑗 ]
(12) and
𝑃3𝑗(𝑐2) = [1 𝑐2 𝑐22 𝑐23]. 𝑀𝐵.
[ 𝛼0,3𝑗
𝛼1,3𝑗 𝛼2,3𝑗 𝛼3,3𝑗 ]
(13)
Combining 𝑃3𝑗(0) = 𝛼0,𝑛𝑗 , 𝑃3𝑗(𝑐1) = 𝛽1𝑗, 𝑃3𝑗(𝑐2) = 𝛽2𝑗, and 𝑃3𝑗(1) = 𝛼𝑛,𝑛𝑗 , we get:
[ 𝑃3𝑗(0)
𝑃3𝑗(𝑐1)
𝑃3𝑗(𝑐2)
𝑃3𝑗(1) ]
=
[ 𝛼0,3𝑗
𝛽1𝑗
𝛽2𝑗
𝛼3,3𝑗 ]
= [
1 0 0 0
1 𝑐1 𝑐12 𝑐13
1 𝑐2 𝑐22 𝑐23
1 1 1 1
] ∙ 𝑀𝐵∙
[ 𝛼0,3𝑗
𝛼1,3𝑗 𝛼2,3𝑗
𝛼3,3𝑗 ]
(𝑗 = 1,2,3,4)
(14) Therefore, we can define:
[ 𝛼0,3𝑗
𝛼1,3𝑗
𝛼2,3𝑗
𝛼3,3𝑗 ]
= [𝑀𝐵]−1∙ [
1 0 0 0
1 𝑐1 𝑐12 𝑐13
1 𝑐2 𝑐22 𝑐23
1 1 1 1
]
−1
∙
[ 𝛼0,3𝑗
𝛽1𝑗
𝛽2𝑗
𝛼3,3𝑗 ]
(𝑗 = 1,2,3,4)
(15) To determine the fixed control points 𝛼𝑖,𝑛𝑗 for (𝑖 =
0,1, ⋯ , 𝑛) of the four fixed Kharitonov's polynomials (four
fixed Bezier curves) that cause the four fixed Kharitonov's polynomials (four fixed Bezier curves) to pass through 𝛼𝑖,𝑛𝑗 , the matrix with 𝑐𝑖’𝑠 for (𝑖 = 0,1, ⋯ , 𝑛) must be invertible,
and this matrix is always invertible if 0 < 𝑐1< 𝑐2< 1.
Finally, the interval control points {[𝑞𝑖−, 𝑞𝑖+]}𝑖=0𝑛 of the parametric interval Bezier curve 𝑄3𝐼(𝑢) that cause the parametric interval Bezier curve 𝑄3𝐼(𝑢) to pass through the regions determined by{[𝑞𝑖−, 𝑞𝑖+]}𝑖=0𝑛 can be obtained from the fixed control points of the four fixed Kharitonov's polynomials (four fixed Bezier curves) 𝛼𝑖,𝑛𝑗 for (𝑖 = 0,1 ,
⋯ , 𝑛) and (𝑗 = 1,2,3,4) as follows:
[𝑞𝑖−, 𝑞 𝑖
+] = [𝑚𝑖𝑛 (𝛼 𝑖,𝑛 𝑗
) , 𝑚𝑎𝑥 (𝛼𝑖,𝑛𝑗 ) ] (𝑖 = 0,1, ⋯ , 𝑛)
(𝑗 = 1,2,3,4)
Algorithm for Parametric Interval Bezier Curve Interpolation
1. Give the values of 𝑢’𝑠(𝑢 = 𝑐𝑖 ) that correspond to the desired interval control points {[𝑞𝑖−, 𝑞𝑖+]}𝑖=0𝑛 and define the desired parametric interval Bezier curve 𝑄𝑛𝐼(𝑢), such that 0 < 𝑐1< 𝑐2< ⋯ ⋯ <
𝑐𝑛−1< 1 (to ensure that the matrix with 𝑐𝑖’𝑠 for
(𝑖 = 1,2, ⋯ , 𝑛 − 1) is invertible) and 𝑐0= 0 and
𝑐𝑛= 1 (because the interval Bezier curve passes
through the regions defined by first and last interval control points).
2. Find the four fixed Kharitonov's polynomials (four fixed Bezier curves) 𝑃𝑛𝑗(𝑢) for (𝑗 = 1,2,3,4)
associated with the given parametric interval Bezier curve 𝑃𝑛𝐼(𝑢).
3. Calculate the fixed control points 𝛼𝑖,𝑛𝑗 for (𝑖 =
0,1 , ⋯ , 𝑛) and (𝑗 = 1,2,3,4) of the four fixed
Kharitonov's polynomials (four fixed Bezier curves) that cause the four fixed Kharitonov's polynomials (four fixed Bezier curves) to pass through 𝛼𝑖,𝑛𝑗 for (𝑖 = 0,1 , ⋯ , 𝑛) and (𝑗 = 1,2,3,4). 4. The desired interval control points {[𝑞𝑖−, 𝑞𝑖+]}𝑖=0𝑛
of the desired parametric interval Bezier curve 𝑄𝑛𝐼(𝑢) that cause the parametric interval Bezier
curve to pass through the regions determined by {[𝑞𝑖−, 𝑞
𝑖+]}𝑖=0𝑛 can be obtained from the fixed
control points of the four fixed Kharitonov's polynomials (four fixed Bezier curves) 𝛼𝑖,𝑛𝑗 for
(𝑖 = 0,1 , ⋯ , 𝑛) and (𝑗 = 1,2,3,4) as follows:
[𝑞𝑖−, 𝑞
𝑖+] = [𝑚𝑖𝑛(𝛼𝑖,𝑛𝑗 ), 𝑚𝑎𝑥(𝛼𝑖,𝑛𝑗 ) ]
(𝑖 = 0,1, ⋯ , 𝑛) (𝑗 = 1,2,3,4)
IV.NUMERICALEXAMPLE
Example: Consider the third order parametric interval Bezier curve 𝑃3𝐼(𝑢) defined by four interval control points
{[𝑝𝑖−, 𝑝𝑖+]}𝑖=03 given as follows:
[𝑝0−, 𝑝
0+] = ([16.0000,17.0000], [14.0000,16.0000])
[𝑝1−, 𝑝1+] = ([24.0000,28.0000], [33.0000,36.0000])
[𝑝2−, 𝑝2+] = ([40.0000,42.0000], [34.0000,39.0000])
[𝑝3−, 𝑝3+] = ([63.0000,68.0000], [18.0000,23.0000])
The problem is to find the desired third order parametric interval Bezier curve 𝑄3𝐼(𝑢), calculated using {[𝑝𝑖−, 𝑝𝑖+]}𝑖=03 and will pass through the regions defined by the interval control points {[𝑞𝑖−, 𝑞𝑖+]}𝑖=03 , and correspond to 𝑢 = 0, 𝑢 =
𝑐1= 0.5, 𝑢 = 𝑐2= 0.75, and 𝑢 = 1, such that 0 < 𝑐1<
𝑐2< 1 is satisfied.
As explained in section II the four fixed Kharitonov's polynomials (four fixed Bezier curves) 𝑃3𝑗(𝑢) for (𝑗 =
1,2,3,4) associated with the given parametric interval Bezier
curve 𝑃3𝐼(𝑢) are obtained, and
{
𝑃31(𝑐1) = (35.2500,31.6250) = 𝛽11, 𝑃31(𝑐2) = (50.0330,31.0160) = 𝛽21 𝑃32(𝑐1) = (36.1250,32.1250) = 𝛽12, 𝑃32(𝑐2) = (48.4859,29.3283) = 𝛽22 𝑃33(𝑐1) = (35.5000,30.5000) = 𝛽13, 𝑃33(𝑐2) = (47.6577,27.2500) = 𝛽23 𝑃34(𝑐1) = (34.6250,30.0000) = 𝛽14, 𝑃34(𝑐2) = (49.2048,28.9377) = 𝛽24}
The fixed control points 𝛼𝑖,𝑛𝑗 for (𝑗 = 1,2,3,4) and
(𝑖 = 0,1,2,3)of the four fixed Kharitonov's polynomials
(four fixed Bezier curves) that cause the four fixed Kharitonov's polynomials (four fixed Bezier curves) to pass through the fixed control points 𝛼𝑖,𝑛𝑗 for (𝑗 = 1,2,3,4) and
(𝑖 = 0,1,2,3) are found as shown in table 1.
Table 1.: The fixed control points of 𝑄3𝑖(𝑢) for (𝑖 = 1,2,3,4).
𝑖 𝛼𝑖,31(𝑥𝑖, 𝑦𝑖) 𝛼𝑖,32(𝑥𝑖, 𝑦𝑖) 𝛼𝑖,33(𝑥𝑖, 𝑦𝑖) 𝛼𝑖,34(𝑥𝑖, 𝑦𝑖)
0 (16.0000,14.0000) (16.0000,14.0000) (19.0000,18.0000) (19.0000,18.0000)
1 (23.9938,32.9987) (27.9946,35.9994) (27.1060,35.1111) (23.1052,32.1104)
2 (42.0062,39.0013) (42.0054,39.0006) (40.2274,34.2222) (40.2282,34.2229)
3 (68.0000,23.0000) (63.0000,18.0000) (63.0000,18.0000) (68.0000,23.0000)
Finally, the interval control points {[𝑞𝑖−, 𝑞𝑖+]}𝑖=03 of the parametric interval Bezier curve 𝑄3𝐼(𝑢) that will pass through the regions determined by four interval control points and correspond to 𝑢 = 0, 𝑢 = 𝑐1= 0.5, 𝑢 = 𝑐2= 0.75 and 𝑢 = 1, are calculated using equation (16) as follows:
[𝑞0−, 𝑞0+] = ([16.0000,19.0000], [14.0000,18.0000])
[𝑞1−, 𝑞1+] = ([23.1052,27.9947], [32.1104,35.9994])
[𝑞2−, 𝑞2+] = ([40.2274,42.0062], [34.2222,39.0013])
[𝑞3−, 𝑞3+] = ([63.0000,68.0000], [18.0000,23.0000])
Simulation results in Fig.1 shows the envelopes of the original parametric interval Bezier curve and the desired parametric interval Bezier curve, respectively.
V.CONCLUSIONS
points is to force the curve to pass through the fixed points, or interpolate the fixed points. An algorithmic approach for parametric interval Bezier curve interpolation based on simple matrix form is presented in this paper. The four fixed Kharitonov's polynomials (four fixed Bezier curves) 𝑃𝑛𝑗(𝑢)
for (𝑗 = 1,2,3,4) associated with the original interval Bezier
curve 𝑃𝑛𝐼(𝑢) are obtained. The fixed control points 𝛼𝑖,𝑛𝑗 for
(𝑖 = 0,1 , ⋯ , 𝑛) and (𝑗 = 1,2,3,4) of the four fixed
Kharitonov's polynomials (four fixed Bezier curves) that cause the four fixed Kharitonov's polynomials (four fixed Bezier curves) to pass through 𝛼𝑖,𝑛𝑗 for (𝑖 = 0,1 , ⋯ , 𝑛) and
(𝑗 = 1,2,3,4) are computed. The desired interval control
points {[𝑞𝑖−, 𝑞𝑖+]}𝑖=0𝑛 of the desired parametric interval Bezier curve 𝑄𝑛𝐼(𝑢) that cause the parametric interval Bezier curve to pass through the regions determined by {[𝑞𝑖−, 𝑞𝑖+]}𝑖=0𝑛 are obtained from the fixed control points of the four fixed Kharitonov's polynomials (four fixed Bezier curves) 𝛼𝑖,𝑛𝑗 for
(𝑖 = 0,1 , ⋯ , 𝑛) and (𝑗 = 1,2,3,4) . Parametric interval
Bezier curve interpolation has important applications in curve and surface design. One of the main and useful applications of these concepts is the treatment of curves and surfaces in terms of control points, a tool extensively used in CAGD.
REFERENCES
[1] W. Zhang, Y. F. Zhang, and Q. J. Ge, “Interference-free tool path generation for 5-axis sculptured surface machining using rational Bezier motions of a flat-end cutter”, International Journal of Production Research, vol. 43, pp. 4103-4124, 2005.
[2] S. Biswas and B. C. Lovell, Bezier and Splines in Image Processing and Machine Vision, Springer, London, 2007.
[3] P. Vani and J. Ramaswamy, “An innovative scheme for effectual fingerprint data compression using Bezier curve representations, International Journal of Computer Science and Information Security, vol. 6, pp. 149-157, 2009.
[4] D. J. Walton and D. S. Meek, “𝐺2 curve design with planar quadratic rational Bezier spiral segments”, International Journal of Computer Mathematics, vol. 90, pp. 325-340, 2013.
[5] M. Aounallah, M. Belkadi, L. Adjlout, and O. Imine, “Numerical shape optimization of a confined cavity in natural convection regime”, Computers & Fluids, vol. 75, pp. 11-21, 2013.
[6] Y. B. Li, T. S. Fu, and Y. B. Gu, “Visual Modeling for Branch and Trunk of Trees Based on Multi-Bezier Curve”, Journal of Computational Information Systems, vol. 8, pp. 10251-10258, 2012. [7] B. Cross and R. J. Cripps, “𝐺3 quintic polynomial approximation for
Generalised Cornu Spiral segments”, Journal of Computational and Applied Mathematics, vol. 236, pp. 3111-3122, 2012.
[8] Q. Q. Hu, “Approximating conic sections by constrained Bezier curves of arbitrary degree”, Journal of Computational and Applied Mathematics, vol. 236, pp. 2813-2821, 2012.
[9] D. Vucina, Z. Lozina, and I. Pehnec, “Computational procedure for optimum shape design based on chained Bezier surfaces parameterization”, Engineering Applications of Artificial Intelligence, vol. 25, pp. 648-667, 2012.
[10] C. H. Isabelle, A. Gudrun, and H. M. Victoria, “Computational procedure for optimum shape design based on chained Bezier surfaces parameterization”, Computer Aided Geometric Design, vol. 26, pp. 725-732, 2009.
[11] Q. Q. Hu and G. J. Wang, “Geometric meanings of the parameters on rational conic segments”, Science in China Series A: Mathematics, vol. 48, pp. 1209-1222, 2005.
[12] O. Ismail, "Degree Elevation of Interval Bezier Curves Using Legendre-Bernstein Basis Transformations", International Journal of
Video & Image Processing and Network Security (IJVIPNS), Vol. 10, No. 6, pp. 6-9, 2010.
[13] O. Ismail, "Degree Elevation of Interval Bezier Curves", International Journal of Video & Image Processing and Network Security (IJVIPNS), Vol. 13, No. 2, pp. 8-11, 2013.
[14] O. Ismail, "Reparametrization and Subdivision of Interval Bezier Curves", International Journal of Video & Image Processing and Network Security (IJVIPNS), Vol. 14, No. 4, pp. 1-6, 2014.
[15] O. Ismail, "A smooth Connection of Interval Bezier Curve Segments", International Journal of Video & Image Processing and Network Security (IJVIPNS), Vol. 15, No. 4, pp. 1-5, 2015.
[16] O. Ismail, "Regularity of Interval Bezier Curves", International Journal of Video & Image Processing and Network Security (IJVIPNS), Vol. 15, No. 6, pp. 1-4, 2015.
[17] O. Ismail, "Degree Reduction of Parametric Interval Bezier Curves", ‘Proc., International Conference on Sciences of Electronic, Technologies of Information and Telecommunications (SETIT 2016)’, Hammamet, Tunisia, 2016.
[18] P. E. Bezier and S. Sioussiou, “Semi-automatic system for defining free-form curves and surfaces”, Computer-aided Design, vol. 15, no. 2, pp. 65-72, 1983.
[19] R. E. Moore, Methods and applications of interval analysis, Society for Industrial and Applied Mathematics, 1979.
[20] V. L. Kharitonov, "Asymptotic stability of an equilibrium position of a family of system of linear differential equations", Differential 'nye Urauneniya, vol. 14, pp. 2086-2088, 1978.
O. Ismail (M’97–SM’04) received the B. E. degree in electrical and
electronic engineering from the University of Aleppo, Syria in 1986. From 1987 to 1991, he was with the Faculty of Electrical and Electronic Engineering of that university. He has an M. Tech. (Master of Technology) and a Ph.D. both in modeling and simulation from the Indian Institute of Technology, Bombay (IITB), in 1993 and 1997, respectively. Dr. Ismail is a Senior Member of IEEE. Life Time Membership of International Journals of Engineering & Sciences (IJENS) andResearchers Promotion Group (RPG). He is an Academic Member of the Athens Institute for Education and Research, belonging to the Computer Research Unit and the Electrical Engineering Research Unit. His main fields of research include computer graphics, computer aided analysis and design (CAAD), computer simulation and modeling, digital image processing, pattern recognition, robust control, modeling and identification of systems with structured and unstructured uncertainties. He has published more than 75 refereed international journals and conferences papers on these subjects. In 1997, he joined the Department of Computer Engineering at the Faculty of Electrical and Electronic Engineering in University of Aleppo, Syria.
In 2004, he joined Department of Computer Science, Faculty of Computer Science and Engineering, Taibah University, K.S.A. as an associate professor for six years.
He has been chosen for inclusion in the special 25th Silver Anniversary Editions of Who’s Who in the World. Published in 2007 and 2010.
Presently, he is working as a professor in Department of Computer Engineering at the Faculty of Electrical and Electronic Engineering, University of Aleppo.
