The Cubic Bezier-Ball Like curve with Shape Parameters

In this paper the basic spotlight is construct new cubic basis functions to generate rational cubic Bezier-Ball Like curve with two shape parameters. The current study the Bezier-Ball Like curve analogous to cubic Bezier curve, Ball curve


Introduction
The curve design is the most important topic of CAGD (Computer Aided Geometric Design) and Computer Graphics.The parametric representation of curves is the most convenient for design specially in polynomial form.The Bezier curve is a parametric curve, a polynomial functions present it in certain parameter θ .In Computer aided Geometric Design (CAGD), a Bezier form is used to present the curve.The number of control points Determine the degree of polynomial.The curve passes through its endpoints and it does not pass through the inner points.Each inner point attracts the curve towards itself and it causes change the direction of the curve.The polygon obtained when the control points are linked with straight lines are called control polygon [1].A rational Bezier curve is special case of nonuniform rational B-spline (NURBS).The rational control points are linked with straight lines are called control polygon .One of the important application of CAGD is construction free form of curve [2].The curve is the important part for the engineers.They use some designing technique of CAGD to determine the shapes of designer.A rational Bezier curve have geometrically meaningful presentations so they are used it in CAD and CAGD [3].The effect with changing the weight and moving the control points on the rational Bezier curve cause a global change in the shape of the curve, for that reason some authors developed methods by combination the shape parameters into the original basis functions, see [4] to [10].These news curve have the same properties of Bezier curve.
The Arabic character is represent in this paper using Bezier-Ball Like curve.
The Arabic script is written from right to left and it is generated using many segments which depend on the shape of letters.The two important things to generate Arabic script are the control points and number of segments.Many authors have been worked in design Arabic fonts.Sarfaz [11] introduced algorithm for automatic capture of Arabic fonts.In this work the rational Bezier-Ball curve is proposed.The beauty of Bezier-Ball Like curve can represent the Bezier curve, Ball curve, and Timmer curve at specific value of shape parameters.The shape of the curve can be modify by change the value of shape parameters and weight.

2
New Basis Functions The cubic basis functions with two shape parameters that satisfies all properties of blending functions are define as a following: Where θϵ [0, 1] and γ, µϵ [−2, 1] so that the function will be positive.
By putting γ = µ = 0 will get the basis function of Ball curve as in equations: By choosing the value of γ = µ = 2 will reduce the basis functions to Timmer basis functions but the value of γ, µ not on the interval [−2, 1] as: For −2 ≤ γ, µ ≤ 1, the basis function (2.1) are non-negative on the interval θϵ [0, 1] but this property does not satisfy for value outside the domain of γ, µ see Figure 1.

•Partition of unity
The sum of Bezier-Ball like Basis Functions is one on the interval θϵ [0, 1].

Properties of Rational Cubic Bezier-Ball like Curve
The properties of rational cubic Bezier-Ball Like curve satisfies the following properties:

• Convex Hull Properties
For value of −2 ≤ γ, µ ≤ 1 the curve confined to the convex hull of its control points and it does not satisfy for values of γ, µ outside the domain.

• Symmetry
That mean the control points Pi , P3−i defined the same rational Bezier-Ball Like cubic curve with different parameterizations.

• Geometric Invariance
The invariance of the shape of rational Bezier-Ball Like cubic curve are protected by the partition of unity property of Bezier-Ball Like basis functions under translation and rotation of its control points.

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The Rational Cubic Bezier-Ball like Curve with Shape Control The weights w1, w2 and the two shape parameters γ, µ control the shape of rational cubic Bezier Ball Like curve.Let the control points are given, taking w1, w2 and µ constants, the curve will become closer to P0P1 such that γ increasing in the range [−2, 1] see Figure4(a).For constants w1, w2 and γ the curve will moves towards P2P3 where µ increasing in the range [−2, 1] as shown in Figure 4(b), and if γ, µ change simultaneously with fixed w1, w2 then the curve will increase toward the control polygon Figure 4(c).By the same manner for fixed values of γ, µ with change the values of w1 and w2.For taking w1 constant with increase w2 from small value to big value the curve will adjacent to right side (moves toward P2P3).And fixed w2 with increase w1 from small to big the curve will become closer to left side (moves towards P0P1) see Figure 4(d).

Approximation of Rational Cubic Bezier-Ball like Curve
In this section, we will illustrate to approximate the rational cubic Bezier-Ball Like curve by choosing diff erent values of γ and µ.For values of γ, µ equal to one will reduce cubic Bezier curve (red), also will produce cubic Ball curve if values of γ, µ equal to zero (green) and for values of γ, µ equal to two will get cubic Timmer curve (blue) see Figure 5.

Applications of Non-Rational Cubic Bezier-Ball like Curve
This section will discuss some applications for non-rational cubic Bezier-  The way for drawing font require the control points for all segment that determine the shape of the letter.After that it is required to apply suitable programming language with non rational cubic Bezier-Ball Like curve for all segment to produce appropriate form of the letter.To adjust the shape of letter must change the inner points for all segments .

Results Discussion
This paper is organized as a follows.In section 2 the new basis functions ‫األسمرية‬ ‫الجامعة‬ ‫مجلة‬ Journal of Alasmarya University 78 with two shape parameters were established.In section 3 the basis functions with specific value of shape parameters were shown.In section 4 discussed the geometric properties of Bezier-Ball Like curve.Section 5, included the rational cubic Bezier-Ball Like curve and geometric properties were discussed in section 6. Section 7 provided a shape control of rational cubic Bezier-Ball Like curve and had shown that the two shape parameters were suitable instrument to control the shape of the curve and its application in font designing was given in section 9 .The result of this study indicate that the important things to generate arabic script were the control points and the number of segments.In addition it was found that by generating the arabic font using non-rational cubic Ball-Bezier curve was appeared similar to the original font.In this study, we investigated the arabic font design apparently very close to the shape of the original font and similar to result that discovered by another authers .

Conclusion
As mentioned above, one of the more significant finding to emerge from this work is that the present study confirms previous finding and contributes additional evidence that suggests a new cubic basis functions which inherits all properties of Bezier curve.Since there is no difference in construct between Bezier-Ball Like and Bezier curve, it is not difficult to adopt proposed curve to a CAD that already use Bezier curve.The designed curve is used to generate Arabic alphabet that given more evidence of the claim.This work can be extended to prove that the rational cubic Ball-Bezier curve is more flexible than the rational quadratic Bezier-Like curve by generating arabic script for two ways to compare which one is better in terms of smoothness and by increasing number of segments.

Figure
‫األسمرية‬ ‫الجامعة‬ ‫مجلة‬Journal of Alasmarya University 70 (1) shows the basis functions with different value of γ, µ in the range [−2, 1].For γ = 1, µ = −1 (dotted lines), and for γ = −1, µ = 1 (solid lines) 3 Basis Function with Specific Value of Shape Parameters At specific value of γ and µ will produced three cases of basis functions of curve.• Case 1 By setting γ = µ = 1 will reduce the basis functions of Bezier curve as follows:

2019 ) Volume 4 ,
Issue 1 , June 2019 75 Ball Like curve such as watermelon and font design using Matlab software.

Figure 4 : 9 . 1 First
Figure 4: The Shape Control of Rational Bezier-Ball Like curve (a) Change the value of γ, (b) Change the value of μ, (c) Change the value of γ, μ, (d) Change the value of w1 and w2.
In this section, we will show some examples of Arabic letters using Matlab software to implement non-rational cubic Bezier-Ball Like curve.Design (Baa) letter via 21 segment of non-rational cubic Bezier-Ball Like curve by using Matlab see Figures 6.

Figure 6 :
Figure 6: Baa letter using non-rational cubic Bezier-Ball Like curve with control polygon

Figure 7 :
Figure 7: Ain letter using non-rational cubic Bezier-Ball Like curve with control polygon