# The Representation of Circular Arc by Using Rational Cubic Timmer Curve

Authors: Muhammad Abbas; Norhidayah Ramli; Ahmad Abd. Majid; Jamaludin Md. Ali
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

In CAD/CAM systems, rational polynomials, in particular the Bézier or NURBS forms, are useful to approximate the circular arcs. In this paper, a new representation method by means of rational cubic Timmer (RCT) curves is proposed to effectively represent a circular arc. The turning angle of a rational cubic Bézier and rational cubic Ball circular arcs without negative weight is still not more than4π/3 and π, respectively. The turning angle of proposed approach is more than Bézier and Ball circular arcs with easier calculation and determination of control points. The proposed method also provides the easier modification in the shape of circular arc showing in several numerical examples.


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## Body

## 1. Introduction

The study of curves plays a significant role in Computer Aided Geometric Design (CAGD) and Computer Graphics (CG) in particular parametric forms because it is easy to model curves interactively [1]. CAGD copes with the representation of free form curves. In parametric form, it is important which basis functions are used to represent the circular arcs. Circular arcs are extensively used in the fields of CAGD and CAD/CAM systems, since circular arcs can be  represented by parametric (rational) polynomials instead of polynomials in explicit form. Faux and Pratt [2] represented only an elliptic segment whose turning angle is less than π by a rational quadratic Bézier curve.Generally, a rational cubic Bézier is used to extend the turning angle for conics. The maximum turning angle of a rational cubic circular arc is not more than4π/3 [3]. Only the negative weight conditions can extend its expressing range to 2π (not equal to 2π) and such conditions are not cooperative in CAD systems because they lose the convex hull property [4]. A rational quartic Bézier curve can express any circular arc whose central angle is less than 2π, and it requires at least rational Bézier curve of degree five to represent a full circle without resorting to negative weights [5]. Fang [6] presented a special representation for conic sections by a rational quartic Bézier curve which has the same weight for all control points but the middle one. G.-J. Wang and G.-Z. Wang [7] presented the necessary and sufficient conditions for the rational cubic Bézier representation of conics by applying coordinate transformation and parameter transformation. Hu and Wang [8] derived the necessary and sufficient conditions on control points and weights for the rational quartic Bézier representation of conics by using two specials kinds, degree reducible and improperly parameterization. Usually rational cubic and rational quartic Bézier curves are used for representation of conic sections or circular arcs representation which can also be found in several papers [9–12]. Hu and Wang [13] constructed necessary and sufficient conditions for conic representation in rational low degree Bézier form and the transformation formula from Bernstein basis to Said-Ball basis.In this work, we present a rational cubic representation of a circular arc using Timmer basis functions. It is able to approximate a circular arc up to2π (but not including 2π) without resorting to negative weights in contrast of rational cubic Bézier and rational cubic Ball circular arcs representation methods. Necessary and sufficient conditions are derived for a representation of RCT circular arc.The remainder of this paper is as follows. The cubic alternative basis functions with geometric properties are given in Section2. The RCT curve with some geometric curve properties is presented in Section 3 and the weight formulation for Timmer circular arc is also part of this section. The RCT curve as conic segment is represented in Section 4. The Timmer circular arc representation method is discussed in Section 5 and some numerical examples of the proposed method are presented in Section 6. Section 7 is devoted to conclusions.


## 2. Bézier-Like Cubic Basis Functions

In this section, the Bézier-like cubic basis functionsBi(u),i=0,1,2,3 are defined in [14, 15] as
(1)B0(u)=(1-u)2(1+(2-m)u),B1(u)=m(1-u)2u,B2(u)=m(1-u)u2,B3(u)=u2(1+(2-m)(1-u))u∈[0,1],
where m is a shape parameter. For m=2, the cubic basis function are the Ball basis functions [14], m=3, the basis functions are Bernstein Bézier basis functions [1], and if m=4, then (1) represents a Timmer cubic basis functions [15, 16]; see Figure 1.(a) Ball cubic basis functions form=2, (b) Bernstein cubic Bézier basis functions for m=3, and (c) Timmer cubic basis function for m=4.
(a)
(b)
(c)Theorem 1 (see [15]).
The Bézier-like cubic basis functionsBi(u),i=0,1,2,3 satisfy the following properties.
(a) Positivity. For0≤m≤3, the basis function (1) are nonnegative on the interval u∈[0,1], but this property is no more satisfied when m=4.
(b) Partition of Unity. The sum of the Bézier-like cubic basis functions is one on the intervalu∈[0,1]. That is,
(2)∑i=03Bi(u)=1.
(c) Monotonicity. For the given value of the parameterm, B0(u) is monotonically decreasing and B3(u) is monotonically increasing.
(d) Symmetry(3)Bi(u;m)=B3-i(1-u;m),i=0,1,2,3.


## 3. A Rational Cubic Bézier-Like Curve

The rational cubic Bézier-like curve with single shape parameterm can be defined as follows.Definition 2.
Given the control pointsPi(i=0,1,2,3) in ℝ2, the rational cubic Bézier-like curve is
(4)r(u)=∑i=03wiPiBi(u)∑i=03wiBi(u),u∈[0,1].In standard representation the weightsw0=w3=1 and the middle weights are positive and in this paper the middle weights are considered to be equal to w1=w2=w. Figure 2 illustrates cubic curves with m=2,3, and 4 with w=1 to represent the ordinary cubic curves.Figure 2
Cubic Said-Ball curve form=2 (red), cubic Bézier curve for m=3 (green), and Timmer PC curve for m=4 (magenta) with arbitrary control points Pi.There are some geometric properties of rational cubic Bézier-like curve as follows.Theorem 3.
The rational cubic Bézier-like curve satisfies the following properties.
(a)End Point Properties. Consider
(5)r(0)=P0,r(1)=P3,r′(0)=m(P1-P0)w,r′(1)=m(P3-P2)w,r′′(0)=2(m2(3-m)P3+mwP2+mw(-2+m-mw)P1fgfgggdd+(-3+m2(-1+w)w+m(1+w))P0),r′′(1)=2(m2(3-m)P0+mwP1+mw(-2+m-mw)P2dfdffffggg+(-3+m2(-1+w)w+m(1+w))P3).(b) Symmetry. The control points Pi and P3-i define the same rational cubic Bézier-like curve in different parameterization; that is, r(u;m,Pi)=r(1-u;m,P3-i)
(c)Geometric Invariance. The partition of unity property of Bézier-like cubic basis functions assures the invariance of the shape of the rational cubic Bézier-like curve under translation and rotation of its control points.
(d)Convex Hull Property. In general, rational cubic curve does not satisfy the convex hull property for m∈ℝ. But if m∈[0,3], it is always confined to the convex hull of its defining control points. For m=4 and w>0, edge P1P2 of the control polygon is tangent to the curve at the midpoint of P1P2 and the curve (4) always lies inside the control polygon see Figure 3.Figure 3
Rational cubic Timmer curve form=4 with different local values of weight w=100 (blue dashed), w=5.5 (blue), w=3.5 (red dashed), w=2 (red), w=1.5 (brown dashed), w=1(brown), w=0.5 (black), and w=0.22 (black dashed).Proposition 4.
The rational cubic Bézier-like curve (4) represents a conic segment if and only if the weight of curve (4) is
(6)w=(m-2m)(P3-P0)·(P2-P1)(P2-P1)·(P2-P1).Proof.
Lets(u)=∑i=03BiPiwi be the rational cubic polynomial in (4). Since the middle weights are equal, the denominator is quadratic, then for s(u) to be a conic, by taking the condition that the numerator of (4) to be quadratic we get
(7)(2-m)(P0-P3)+m(P1-P2)w=0,(P2-P1)w=m-2m(P3-P0),w=(m-2m)(P3-P0)(P2-P1).
Showing thatP2-P1∥P3-P0,
(8)w=(m-2m)(P3-P0)·(P2-P1)(P2-P1)·(P2-P1).
Form=4,
(9)w=12(P3-P0)·(P2-P1)(P2-P1)·(P2-P1).


## 4. A Rational Cubic Timmer (RCT) Curve as a Conic Segment

In this section, we illustrate simple determination of middle weights for a curve to interpolate the intermediate pointS. Sarfraz [17] has shown the construction using rational Bézier cubic. It is easy to determine the weights using RCT instead of Bézier because S is the intermediate point of RCT curve (4) at point t=1/2 such that
(10)r(0.5)=w1P1+w2P2w1+w2.SinceS is known intermediate point of Timmer; thus from (9), we obtain
(11)S=w1P1+w2P2w1+w2.From (11), we get
(12)w1w2=k,
where
(13)k=(P2-S)·(P2-S)(S-P1)·(P2-S).As long as the ratio does not change the curve passes throughS. Figure 4 illustrates the different RCT curves.Figure 4
Different rational cubic Timmer curves with different values ofw2.From (12), if S is the midpoint, then the ratio is one. If w is equal to (9), then the RCT is a conic. Figure 5 shows that the family of RCT with S is the midpoint of P1P2.Figure 5
Family of rational cubic Timmer with midpointS and red curve is a conic.Sometimes only the end points, end tangents, and the intermediate point are given. The middle control points are obtained by taking the intersection points when the line throughS is parallel to the line defined by two end points intersecting with end tangents.


## 5. A Representation of Circular Arc

In this section, the representation of circular arc using RCT curve (4) is discussed. Let r(u)=(x,y) be a RCT curve centered at C with θ as a turning angle θ∈(0,2π) with radius r=csc⁡α, where α=π-θ/2  is given. Let T0 and T1 be the end point tangents to this circular arc. Since P2-P1∥P3-P0, referring to Figure 6, the control points of r(u) are
(14)P0(-1,0),P1(-rcotα2,tan⁡(π-α2)),P2(rcotα2,tan⁡(π-α2)),P3(1,0),
with equal internal weights
(15)w=(m-2m)(P3-P0)·(P2-P1)(P2-P1)·(P2-P1).Figure 6
Rational cubic Timmer circular arc form=4.The parametric coordinates of the curve (4) are
(16)x(u)=-1+2u1+2(-1+u)u+(-2+m)(-1+u)uCos⁡⁡α,y(u)=-(-2+m)(-1+u)uSin⁡⁡α1+2(-1+u)u+(-2+m)(-1+u)uCos⁡⁡α.Theorem 5.
A RCT curver(u)=(x,y) with control points Pi,i=0,1,2,3, and weight w represents the circular arc with central angle θ∈(0,2π) if and only if x2(u)+(y(u)-cot⁡α)2=csc2⁡α.Proof.
Letr(u) be a circular arc with control points Pi,i=0,1,2,3. The coordinates of any point (x(u),y(u)) on the curve (4) with equal internals weights w are given by
(17)x(u)=(1-2u)2(1-u)u(1+Cos⁡⁡α)-1,y(u)=-2u(1-u)Sin⁡⁡α2u(1-u)(1+Cos⁡⁡α)-1.
Then(18)x2(u)+(y(u)-cot⁡α)2=(1-2u2u(1-u)(1+cos⁡⁡α)-1)2+(-2u(1-u)sin⁡α2u(1-u)(1+cos⁡⁡α)-12u(1-u)sinα2u(1-u)(1+cos⁡α)-1ghgh-cot⁡α)2=(×sin2(2u(1-u)(1+cos⁡α)-1))2(1-2u)2sin2⁡αh+(2u(1-u)sin2⁡α+cos⁡⁡αhgh×sin2(2u(1-u)(1+cos⁡⁡α)-1))2)h×(sin2(2u(1-u)hfgffff×(1+cos⁡⁡α)-1)2sin2⁡α)-1=(2u(1-u)(1+cos⁡α)-1)2h×(×(1+cos⁡α)-1)2(2u(1-u)hgffgg×(1+cos⁡α)-1)2sin2⁡α)-1=csc2⁡α=r2.Proposition 6.
Givenm=4 and θ∈(0,2π) as the central angle of the circular arc r(u) with control points Pi, then the weight is w=sin2⁡(α/2).
This is the outcome of (9).Proposition 7.
Forw=0 the RCT curve (4) becomes a straight line which is parallel to both x-axis and P0P3.


## 6. Numerical Examples

We apply our representation method with examples that are shown in Figures7 and 8 using different values of turning angle θ and values of weight w.Figure 7
The representation of circular arcs with different turning angles and weightsw=0.5 (blue), w=0.413 (blue dashed), w=0.25 (red), w=0.178 (red dashed), w=0.116 (green dashed), w=0.078 (green), and w=0.066 (black).Figure 8
RCT curve as a circular arc with weightw=0.00017 and turning angle θ=119π/60.


## 7. Conclusion

In this paper, the representation of circular arcs using RCT curves was presented. The largest turning angle of a rational cubic Bézier and rational cubic Ball circular arcs is still not more than4π/3 and π, respectively. Therefore, we have increased the range for representation of circular arc up to 2π (but not including 2π) without resorting to negative weights using RCT curve. Conic segments have also represented with RCT curves, thus, studying the necessary and sufficient conditions for RCT representation of circular arc is worthy.



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