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 than and , 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.
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 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 than [3]. Only the negative weight conditions can extend its expressing range to (not equal to ) 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 , 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 to (but not including ) 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 Section 2. 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 functions are defined in [14, 15] as where is a shape parameter. For , the cubic basis function are the Ball basis functions [14], , the basis functions are Bernstein Bézier basis functions [1], and if , then (1) represents a Timmer cubic basis functions [15, 16]; see Figure 1.
(a)
(b)
(c)
Theorem 1 (see [15]). The Bézier-like cubic basis functions satisfy the following properties.
(a) Positivity. For , the basis function (1) are nonnegative on the interval , but this property is no more satisfied when .
(b) Partition of Unity. The sum of the Bézier-like cubic basis functions is one on the interval . That is,
(c) Monotonicity. For the given value of the parameter , is monotonically decreasing and is monotonically increasing.
(d) Symmetry
3. A Rational Cubic Bézier-Like Curve
The rational cubic Bézier-like curve with single shape parameter can be defined as follows.
Definition 2. Given the control points in , the rational cubic Bézier-like curve is
In standard representation the weights and the middle weights are positive and in this paper the middle weights are considered to be equal to . Figure 2 illustrates cubic curves with , and 4 with to represent the ordinary cubic curves.
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
(b) Symmetry. The control points and define the same rational cubic Bézier-like curve in different parameterization; that is,
(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 . But if , it is always confined to the convex hull of its defining control points. For and , edge of the control polygon is tangent to the curve at the midpoint of and the curve (4) always lies inside the control polygon see Figure 3.
Proposition 4. The rational cubic Bézier-like curve (4) represents a conic segment if and only if the weight of curve (4) is
Proof. Let be the rational cubic polynomial in (4). Since the middle weights are equal, the denominator is quadratic, then for to be a conic, by taking the condition that the numerator of (4) to be quadratic we get
Showing that ,
For ,
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 point . Sarfraz [17] has shown the construction using rational Bézier cubic. It is easy to determine the weights using RCT instead of Bézier because is the intermediate point of RCT curve (4) at point such that
Since is known intermediate point of Timmer; thus from (9), we obtain
From (11), we get where
As long as the ratio does not change the curve passes through . Figure 4 illustrates the different RCT curves.
From (12), if is the midpoint, then the ratio is one. If is equal to (9), then the RCT is a conic. Figure 5 shows that the family of RCT with is the midpoint of .
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 through 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 be a RCT curve centered at with as a turning angle with radius , where is given. Let and be the end point tangents to this circular arc. Since , referring to Figure 6, the control points of are with equal internal weights
The parametric coordinates of the curve (4) are
Theorem 5. A RCT curve with control points , and weight represents the circular arc with central angle if and only if .
Proof. Let be a circular arc with control points . The coordinates of any point on the curve (4) with equal internals weights are given by
Then
Proposition 6. Given and as the central angle of the circular arc with control points , then the weight is .
This is the outcome of (9).
Proposition 7. For the RCT curve (4) becomes a straight line which is parallel to both -axis and .
6. Numerical Examples
We apply our representation method with examples that are shown in Figures 7 and 8 using different values of turning angle and values of weight .
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 than and , respectively. Therefore, we have increased the range for representation of circular arc up to (but not including ) 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are indebted to the anonymous reviewers for their helpful, valuable suggestions and comments for the improvement of the paper. The authors also would like to extend the gratitude to the School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia, for the financial support from FRGS Grant no. 203/PMATHS/6711324.