Abstract. A class of cubic Bézier-type curves based on the blending of algebraic and trigonometric polynomials, briefly CAT Bézier curves, is presented in this ...
The CAT Bézier Curves Jin Xie1, XiaoYan Liu2, and LiXiang Xu1 1
2
Department of Mathematics and Physics, Hefei University, 230601Hefei, China Department of Mathematics and Physics, University of La Verne, 91750 La Verne, USA
Abstract. A class of cubic Bézier-type curves based on the blending of algebraic and trigonometric polynomials, briefly CAT Bézier curves, is presented in this paper. The CAT Bézier curves retain the main superiority of cubic Bézier curves. The CAT Bézier curves can approximate the Bézier curves from the both sides, and the shapes of the curves can be adjusted totally or locally. With the shape parameters chosen properly, the introduced curves can represent some transcendental curves exactly. Keywords: C-Bézier curve, CAT-Bézier curve, shape parameter, local and total control, transcendental curve.
1 Introduction In CAGD, the cubic Bézier curves, have gained wide-spread applications, see [1]. However, the positions of these curves are fixed relative to their control polygons. Although the weights in the non-uniform rational B-spline (NURBS) curves, have an influence on adjustment of the shapes of the curves, see [2]. For given control points, changing the weights to adjust the shape of a curve is quite opaque to the user. On the other hand, it is also noticed that those curves based on algebraic polynomials have many shortcomings, especially in representing transcendental curves such as the cycloid and helix, etc. Hence, some author proposed new methods in the space of mixed algebraic and non-algebraic polynomials, see [3-10]. Pottman and Wagner [4] investigated helix splines. GB-splines were constructed in [5] for tension generalized splines allowing the tension parameters to vary from interval to interval and the main results were extended to GB-splines of arbitrary order in [6]. Han, Ma and Huang [7] studied the cubic trigonometric Bézier curve with two shape parameters, the shape of which can be adjusted locally by shape parameters. Zhang [8, 9] constructed the C-B curves based on the space span{1, t, sint, cost}, which can precisely represent some transcendental curves such as the helix and cycloid, but can only approximate the curves from the single side. Wang, Chen and Zhou studied non-uniform algebraictrigonometric B-splines (k≥3) for space span{1,t,t2…tk-2, sint, cost } in [10]. Note that these existing methods can deal with some polynomial curves and transcendental curves precisely. But the forms of these curves are adjusted by shape parameters with complicated procedures. In this paper, we present a class of cubic Bézier-type curves with two shape parameters based on the blending of algebraic and trigonometric polynomials. This approach has the following feature: 1.The introduced curves can approximate the S. Lin and X. Huang (Eds.): CSEE 2011, Part II, CCIS 215, pp. 125–130, 2011. © Springer-Verlag Berlin Heidelberg 2011
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Bézier curve from the both sides, and the change range of the curves is wider than that of C-Bézier curves. 2. The shape of the curves can be adjusted totally or locally. 3. With the shape parameters and control points chosen properly, the CAT Bézier curves can be used to represent some transcendental curves such as helix and cycloid.
2 The CAT Bézier Base Functions and Their Related Propositions Definition 2.1. For two arbitrarily selected real values λ and μ ,where λ ,μ ≤
π2
=
π2 −8 5.27898, the following four functions in variable t with are called CAT-Bézier base functions: ⎧ π ( 3 − 2 λ ) π − 3 π t + 3 π λ t 2 − π λ t 3 + 6 ( λ − 1 ) c o s ⎛⎜ t ⎞⎟ ⎪ ⎝ 2 ⎠, ⎪ b 0 ,3 ( t ) = 3π − 6 + ( 6 − 2 π ) λ ⎪ ⎪ ⎛ ⎛π ⎞ ⎛ π ⎞⎞ ⎪ 2 − π + π t − 2 ⎜ s in ⎜ t ⎟ − c o s ⎜ t ⎟ ⎟ ⎪ ⎝ 2 ⎠ ⎝ 2 ⎠⎠ ⎝ − b 0 ,3 ( t ) , ⎪ b1 ,3 ( t ) = ⎪ 4 −π ⎨ ⎛ ⎛π ⎞ ⎛π ⎞⎞ ⎪ 2 − π t + 2 ⎜ s in ⎜ t ⎟ − c o s ⎜ t ⎟ ⎟ ⎪ ⎝ 2 ⎠ ⎝ 2 ⎠⎠ ⎝ − b 3 ,3 ( t ) , ⎪ b 2 ,3 ( t ) = 4 −π ⎪ ⎪ ⎛π ⎞ 3 (1 − μ ) π t + π μ t 3 + 6 ( μ − 1 ) s in ⎜ t ⎟ ⎪ ⎝ 2 ⎠. ⎪ b (t ) = ⎪ 3 ,3 3π − 6 + ( 6 − 2 π ) μ ⎩
(1)
It can be verified by straight calculations that these CAT Bézier base functions possess the properties similar to those of the cubic Bernstein base functions. (a) Properties at the endpoints: ( j) ⎧⎪b0,3 ( 0) = 1 ⎪⎧bi ,3 ( 0) = 0 , , (2) ⎨ ⎨ ( j) ⎪⎩b3−i ,3 (1) = 0 ⎩⎪b3,3 (1) = 1
where j = 0,1,", i − 1, i = 1,2,3 ,and bi(,30) ( t ) = bi ,3 ( t ) . (b) Symmetry:
bi ,3 ( t; λ , μ ) = b3− i ,3 (1-t; λ , μ ) ,for i = 0,1, 2,3 .
(3)
(c) Partition of unity: 3
∑ b ( t ) =1 . i=0
(4)
i ,3
Proposition 2.1. These CAT Bézier base functions are nonnegative for λ ,μ ≤
namely,
bi ,3 ( t ) ≥ 0 , i = 0,1,2,3
Proof: By simple computation, for t ∈ [0 ,1] and λ ≤
non-negativity of b0 ,3 ( t ) can be easily proved.
π2 π2 −8
π2 π2 −8
,
(5)
, we have b0′ ,3 ( t ) ≤ 0 , so the
The CAT Bézier Curves
For t ∈ [0 ,1] and λ < 3π − 6 , since b1,3 ( 0 ) = b1,3 (1) = b′1,3 (1) = 0 , b1′,3 ( 0) = 2π − 6
and b1′′,3 (1) =
π2 8 − 2π
127
3π >0 3 π 6 − − ( ) ( 2π − 6) λ
,
> 0 , if b1,3 ( t ) fails to retain the same sign in the interval (0,1), then
b1′,3 ( t ) has at least four zeros in [0,1]. So ⎛ π3 ⎞ π 6 (1 − λ ) + πλ 6πλ π − b1′′′,3 ( t ) = ⎜ sin t ⎟⎟ cos t + ⎜ 2 (16 − 4π ) ( ( 3π − 6) − ( 2π − 6) λ ) 2 ⎝ 16 − 4π ( 3π − 6) − ( 2π − 6) λ ⎠
has at least two zeros in (0,1), which contradicts the properties of trigonometric functions. Since b1,3 ( t ) retains the same sign and b1,3 ( 0.5) = 24 2 ( λ − 1) + (12 − 11λ ) π > 0 8 ( ( 3π − 6 ) − ( 2π − 6 ) λ )
for λ < 3π − 6 , the property of non-negativity of b1,3 ( t ) is proved. By symmetry, the 2π − 6
non-negativity of b3,3 ( t ) and b2 ,3 ( t ) can be proved similarly. , bi ,3 ( t ) ≥ 0 , i = 0 ,1,2 ,3 . π2 −8 Fig.1 shows the figures of the four CAT-Bézier base functions and C-Bézier base functions. So, for t ∈ [0 ,1] and λ ,μ ≤
π2
(a)
(b)
(c)
(d)
Fig. 1. The figures of four CAT-Bézier base functions
3 The CAT Bézier Curves and Their Related Properties 3.1 Construction of the CAT-Bézier Curves Definition 3.1. Given points Pi ( i = 0 ,1,2 ,3) in R 2 or R3 , then 3
r ( t ) = ∑ Pibi ,3 ( t ) i=0
(6)
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is called a CAT Bézier curve, where t ∈ [0 ,1] , λ , μ ≤ 5.27898 , bi ,3 ( t ) , i = 0,1,2,3 are the CAT Bézier base functions. From the definition of the base functions, some properties of the CAT-Bézier curves can be obtained as follows: Theorem 3.1. The CAT-Bézier curve (6) has the following properties: (a) Terminal properties :
⎧r ( 0) = P0 ,r (1) = P3 , ⎪ 3π 3π ⎨ ⎪r′( 0) = 3π − 6 + ( 6 − 2π ) λ ( P1 − P0 ) ,r′(1) = 3π − 6 + ( 6 − 2π ) μ ( P3 − P2 ) . ⎩
(7)
(b) Symmetry: P0 , P1 , P2 , P3 and P3 , P2 , P1 , P0 determine the same CAT-Bézier curve in different parameterizations, i.e. r ( t,λ , μ ; P0 , P1 , P2 , P3 ) = r (1 − t,λ , μ ; P3 , P2 , P1 , P0 ) . (8) (c) Geometric invariance: The shape of a CAT Bézier curve is independent of the choice of coordinates, namely, (3.1) satisfies the following two equations: (9) r ( t,λ , μ ; P0 +q , P1 +q , P2 +q , P3 +q ) = r ( t, λ , μ ; P0 , P1 , P2 , P3 ) +q , r ( t ,λ , μ ; P0 * T , P1 * T , P2 * T , P3 * T ) = r ( t ,λ , μ ; P0 , P1 , P2 , P3 ) * T ,
(10)
where q is an arbitrary vector in R or R , and T is an arbitrary d × d matrix, d = 2 or 3. (d) Convex hull property: The entire CAT-Bézier curve segment must lie inside its control polygon spanned by P0 , P1 , P2 , P3 . 2
3
3.2 Shape Control of the CAT-Bézier Curves
By introducing the shape parameters λ and μ , the CAT-Bézier curves possess the better representation ability than C-Bézier curves. As shown in Fig.2, when the control polygon is fixed, by letting λ be equal to μ and adjusting the shape parameter from -∞ to 5.27898, the CAT-Bézier curves can range from below the C-Bézier curves to above the cubic Bézier curves. And, the shape parameters are of the property that the larger the shape parameter is, the more closely the curves approximate the control polygon. For t ∈ [0 ,1] , we rewrite (6) as follows: ⎛ ⎛ ⎛π ⎞ ⎛π ⎞ ⎛ π ⎞⎞ ⎛ π ⎞⎞ 2 − π (1 − t ) + 2 ⎜ c os ⎜ t ⎟ − sin ⎜ t ⎟ ⎟ 2 − π t + 2 ⎜ sin ⎜ t ⎟ − cos ⎜ t ⎟ ⎟ ⎝2 ⎠ ⎝ 2 ⎠⎠ ⎝2 ⎠ ⎝ 2 ⎠⎠ ⎝ ⎝ r (t ) = P1 + P2 4 −π 4 −π + b0 ,3 ( t; λ )( P0 − P1 ) + b3 ,3 ( t; μ )( P3 − P2 ) .
(11)
Obviously, shape parameters λ and μ only affect curves on the control edge P0P1 and P2P3 respectively. In fact, from (11), we can adjust the shape of the CAT-Bézier curves locally: as μ increases and λ fixes, the curve moves in the direction of the edge P2 P3; as μ decreases and λ fixes, the curve moves in the opposite direction to the edge P2P3, as shown in Fig.3. The same effects on the edge P0 P1 are produced by the shape parameter λ .
The CAT Bézier Curves
Fig. 2. Adjusting the curves totally
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Fig. 3. Adjusting the curves locally
4 The Applications of the CAT-Bézier Curves In this section, we can represent some transcendental curves by means of CAT-Bézier curves precisely. Proposition 4.1. Let P0, P1, P2 and P3 be four control points as follows, ⎛ ( 2 − π )2 ⎞ ⎛ 2 −π 4 −π ⎞ a,0 ⎟ , P2 = ⎜ a, a ⎟ , P3 = ( 0,a )( a ≠ 0 ) . P0 = ( 0,0 ) , P1 = ⎜ − ⎜ ⎟ 2 π 2 ⎝ π ⎠ ⎝ ⎠ Then the corresponding CAT-Bézier curve with the shape parameters λ = μ = 0 and t ∈ [ 0 ,1] represents an arc of cycloid. Proof. Substituting
⎛ ( 2 − π )2 ⎞ ⎛ 2 −π 4 −π ⎞ a,0 ⎟ , P2 = ⎜ a, a ⎟ ,P3 = ( 0,a ) into P0 = ( 0,0) , P1 = ⎜ − ⎜ ⎟ 2 π 2 ⎝ π ⎠ ⎝ ⎠
(6) yields
the following coordinates of the CAT-Bézier curve,
， ，
⎧ π ⎞ ⎛ ⎪ x ( t ) = a ⎜ t − sin 2 t ⎟ ⎪ ⎝ ⎠ ⎨ π ⎛ ⎪ y ( t ) = a 1 − cos t ⎞ ⎜ ⎟ ⎪⎩ 2 ⎠ ⎝
(12)
which is a parametric equation of a cycloid. Proposition 4.2. Let P0, P1, P2 and P3 be four properly chosen control points such that π −2 ⎞ ⎛π − 2 ⎛ π − 2 2b ⎞ a,a, b ⎟ , P2 = ⎜ a, a, ⎟ , P3 = ( a,0 ,b )( a ≠ 0,b ≠ 0 ) . P0 = ( 0,a,0 ) , P1 = ⎜ π 2 π ⎠ ⎝ 2 ⎠ ⎝
Then the corresponding CAT-Bézier curve with the shape parameters λ = μ = 0 and t ∈ [ 0 ,1] represents an arc of a helix. Proof. Substituting P0 = ( 0,a,0 ) , P1 = ⎛⎜ π − 2 a,a, π − 2 b ⎞⎟ , P2 = ⎛⎜ a, π − 2 a, 2b ⎞⎟ , P3 = ( a,b ) into ⎝
2
π
⎠
⎝
2
π ⎠
(6) yields the coordinates of the CAT-Bézier curve π ⎧ ⎪ x (t ) = a c o s 2 t , ⎪ π ⎪ ⎨ y ( t ) = a s in t , 2 ⎪ ⎪ z (t ) = b t , ⎪ ⎩
which is the parametric equation of a helix.
(13)
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Remark: By selecting proper control points and shape parameters, one can show that the segments of sine, cosine curve and ellipse can also be represented via CAT-Bézier curves.
5 Conclusions As mentioned above, the CAT-Bézier curves have all the properties of C-Bézier curves. What is more, the introducing curves can precisely represent some transcendental curves. Also, we can adjust the shapes of the CAT-Bézier curves totally or locally. Because there is hardly any difference in structure between a CATBézier curve and a cubic Bézier curve, it is not difficult to adapt a CAT-Bézier curve to a CAD/CAM system that already uses the cubic Bézier curves. Acknowledgments. This work is supported by the National Nature Science Foundation of China (No.61070227), the Key Project Foundation of Teaching Research of Department of Anhui Province (No.20100935).
References 1. Farin, G.: NURBS curves and surfaces. A.K.Peter, Wellesley (1995) 2. Shi, F.Z.: Computer aided geometric design & non-uniform rational B-spline. Higher Education Press, Beijing (2001) 3. Peña, J.M.: Shape Preserving Representations for Trigonometric Polynomial Curves. Comput. Aided Geom. Design 14, 5–11 (1997) 4. Pottman, H., Wagner, M.G.: Helix splines as an example of affine tchebycheffian splines. Adv. Comput. Math. 2, 123–142 (1994) 5. Kvasov, B.I.: GB-splines and their properties. Ann. Numer. Math. 3, 139–149 (1996) 6. Kvasov, B.I., Sattayatham, P.: GB-splines of arbitrary order. J.Comput.Appl. Math. 104, 63–88 (1999) 7. Han, X.L., Ma, Y., Huang, X.: The cubic trigonometric Bézier curve with two shape parameters. Appl. Math. Lett. 22, 226–231 (2009) 8. Zhang, J.W.: C-curves:An Extension of Cubic Curves. Comput. Aided Geom. Design 13, 199–217 (1996) 9. Zhang, J.W.: Two different forms of C-B-splines. Comput. Aided Geom. Design 14, 31– 41 (1997) 10. Wang, G.Z., Chen, Q.Y., Zhou, M.H.: NUAT-spline curves. Comput. Aided Geom. Design 14, 31–41 (1997)
