Geometric modeling with quasi-Hermite curves ... - Semantic Scholar
Geometric modeling with quasi-Hermite curves and surfaces Sugen Chen School of Mathematics & Computational Science, Anqing Teachers College, Anqing 246011, China
Abstract A class of quasi-Hermite base function is established in space Γ = {1, t, sin t, cos t, cos 2t}. The corresponding quasi-Hermite curves with a shape parameter α are defined by the introduced base function. The curves can easily be adjusted by using the shape parameter α. With the parameter chosen properly , the defined curves can precisely be used to represent straight line segment, circular arcs, elliptic arcs, cycloid, sine and cosine curves. And quasiCoons surface is defined by quasi-Hermite base function in stand of Hermite base function. At last, the quasi-bicubic Coons surfaces is discussed especially, and the surfaces can represent spherical surfaces, ellipsoid, cylinder, anchor ring and circular conical surface exactly.
1. Introduction B´ezier curves and uniform B-spline curves are powerful tools for constructing free form curves and surfaces, and Hermite curves are also useful in interpolation. But they cannot present circular arcs, cycloid and other transcendental curves precisely. On the other hand, how to present complicate free form surface is a difficult problem in plane, car and other products design. Coons patches as a method to generate interpolation surfaces have important effect in curve and surface modeling. Traditional Coons patch is a powerful tool in surface’s presentation and design, but it is powerless in present quadric surface which is often encountered in products design. Recently, many bases are presented in other new space( [3], [4], [5], [6], [7], [9]). The corresponding curves and tensor product surfaces are defined, they can present sphere and elliptic surface and other quadric surfaces. C-Coons patch( [2]) is constructed by C-Hermite blending functions. QCT-Coons patch ( [1]) is constructed by trigonometric function. The main contribution of this paper is to develop a method based on a quasi-Hermite base function. This approach has the following features: First, we establish quasi-Hermite base function in space Γ = {1, t, sin t, cos t, cos 2t} with a parameter α. With the shape parameter chosen properly, the quasi-Hermite curves can precisely be used to represent
_____________________________
978-1-4244-3701-6/09/$25.00 ©2009 IEEE
Benyue Su School of Computer & Information, Institute of Computer Applications, Anqing Teachers College, Anqing 246011, China [email protected]
straight line, circular arcs and some transcendental curves. Second, quasi-Coons surfaces are defined by using the quasiHermite base functions, then we discussed quasi-bicubic Coons surface especially, which can represent spherical surfaces, ellipsoid, cylinder, anchor ring and circular conical surface exactly. The rest of this paper is organized as follows. Section 2 defines the quasi-Hermite base function and the corresponding curves. In section 3, we define the quasi-Hermite curves in B´ezier-type, and present the representations of circular arcs, elliptic arcs, cycloid, etc. by using the introduced curves. The quasi-Coons surface is defined in section 4 and we also present shape modeling by the introduced surfaces in this section. In section 5, we discuss the quasi-bicubic Coons surface especially, and represent spherical surfaces, ellipsoid, cylinder, anchor ring and circular conical surface exactly by chosen parameter α = 0. Finally, we conclude the paper in section 6.
2. Quasi-Hermite base function and quasiHermite curve Definition 1 For an arbitrarily selected real value of α, and t ∈ [0, π2 ], H0 (t), H1 (t), G0 (t) and G1 (t) are called quasi-Hermite blending functions which can be defined to be (Fig.1) F0 (t) = F1 (t) = G0 (t) = G1 (t) =
π−2−πα 2−4α − 2−4α π−4 π−4 t + π−4 sin t − 2−4α π−4 cos t + α cos 2t, −2+πα 2−4α 2−4α π−4 + π−4 t − π−4 sin t + 2−4α π−4 cos t − α cos 2t, 2−πα 2−4α π−2−4α sin t π−4 − π−4 t + π−4 2−4α − π−4 cos t + α cos 2t, π−2−πα 2−4α − 2−4α π−4 π−4 t + π−4 sin t cos t + α cos 2t. − π−2−4α π−4
(1)
Definition 2 Let q0 , q1 be given points, and q0 , q1 be the corresponding tangent vectors. Given an arbitrarily real value α, the quasi-Hermite curve is defined as (Fig.2)
we can deduce the following definition.
1.2
1
Definition 3 Let P0 , P1 , P2 , P3 be the control points, the quasi-Hermite curve is defined as
F0(t)
0.8
F1(t) G0(t)
0.6
G1(t)
0.4
0.2
0
−0.2
−0.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Fig.1 Quasi−Hermite base functions with different parameter α
3
QHC(t) = P0 · B0 (t) + P1 · B1 (t) + P2 · B2 (t) + P3 · B3 (t), (5) where B0 (t) = F0 (t) − G0 (t) = 1 − sin t, B1 (t) = −G1 (t) = 2−4α 2−4α π−2−4α cos t−α cos 2t, = − π−2−πα π−4 + π−4 t− π−4 sin t+ π−4 B2 (t) = G0 (t) = 2−4α π−2−4α = 2−πα sin t − 2−4α π−4 − π−4 t + π−4 π−4 cos t + α cos 2t, B3 (t) = F1 (t) + G1 (t) = 1 − cos t,
α=0 2.8
From direct calculation, it’s easy to know that B0 (t), B1 (t), B2 (t), B3 (t) have the properties: π π B0 (t) = B3 ( − t), B1 (t) = B2 ( − t), (6) 2 2 Proposition 1 (symmetry) From Eq(5) and Eq(6), for the same α, both P0 , P1 , P2 , P3 and P3 , P2 , P1 , P0 define the same curve in a different parameterization:
α=10
2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
Fig.2 Quasi−Hermite curve with different parameter α
QHC(t) = q0 ·F0 (t)+q1 ·F1 (t)+q0 ·G0 (t)+q1 ·G1 (t). (2) From the definition 1 and definition 2, we know that the base functions and the corresponding quasi-Hermite curves have the properly: F0 (0) = 1, F1 (0) = 0, G0 (0) = 0, G1 (0) = 0, F0 ( π2 ) = 0, F1 ( π2 ) = 1, G0 ( π2 ) = 0, G1 ( π2 ) = 0, F0 (0) = 0, F1 (0) = 0, G0 (0) = 1, G1 (0) = 0, F0 ( π2 ) = 0, F1 ( π2 ) = 0, G0 ( π2 ) = 0, G1 ( π2 ) = 1, (3) and QHC(0) = q0 , QHC( π2 ) = q1 , (4) QHC (0) = q0 , QHC ( π2 ) = q0
3. B´ezier-type Quasi-Hermite curve and applications From definition 2, we know that q0 , q1 and q0 , q1 are interpolation points and corresponding tangent vectors. It’s not convenient for curve design. In order to avoid this drawback, we define the quasi-Hermite curve in B´ezier-type, Let q0 = P0 , q1 = P3 , q0 = P2 − P0 , q1 = P3 − P1 , then
π QHC(t, α, P0 , P1 , P2 , P3 ) = QHC( −t, α, P3 , P2 , P1 , P0 ), 2 (7) Let Pix denote the projection of point Pi at x-axis and Piy denote the projection of point Pi at y-axis, i = 0, 1, 2, 3. Then we can conclude following results to be (1) Suppose P0 , P1 , P2 , P3 are four control points, If they are collinear or degenerate to two points with multiple knots, then their corresponding quasi-Hermite curve with parameter α represents a line segment. (2) Let P1x = P2x = P3x , P0y = P2y and P1y = P3y , then their corresponding quasi-Hermite curve with parameter α = 12 represents a segment of parabola. (Fig.3) (3) Let P0x = P1x = P2x = P3x and P0y = P1y = P2y = P3y , then their corresponding quasi-Hermite curves with parameter α = 12 or α = 0 represent a segment of elliptic arc. (Fig.4) (4) Let P1x = P2x , −2 P1x = π−4 P1x + π−2 π−4 P2x , P3x =
π−2 π−4 P1x
−
2 π−4 P2x ,
and
−2 Poy = π−4 P1y + π−2 π−4 P2y , P1y = P2y , P1y = P3y , then their corresponding quasi-Hermite curve with parameter α = 0 represents a segment of cosine curve. (Fig.5)
(5) Let P1x = P2x , −2 P1x = π−4 P1x + π−2 π−4 P2x , P3x =
π−2 π−4 P1x
−
2 π−4 P2x ,
and
Fig.5 The representation of cosine curve by quasi−Hermite curve
Fig.3 The representation of parabola by quasi−Hermite curv
Fig.4 The representation of circular and elliptic arc by quasi−Hermite curve
Fig.6 The representation of sine curve by quasi−Hermite curve
2 P0y = P1y , P1y = P2y , P3y = π−2 π−4 P1y − π−4 P2y , then their corresponding quasi-Hermite curve with parameter α = 0 represents a segment of sine curve. (Fig.6)
(6) Let P0x = P2x = 0, P3x =
π−2 π−4 P1x ,
and
2 P1x , P0y = P1y = P2y = 0, P3y = π−4 then their corresponding quasi-Hermite curve with parameter α = 0 represents a segment of cycloid curve. (Fig.7)
Fig.7 The representation of cycloid by quasi−Hermite curve
4. Quasi-Coons surface Given four boundary curve r(u, j), r(i, v) , rv (u, j) and ru (i, v), i, j = 0, π2 , we construct quasi-Coons surface as following: First, we construct surface r1 (u, v) along u-direction based on boundary curves r(0, v), r( π2 , v) and ru (0, v), ru ( π2 , v) as r1 (u, v) =
⎛
⎞ F0 (v) ⎜ F1 (v) ⎟ ⎟ r(u, 0) r(u, π2 ) rv (u, 0) rv (u, π2 ) ⎜ ⎝ G0 (v) ⎠ , G1 (v))
⎞
(8)
Second, we construct surface r2 (u, v) along u-direction based on boundary curves r(u, 0), r(u, π2 ) and rv (u, 0), rv (u, π2 ) as
(9)
Third, we construct surface r3 (u, v) as r3 (u, v) = QHU × P × QHV
⎛
r(0, v) ⎜ r( π , v) ⎟ 2 ⎟ F0 (u) F1 (u) G0 (u) G1 (u) ⎜ ⎝ ru (0, v) ⎠ , ru ( π2 , v)
r2 (u, v) =
where QHU = (F0 (u), F1 (u), G0 (u), G1 (u)), QHV = (F0 (v), F1 (v), G0 (v), G1 (v))T ,
(10)
r(0, v) = r( π2 , v) = r(u, 0) = 1.5 1.5
1.5 1
0
2
1.6 1.4
2
−0.5
1.4 0.5
1.2
1.2 1 0.8 0.6 0.4
1
1
0.8 1.5 2 2.5 0
0.4 0.2
2
−0.5
1.8
0
0 −0.5
2 1.8
1.6
−0.5
1.4 0
1.2 1
0.5
0.8
1
0.6
0.2 2.5 0
0
1.8 1.6
0
r(u, π2 ) =
0.5
0.5
−0.5
1.8
1
1
0.5
0.5
1.6
1.2 1 1.5
0.4 2
0.6 2
0.4
0.2 2.5 0
ru ( π2 , v) =
0.8
0.6 1.5
ru (0, v) =
1.4 1
0.2 2.5 0
rv (u, 0) =
Fig.8 Quasi−Coons surface with different parameter α based on same boundary curves
rv (u, π2 ) = ⎛ ⎜ ⎜ P =⎜ ⎝
P (0, 0) P (0, π2 ) Pv 0, 0) Pv (0, π2 ) π π π π P ( 2 , 0) P ( 2 , 2 ) Pv ( 2 , 0) Pv ( π2 , π2 ) Pu (0, 0) Pu (0, π2 ) Puv (0, 0) Puv (0, π2 ) Pu ( π2 , 0) Pu ( π2 , π2 ) Puv ( π2 , 0) Puv ( π2 , π2 )
⎞
At last, From Eq.(8), Eq.(9) and Eq.(10), we define quasi-Coons surface to be Definition 4 surface, if
Surface r(u, v) is called as quasi-Coons
r(u, v) = r1 (u, v) + r2 (u, v) − r3 (u, v)
Definition 5
⎟ ⎟ ⎟. ⎠
(11)
Remark: Traditional Coons patch is powerful in surface’s presentation and design, but when the boundary curves and cross boundary derivatives are given, then the shape of Coons patch is determined. From Eq.(11), we know that quasi-Coons surface has shape parameter α, so it can be used to change the shape of surface when the boundary curves are given (Fig.8).
5. Quasi-bicubic Coons surface and applications In section 4, we found that it was not convenient to apply the quasi-Coons surface defined by Eq.(11) because of so many boundary information. In order to simplify the expression of Eq.(11), we suppose that α is fixed with α = 0 in this section, and define boundary curves and cross boundary derivative by using the information of corner points and base functions defined by Eq.(1) as follows.
F0 (v)r(0, 0) + F1 (v)r(0, π2 )+ G0 (v)rv (0, 0) + G1 (v)rv (0, π2 ), F0 (v)r( π2 , 0) + F1 (v)r( π2 , π2 )+ G0 (v)rv ( π2 , 0) + G1 (v)rv ( π2 , π2 ), F0 (u)r(0, 0) + F1 (u)r( π2 , 0)+ G0 (u)ru (0, 0) + G1 (u)ru ( π2 , 0), F0 (u)r(0, π2 ) + F1 (v)r( π2 , π2 )+ G0 (u)ru (0, π2 ) + G1 (u)ru ( π2 , π2 ), F0 (v)ru (0, 0) + F1 (v)ru (0, π2 )+ G0 (v)ruv (0, 0) + G1 (v)ruv (0, π2 ), F0 (v)ru ( π2 , 0) + F1 (v)ru ( π2 , π2 )+ G0 (v)ruv ( π2 , 0) + G1 (v)ruv ( π2 , π2 ), F0 (u)rv (0, 0) + F1 (u)rv ( π2 , 0)+ G0 (u)ruv (0, 0) + G1 (u)ruv ( π2 , 0), F0 (u)rv (0, π2 ) + F1 (v)rv ( π2 , π2 )+ G0 (u)ruv (0, π2 ) + G1 (u)ruv ( π2 , π2 ), Given the four corner points information
P (0, 0), P (0, π2 ), P ( π2 , 0), P ( π2 , π2 ), Pv (0, 0), Pv (0, π2 ), Pv ( π2 , 0), Pv ( π2 , π2 ), Pu (0, 0), Pu (0, π2 ), Pu ( π2 , 0), Pu ( π2 , π2 ), Puv (0, 0), Puv (0, π2 ), Puv ( π2 , 0), Puv ( π2 , π2 ). From Eq.(11) and Eq.(8) formula above, the quasi-bicubic Coons surface is defined as: QHS(u, v) = QHU × P × QHV
(12)
where QHU = (F0 (u), F1 (u), G0 (u), G1 (u)), QHV = (F0 (v), F1 (v), G0 (v), G1 (v))T , ⎛ ⎜ ⎜ P =⎜ ⎝
⎞
P (0, 0) P (0, π2 ) Pv 0, 0) Pv (0, π2 ) π π π π P ( 2 , 0) P ( 2 , 2 ) Pv ( 2 , 0) Pv ( π2 , π2 ) Pu (0, 0) Pu (0, π2 ) Puv (0, 0) Puv (0, π2 ) Pu ( π2 , 0) Pu ( π2 , π2 ) Puv ( π2 , 0) Puv ( π2 , π2 )
Case 4.1 Given the information matrix ⎛ (0, 0, 0) (a, 0, 0) (a, 0, 0) (0, 0, −c) ⎜ (0, 0, c) (0, b, 0) (0, b, 0) (0, 0, −c) ⎜ P =⎜ ⎝ (0, 0, 0) (0, b, 0) (0, b, 0) (0, 0, 0) (0, 0, 0) (−a, 0, 0) (−a, 0, 0) (0, 0, 0)
⎟ ⎟ ⎟. ⎠
⎞ ⎟ ⎟ ⎟, ⎠
a, b, c > 0, then, quasi-bicubic Coons surface represents a part of ellipsoid. Especially it can represent a part of spherical
Sphere
One−eighth Sphere
1.6 1.5
1
1.4
0.5
1.2
1
1
0
0.8
−0.5
0.5
0.6
−1
0.4 0 1
−1.5 1
0.2 1
1 0.5
0
0.5
−2
−1
0 0
1
2
Fig.11 The representation of circular cylinder surface by quasi−bicubic Coons surface
−1 −1
0 0
−2
0
0
Fig.9 Sphere modeled by quasi−bicubic Coons surface
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −2
−1
0
1
2
2
1
0
−1
−2
Fig.10 The representation of Circular conical surface by quasi−bicubic Coons surface
surfaces when a = b = c. (Fig.9)
⎛ ⎜ ⎜ P =⎜ ⎝
Case 4.2 Given the information matrix ⎛ ⎜ ⎜ P =⎜ ⎝
a=
(0, 0, 0) (a, 0, a) (1, 0, 1) (1, 0, 1) (0, 0, 0) (0, a, a) (0, 1, 1) (0, 1, 1) (0, 0, 0) (0, 0, 0) (0, 1, 0) (0, 1, 0) (0, 0, 0) (−a, 0, 0) (−1, 0, 0) (−1, 0, 0)
⎞ ⎟ ⎟ ⎟, ⎠
(a, 0, b) (0, 0, b) (−b, 0, 0) (0, a, b) (0, 0, b) (0, −b, 0) (0, a, 0) (0, 0, 0) (0, −b, 0) (−a, 0, 0) (0, 0, 0) (b, 0, 0)
⎞ ⎟ ⎟ ⎟, ⎠
a > b > 0, then, quasi-bicubic Coons surface represents a part of circular conical surface. (Fig.12)
6. Conclusion
π 2.
Then, quasi-bicubic Coons surface represents a part of circular conical surface. (Fig.10) Case 4.3 Given the information matrix ⎛ ⎜ ⎜ P =⎜ ⎝
a > 0, b =
(a, 0, 0) (a, 0, b) (0, 0, 1) (0, 0, 1) (0, a, 0) (0, a, b) (0, 0, 1) (0, 0, 1) (0, a, 0) (0, a, 0) (0, 0, 0) (0, 0, 0) (−a, 0, 0) (−a, 0, 0) (0, 0, 0) (0, 0, 0)
⎞ ⎟ ⎟ ⎟, ⎠
π 2,
then, quasi-bicubic Coons surface represents a part of circular cylindrical surface. (Fig.11) Case 4.4 Given the information matrix
(a + b, 0, 0) (0, a + b, 0) (0, a + b, 0) (−a − b, 0, 0)
We have established quasi-Hermite curves by trigonometric polynomials, which inherit similar properties of Hermite curves. The quasi-Hermite curves and quasi-Coons surfaces can be used to represent circular arcs, elliptic arcs, cycloid, sine curves, cosine curves and spherical surfaces, ellipsoid, cylinder, anchor ring , circular conical surface exactly, which indicate that the quasi-Hermite curves and surfaces are more powerful than traditional Hermite curves and Coons surfaces on expression of special curves and surfaces. So this paper maybe offer a useful tool in CAGD.
Acknowledgment This work was completed with the support by the National Natural Science Foundation of China under Grant No. 60773128 and in part by the Natural Science Foundation of Anhui Provincial Education Department under Grant No. KJ2009A123.
References [1] J. C. Li, L. Z. Song, and M. W. Zhang, ” Expression and application of QCT-Coons patch,” Computer Engineeing and Application (in Chinese), 43(21), pp. 87-89, 2007. [2] Y. Li, W. C. Tang, and H. C. Liu, ” C-Coons patch and its property,” Journal of Computer-Aided Design & Computer Graphics (in Chinese), 15(9), pp. 1177-1180, 2003. [3] J. M. Pe˜ na, ” Shape preserving representations for trigonametric polynomial curves,” Computer Aided Geometric Design, 14(1), pp. 5-11, 1997. [4] J. S´anchez-Reyes, ” Harmonic rational B´ezier curves, p-B´ezier curves and trigonometric polynomials,” Computer Aided Geometric Design, 15, pp. 909-923, 1998. [5] B. Y. Su, J. Q. Tan, ” A family of quasi-cubic blended splines and applications,” J. Zhejiang Univ. SCIENCE A, 7, pp. 15501560, 2006. [6] B. Y. Su, J. Q. Tan, ” Geometric modeling for interpolation surfaces based on blended coordinate system,” Lecture Notes in Computer Science, 4270, pp. 222-231, 2006. [7] G. Walz, ” Trigonometric B´ ezier and Stancu polynomials over intervals and triangles,” Computer Aided Geometric Design, 14(4), pp. 393-397, 1997. [8] S. H. Yoon, M. S. Kim, ” Sweep-based Freeform Deformations,” Computer Graphics Forum (Eurographics 2006), 25, pp. 487-496, 2006. [9] J. W. Zhang, ” C-curves: An extension of cubic curves,” Computer Aided Geometric Design, 13(3), pp. 199-217, 1996.
Abstract A class of quasi-Hermite base function is established in space Γ = {1, t, sin t, cos t, cos 2t}. The corresponding quasi-Hermite curves with a shape parameter α are defined by the introduced base function. The curves can easily be adjusted by using the shape parameter α. With the parameter chosen properly , the defined curves can precisely be used to represent straight line segment, circular arcs, elliptic arcs, cycloid, sine and cosine curves. And quasiCoons surface is defined by quasi-Hermite base function in stand of Hermite base function. At last, the quasi-bicubic Coons surfaces is discussed especially, and the surfaces can represent spherical surfaces, ellipsoid, cylinder, anchor ring and circular conical surface exactly.
1. Introduction B´ezier curves and uniform B-spline curves are powerful tools for constructing free form curves and surfaces, and Hermite curves are also useful in interpolation. But they cannot present circular arcs, cycloid and other transcendental curves precisely. On the other hand, how to present complicate free form surface is a difficult problem in plane, car and other products design. Coons patches as a method to generate interpolation surfaces have important effect in curve and surface modeling. Traditional Coons patch is a powerful tool in surface’s presentation and design, but it is powerless in present quadric surface which is often encountered in products design. Recently, many bases are presented in other new space( [3], [4], [5], [6], [7], [9]). The corresponding curves and tensor product surfaces are defined, they can present sphere and elliptic surface and other quadric surfaces. C-Coons patch( [2]) is constructed by C-Hermite blending functions. QCT-Coons patch ( [1]) is constructed by trigonometric function. The main contribution of this paper is to develop a method based on a quasi-Hermite base function. This approach has the following features: First, we establish quasi-Hermite base function in space Γ = {1, t, sin t, cos t, cos 2t} with a parameter α. With the shape parameter chosen properly, the quasi-Hermite curves can precisely be used to represent
_____________________________
978-1-4244-3701-6/09/$25.00 ©2009 IEEE
Benyue Su School of Computer & Information, Institute of Computer Applications, Anqing Teachers College, Anqing 246011, China [email protected]
straight line, circular arcs and some transcendental curves. Second, quasi-Coons surfaces are defined by using the quasiHermite base functions, then we discussed quasi-bicubic Coons surface especially, which can represent spherical surfaces, ellipsoid, cylinder, anchor ring and circular conical surface exactly. The rest of this paper is organized as follows. Section 2 defines the quasi-Hermite base function and the corresponding curves. In section 3, we define the quasi-Hermite curves in B´ezier-type, and present the representations of circular arcs, elliptic arcs, cycloid, etc. by using the introduced curves. The quasi-Coons surface is defined in section 4 and we also present shape modeling by the introduced surfaces in this section. In section 5, we discuss the quasi-bicubic Coons surface especially, and represent spherical surfaces, ellipsoid, cylinder, anchor ring and circular conical surface exactly by chosen parameter α = 0. Finally, we conclude the paper in section 6.
2. Quasi-Hermite base function and quasiHermite curve Definition 1 For an arbitrarily selected real value of α, and t ∈ [0, π2 ], H0 (t), H1 (t), G0 (t) and G1 (t) are called quasi-Hermite blending functions which can be defined to be (Fig.1) F0 (t) = F1 (t) = G0 (t) = G1 (t) =
π−2−πα 2−4α − 2−4α π−4 π−4 t + π−4 sin t − 2−4α π−4 cos t + α cos 2t, −2+πα 2−4α 2−4α π−4 + π−4 t − π−4 sin t + 2−4α π−4 cos t − α cos 2t, 2−πα 2−4α π−2−4α sin t π−4 − π−4 t + π−4 2−4α − π−4 cos t + α cos 2t, π−2−πα 2−4α − 2−4α π−4 π−4 t + π−4 sin t cos t + α cos 2t. − π−2−4α π−4
(1)
Definition 2 Let q0 , q1 be given points, and q0 , q1 be the corresponding tangent vectors. Given an arbitrarily real value α, the quasi-Hermite curve is defined as (Fig.2)
we can deduce the following definition.
1.2
1
Definition 3 Let P0 , P1 , P2 , P3 be the control points, the quasi-Hermite curve is defined as
F0(t)
0.8
F1(t) G0(t)
0.6
G1(t)
0.4
0.2
0
−0.2
−0.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Fig.1 Quasi−Hermite base functions with different parameter α
3
QHC(t) = P0 · B0 (t) + P1 · B1 (t) + P2 · B2 (t) + P3 · B3 (t), (5) where B0 (t) = F0 (t) − G0 (t) = 1 − sin t, B1 (t) = −G1 (t) = 2−4α 2−4α π−2−4α cos t−α cos 2t, = − π−2−πα π−4 + π−4 t− π−4 sin t+ π−4 B2 (t) = G0 (t) = 2−4α π−2−4α = 2−πα sin t − 2−4α π−4 − π−4 t + π−4 π−4 cos t + α cos 2t, B3 (t) = F1 (t) + G1 (t) = 1 − cos t,
α=0 2.8
From direct calculation, it’s easy to know that B0 (t), B1 (t), B2 (t), B3 (t) have the properties: π π B0 (t) = B3 ( − t), B1 (t) = B2 ( − t), (6) 2 2 Proposition 1 (symmetry) From Eq(5) and Eq(6), for the same α, both P0 , P1 , P2 , P3 and P3 , P2 , P1 , P0 define the same curve in a different parameterization:
α=10
2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
Fig.2 Quasi−Hermite curve with different parameter α
QHC(t) = q0 ·F0 (t)+q1 ·F1 (t)+q0 ·G0 (t)+q1 ·G1 (t). (2) From the definition 1 and definition 2, we know that the base functions and the corresponding quasi-Hermite curves have the properly: F0 (0) = 1, F1 (0) = 0, G0 (0) = 0, G1 (0) = 0, F0 ( π2 ) = 0, F1 ( π2 ) = 1, G0 ( π2 ) = 0, G1 ( π2 ) = 0, F0 (0) = 0, F1 (0) = 0, G0 (0) = 1, G1 (0) = 0, F0 ( π2 ) = 0, F1 ( π2 ) = 0, G0 ( π2 ) = 0, G1 ( π2 ) = 1, (3) and QHC(0) = q0 , QHC( π2 ) = q1 , (4) QHC (0) = q0 , QHC ( π2 ) = q0
3. B´ezier-type Quasi-Hermite curve and applications From definition 2, we know that q0 , q1 and q0 , q1 are interpolation points and corresponding tangent vectors. It’s not convenient for curve design. In order to avoid this drawback, we define the quasi-Hermite curve in B´ezier-type, Let q0 = P0 , q1 = P3 , q0 = P2 − P0 , q1 = P3 − P1 , then
π QHC(t, α, P0 , P1 , P2 , P3 ) = QHC( −t, α, P3 , P2 , P1 , P0 ), 2 (7) Let Pix denote the projection of point Pi at x-axis and Piy denote the projection of point Pi at y-axis, i = 0, 1, 2, 3. Then we can conclude following results to be (1) Suppose P0 , P1 , P2 , P3 are four control points, If they are collinear or degenerate to two points with multiple knots, then their corresponding quasi-Hermite curve with parameter α represents a line segment. (2) Let P1x = P2x = P3x , P0y = P2y and P1y = P3y , then their corresponding quasi-Hermite curve with parameter α = 12 represents a segment of parabola. (Fig.3) (3) Let P0x = P1x = P2x = P3x and P0y = P1y = P2y = P3y , then their corresponding quasi-Hermite curves with parameter α = 12 or α = 0 represent a segment of elliptic arc. (Fig.4) (4) Let P1x = P2x , −2 P1x = π−4 P1x + π−2 π−4 P2x , P3x =
π−2 π−4 P1x
−
2 π−4 P2x ,
and
−2 Poy = π−4 P1y + π−2 π−4 P2y , P1y = P2y , P1y = P3y , then their corresponding quasi-Hermite curve with parameter α = 0 represents a segment of cosine curve. (Fig.5)
(5) Let P1x = P2x , −2 P1x = π−4 P1x + π−2 π−4 P2x , P3x =
π−2 π−4 P1x
−
2 π−4 P2x ,
and
Fig.5 The representation of cosine curve by quasi−Hermite curve
Fig.3 The representation of parabola by quasi−Hermite curv
Fig.4 The representation of circular and elliptic arc by quasi−Hermite curve
Fig.6 The representation of sine curve by quasi−Hermite curve
2 P0y = P1y , P1y = P2y , P3y = π−2 π−4 P1y − π−4 P2y , then their corresponding quasi-Hermite curve with parameter α = 0 represents a segment of sine curve. (Fig.6)
(6) Let P0x = P2x = 0, P3x =
π−2 π−4 P1x ,
and
2 P1x , P0y = P1y = P2y = 0, P3y = π−4 then their corresponding quasi-Hermite curve with parameter α = 0 represents a segment of cycloid curve. (Fig.7)
Fig.7 The representation of cycloid by quasi−Hermite curve
4. Quasi-Coons surface Given four boundary curve r(u, j), r(i, v) , rv (u, j) and ru (i, v), i, j = 0, π2 , we construct quasi-Coons surface as following: First, we construct surface r1 (u, v) along u-direction based on boundary curves r(0, v), r( π2 , v) and ru (0, v), ru ( π2 , v) as r1 (u, v) =
⎛
⎞ F0 (v) ⎜ F1 (v) ⎟ ⎟ r(u, 0) r(u, π2 ) rv (u, 0) rv (u, π2 ) ⎜ ⎝ G0 (v) ⎠ , G1 (v))
⎞
(8)
Second, we construct surface r2 (u, v) along u-direction based on boundary curves r(u, 0), r(u, π2 ) and rv (u, 0), rv (u, π2 ) as
(9)
Third, we construct surface r3 (u, v) as r3 (u, v) = QHU × P × QHV
⎛
r(0, v) ⎜ r( π , v) ⎟ 2 ⎟ F0 (u) F1 (u) G0 (u) G1 (u) ⎜ ⎝ ru (0, v) ⎠ , ru ( π2 , v)
r2 (u, v) =
where QHU = (F0 (u), F1 (u), G0 (u), G1 (u)), QHV = (F0 (v), F1 (v), G0 (v), G1 (v))T ,
(10)
r(0, v) = r( π2 , v) = r(u, 0) = 1.5 1.5
1.5 1
0
2
1.6 1.4
2
−0.5
1.4 0.5
1.2
1.2 1 0.8 0.6 0.4
1
1
0.8 1.5 2 2.5 0
0.4 0.2
2
−0.5
1.8
0
0 −0.5
2 1.8
1.6
−0.5
1.4 0
1.2 1
0.5
0.8
1
0.6
0.2 2.5 0
0
1.8 1.6
0
r(u, π2 ) =
0.5
0.5
−0.5
1.8
1
1
0.5
0.5
1.6
1.2 1 1.5
0.4 2
0.6 2
0.4
0.2 2.5 0
ru ( π2 , v) =
0.8
0.6 1.5
ru (0, v) =
1.4 1
0.2 2.5 0
rv (u, 0) =
Fig.8 Quasi−Coons surface with different parameter α based on same boundary curves
rv (u, π2 ) = ⎛ ⎜ ⎜ P =⎜ ⎝
P (0, 0) P (0, π2 ) Pv 0, 0) Pv (0, π2 ) π π π π P ( 2 , 0) P ( 2 , 2 ) Pv ( 2 , 0) Pv ( π2 , π2 ) Pu (0, 0) Pu (0, π2 ) Puv (0, 0) Puv (0, π2 ) Pu ( π2 , 0) Pu ( π2 , π2 ) Puv ( π2 , 0) Puv ( π2 , π2 )
⎞
At last, From Eq.(8), Eq.(9) and Eq.(10), we define quasi-Coons surface to be Definition 4 surface, if
Surface r(u, v) is called as quasi-Coons
r(u, v) = r1 (u, v) + r2 (u, v) − r3 (u, v)
Definition 5
⎟ ⎟ ⎟. ⎠
(11)
Remark: Traditional Coons patch is powerful in surface’s presentation and design, but when the boundary curves and cross boundary derivatives are given, then the shape of Coons patch is determined. From Eq.(11), we know that quasi-Coons surface has shape parameter α, so it can be used to change the shape of surface when the boundary curves are given (Fig.8).
5. Quasi-bicubic Coons surface and applications In section 4, we found that it was not convenient to apply the quasi-Coons surface defined by Eq.(11) because of so many boundary information. In order to simplify the expression of Eq.(11), we suppose that α is fixed with α = 0 in this section, and define boundary curves and cross boundary derivative by using the information of corner points and base functions defined by Eq.(1) as follows.
F0 (v)r(0, 0) + F1 (v)r(0, π2 )+ G0 (v)rv (0, 0) + G1 (v)rv (0, π2 ), F0 (v)r( π2 , 0) + F1 (v)r( π2 , π2 )+ G0 (v)rv ( π2 , 0) + G1 (v)rv ( π2 , π2 ), F0 (u)r(0, 0) + F1 (u)r( π2 , 0)+ G0 (u)ru (0, 0) + G1 (u)ru ( π2 , 0), F0 (u)r(0, π2 ) + F1 (v)r( π2 , π2 )+ G0 (u)ru (0, π2 ) + G1 (u)ru ( π2 , π2 ), F0 (v)ru (0, 0) + F1 (v)ru (0, π2 )+ G0 (v)ruv (0, 0) + G1 (v)ruv (0, π2 ), F0 (v)ru ( π2 , 0) + F1 (v)ru ( π2 , π2 )+ G0 (v)ruv ( π2 , 0) + G1 (v)ruv ( π2 , π2 ), F0 (u)rv (0, 0) + F1 (u)rv ( π2 , 0)+ G0 (u)ruv (0, 0) + G1 (u)ruv ( π2 , 0), F0 (u)rv (0, π2 ) + F1 (v)rv ( π2 , π2 )+ G0 (u)ruv (0, π2 ) + G1 (u)ruv ( π2 , π2 ), Given the four corner points information
P (0, 0), P (0, π2 ), P ( π2 , 0), P ( π2 , π2 ), Pv (0, 0), Pv (0, π2 ), Pv ( π2 , 0), Pv ( π2 , π2 ), Pu (0, 0), Pu (0, π2 ), Pu ( π2 , 0), Pu ( π2 , π2 ), Puv (0, 0), Puv (0, π2 ), Puv ( π2 , 0), Puv ( π2 , π2 ). From Eq.(11) and Eq.(8) formula above, the quasi-bicubic Coons surface is defined as: QHS(u, v) = QHU × P × QHV
(12)
where QHU = (F0 (u), F1 (u), G0 (u), G1 (u)), QHV = (F0 (v), F1 (v), G0 (v), G1 (v))T , ⎛ ⎜ ⎜ P =⎜ ⎝
⎞
P (0, 0) P (0, π2 ) Pv 0, 0) Pv (0, π2 ) π π π π P ( 2 , 0) P ( 2 , 2 ) Pv ( 2 , 0) Pv ( π2 , π2 ) Pu (0, 0) Pu (0, π2 ) Puv (0, 0) Puv (0, π2 ) Pu ( π2 , 0) Pu ( π2 , π2 ) Puv ( π2 , 0) Puv ( π2 , π2 )
Case 4.1 Given the information matrix ⎛ (0, 0, 0) (a, 0, 0) (a, 0, 0) (0, 0, −c) ⎜ (0, 0, c) (0, b, 0) (0, b, 0) (0, 0, −c) ⎜ P =⎜ ⎝ (0, 0, 0) (0, b, 0) (0, b, 0) (0, 0, 0) (0, 0, 0) (−a, 0, 0) (−a, 0, 0) (0, 0, 0)
⎟ ⎟ ⎟. ⎠
⎞ ⎟ ⎟ ⎟, ⎠
a, b, c > 0, then, quasi-bicubic Coons surface represents a part of ellipsoid. Especially it can represent a part of spherical
Sphere
One−eighth Sphere
1.6 1.5
1
1.4
0.5
1.2
1
1
0
0.8
−0.5
0.5
0.6
−1
0.4 0 1
−1.5 1
0.2 1
1 0.5
0
0.5
−2
−1
0 0
1
2
Fig.11 The representation of circular cylinder surface by quasi−bicubic Coons surface
−1 −1
0 0
−2
0
0
Fig.9 Sphere modeled by quasi−bicubic Coons surface
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −2
−1
0
1
2
2
1
0
−1
−2
Fig.10 The representation of Circular conical surface by quasi−bicubic Coons surface
surfaces when a = b = c. (Fig.9)
⎛ ⎜ ⎜ P =⎜ ⎝
Case 4.2 Given the information matrix ⎛ ⎜ ⎜ P =⎜ ⎝
a=
(0, 0, 0) (a, 0, a) (1, 0, 1) (1, 0, 1) (0, 0, 0) (0, a, a) (0, 1, 1) (0, 1, 1) (0, 0, 0) (0, 0, 0) (0, 1, 0) (0, 1, 0) (0, 0, 0) (−a, 0, 0) (−1, 0, 0) (−1, 0, 0)
⎞ ⎟ ⎟ ⎟, ⎠
(a, 0, b) (0, 0, b) (−b, 0, 0) (0, a, b) (0, 0, b) (0, −b, 0) (0, a, 0) (0, 0, 0) (0, −b, 0) (−a, 0, 0) (0, 0, 0) (b, 0, 0)
⎞ ⎟ ⎟ ⎟, ⎠
a > b > 0, then, quasi-bicubic Coons surface represents a part of circular conical surface. (Fig.12)
6. Conclusion
π 2.
Then, quasi-bicubic Coons surface represents a part of circular conical surface. (Fig.10) Case 4.3 Given the information matrix ⎛ ⎜ ⎜ P =⎜ ⎝
a > 0, b =
(a, 0, 0) (a, 0, b) (0, 0, 1) (0, 0, 1) (0, a, 0) (0, a, b) (0, 0, 1) (0, 0, 1) (0, a, 0) (0, a, 0) (0, 0, 0) (0, 0, 0) (−a, 0, 0) (−a, 0, 0) (0, 0, 0) (0, 0, 0)
⎞ ⎟ ⎟ ⎟, ⎠
π 2,
then, quasi-bicubic Coons surface represents a part of circular cylindrical surface. (Fig.11) Case 4.4 Given the information matrix
(a + b, 0, 0) (0, a + b, 0) (0, a + b, 0) (−a − b, 0, 0)
We have established quasi-Hermite curves by trigonometric polynomials, which inherit similar properties of Hermite curves. The quasi-Hermite curves and quasi-Coons surfaces can be used to represent circular arcs, elliptic arcs, cycloid, sine curves, cosine curves and spherical surfaces, ellipsoid, cylinder, anchor ring , circular conical surface exactly, which indicate that the quasi-Hermite curves and surfaces are more powerful than traditional Hermite curves and Coons surfaces on expression of special curves and surfaces. So this paper maybe offer a useful tool in CAGD.
Acknowledgment This work was completed with the support by the National Natural Science Foundation of China under Grant No. 60773128 and in part by the Natural Science Foundation of Anhui Provincial Education Department under Grant No. KJ2009A123.
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