Volume 2011, Issue 1 349315

Research Article

Open Access

Geometric Information and Rational Parametrization of Nonsingular Cubic Blending Surfaces

Corresponding Author

Minghao Guo

[email protected]

Key Laboratory of Symbolic Computation and Knowledge Engineering (Ministry of Education), School of Mathematics, Jilin University, Changchun 130012, China jlu.edu.cn

Search for more papers by this author

Tieru Wu,

Tieru Wu

Key Laboratory of Symbolic Computation and Knowledge Engineering (Ministry of Education), School of Mathematics, Jilin University, Changchun 130012, China jlu.edu.cn

Search for more papers by this author

Shugong Zhang,

Shugong Zhang

Key Laboratory of Symbolic Computation and Knowledge Engineering (Ministry of Education), School of Mathematics, Jilin University, Changchun 130012, China jlu.edu.cn

Search for more papers by this author

Minghao Guo,

Corresponding Author

Minghao Guo

[email protected]

Key Laboratory of Symbolic Computation and Knowledge Engineering (Ministry of Education), School of Mathematics, Jilin University, Changchun 130012, China jlu.edu.cn

Search for more papers by this author

Tieru Wu,

Tieru Wu

Key Laboratory of Symbolic Computation and Knowledge Engineering (Ministry of Education), School of Mathematics, Jilin University, Changchun 130012, China jlu.edu.cn

Search for more papers by this author

Shugong Zhang,

Shugong Zhang

Key Laboratory of Symbolic Computation and Knowledge Engineering (Ministry of Education), School of Mathematics, Jilin University, Changchun 130012, China jlu.edu.cn

Search for more papers by this author

First published: 18 September 2011

https://doi.org/10.1155/2011/349315

Academic Editor: Ke Chen

Share a link

Email
Wechat
Bluesky

Abstract

The techniques for parametrizing nonsingular cubic surfaces have shown to be of great interest in recent years. This paper is devoted to the rational parametrization of nonsingular cubic blending surfaces. We claim that these nonsingular cubic blending surfaces can be parametrized using the symbolic computation due to their excellent geometric properties. Especially for the specific forms of these surfaces, we conclude that they must be F₃, F₄, or F₅ surfaces, and a criterion is given for deciding their surface types. Besides, using the algorithm proposed by Berry and Patterson in 2001, we obtain the uniform rational parametric representation of these specific forms. It should be emphasized that our results in this paper are invariant under any nonsingular real projective transform. Two explicit examples are presented at the end of this paper.

1. Introduction

In the areas of computer graphics and geometric modeling, we may sometimes encounter the problems about cubic surfaces. A real nonsingular cubic surface V(f) can be defined either by the implicit form, that is, the real zeros of a polynomial equation f(x₁, x₂, x₃) = 0 of degree 3 in R³, or by the rational parametric form, that is,

()

where f₁, …, f₄ are homogeneous polynomials and y₁, y₂, and y₃ are parameters. Both forms are appropriate for solving different types of problems. For example, in the geometric modeling systems, the implicit form offers many advantages when performing geometric operations like union, intersection, blending, and warping. Besides, the rational parametric form is preferred in the free-form surface modeling, because it guarantees manifoldness and provides a way of generating points within the surface. So, the automatic transition between these two forms is very important.

Generating a rational parametric representation of an algebraic surface (called parametrization for short) is always difficult, even impossible sometimes, while the reverse process called implicitization is always solvable (such as using Gröbner basis method). Most of the parametrization algorithms for nonsingular cubic surfaces are based on geometric information of nonsingular cubic surfaces, such as the existence of 27 lines on nonsingular cubic surfaces [1]. In 1987, Sederberg and Snively parametrized cubic surfaces in terms of biquadratic polynomials using pairs of skew lines on the surfaces in [2]. This method is further developed by Bajaj et al. in [3] and Polo-Blanco and Top in [4]. In 2001, Berry and Patterson unified the implicitization and parametrization of a nonsingular cubic surface using Hilbert-Burch theorem in [5]. In 2007, Chen et al. presented an alternative method for parametrizing quadric and cubic surfaces based on the theory of μ-basis in [6].

This paper is devoted to nonsingular cubic blending surfaces in [7–9], which meet two quadratic surfaces with G¹ continuity. The blending surfaces have attracted more and more attention in recent years and have been used in particular for filling surface holes, smoothing corners and edges, and making computer animation. Wu and Cheng in [10] discussed the parametrization of the special blending surfaces defined by

()

where b₁, b₂, d₁, d₂, and r are real numbers. We further develop their results in a more general way in this paper. We consider successively two classes of these nonsingular cubic blending surfaces: the specific forms and the general forms.

(i)
For the specific forms, we conclude that they must be F₃, F₄, or F₅ surfaces, and present a criterion to decide their surface types. Besides, using the algorithm proposed in [5], we obtain their uniform rational parametric representation.
(ii)
For the general forms, although they do not have the analogous concise properties as the specific forms do, we can still come to the conclusion that their rational parametrizations can be computed using the symbolic computation.

Additionally, it should be pointed out that our results in this paper are invariant under any nonsingular real projective transform.

The rest of the paper is organized as follows. Section 2 recalls some geometric information of nonsingular cubic surfaces and introduces the constructions for nonsingular cubic blending surfaces. Section 3 is devoted to the specific forms of nonsingular cubic blending surfaces, and we study in detail the geometric information on them and their uniform rational parametric representation. Section 4 discusses the analogous geometric properties of the general forms of nonsingular cubic blending surfaces as the specific forms. Two explicit examples are presented in Section 5. Finally, we conclude the paper in Section 6.

2. Notations and Preliminaries

For the convenience of applications, we always consider the problem in R[x₁, x₂, x₃]. Let V(f) denote the algebraic surface determined by the polynomial equation f(x₁, x₂, x₃) = 0. Assume that F is a set of polynomials, and denote by V(F) the set of solutions of the system of all polynomials in F. Let 〈g, h〉 be the ideal generated by the polynomials g and h.

2.1. Geometric Information of Nonsingular Cubic Surfaces

In 1849, Cayley and Salmon published the famous theorem that there are 27 lines lying completely on a nonsingular cubic surface. Every line on a nonsingular cubic surface is met by ten others. A plane containing three of the lines is called a tritangent plane. There are 45 such planes on a nonsingular cubic surface. Nonsingular cubic surfaces can be divided into 5 species F₁, F₂, …, F₅ with respect to the number of real lines and real components. See Table 1 for more details.

Table 1. Some geometric information of nonsingular cubic surfaces (derived from [14]).

Surface type	F₁	F₂	F₃	F₄	F₅
Number of real lines	27	15	7	3	3
Number of real tritangent planes	45	15	5	7	13
Number of real components	1	1	1	1	2

Remark 2.1. Most of the parametrization algorithms lose effectiveness for F₅ surfaces, since the F₅ surfaces have no real one-to-one parametrization. Thus, we will not address nonsingular cubic blending surfaces of type F₅ in this paper.

The computation of lines on a nonsingular cubic surface is our starting point for the analysis and parametrization of the surface, since we could know other geometric information of the surface from Table 1 once the number of real lines on the surface is verified.

Sederberg showed how to compute lines on a nonsingular cubic surface in [11]. Assume that we have a nonsingular cubic surface given by its implicit equation f(x₁, x₂, x₃) = 0. The parametrization of a line with unknown coefficients is l(t): = (t, x₂₀ + x₂₁t, x₃₀ + x₃₁t). Substituting the parametric equation of the line into the implicit equation of the surface yields an equation f(l(t)) of degree 3 in the parameter t. If this equation is identically zero, that is, all the coefficients vanish simultaneously, it guarantees that the line lies entirely on the surface. In this way, the problem of finding a line on a nonsingular cubic surface is transformed into the problem of solving a system of four nonlinear equations in four unknowns. We can find at least one solution of this system using Gröbner basis method or some other techniques.

If one line l(t) is known, we can use this line to find other lines on the surface. We take a pencil of planes through l(t) and intersect it with the surface. The intersection consists of l(t) and a residual conic 𝒞. The condition for 𝒞 to degenerate into a pair of lines is that the determinant of its Hessian vanishes. The determinant of the Hessian of 𝒞 is a polynomial of degree 5. Each of the 5 roots corresponds to a plane in which the residual intersection is degenerate. After computing the roots of this polynomial, it is possible to get other lines on the surface.

Remark 2.2. It is relatively expensive to compute one line on a nonsingular cubic surface. Besides, since quintic equations can only be solved numerically, not analytically, we could not obtain exact lines on the surface. However, nonsingular cubic blending surfaces in this paper just avoid these obstacles.

2.2. Nonsingular Cubic Blending Surfaces

In this subsection, we briefly introduce the constructions of nonsingular cubic blending surfaces. Suppose that we are given two quadratic surfaces V(g₁), V(g₂), and two associated clipping planes V(h₁), V(h₂). Without loss of generality, we assume that V(h₁) and V(h₂) intersect. Denote h₃ = x_i, where i ∈ {1,2, 3} such that V(h₁, h₂, h₃) is an isolated point in R³.

Wu et al. in [12] concluded that if the coefficients of g₁, g₂, h₁, and h₂ satisfy some certain conditions, then there exists a cubic blending surface V(f) which is tangent to V(g_i) along V(g_i, h_i) (i = 1,2). They expanded g₁ and g₂ with respect to h₁, h₂, and h₃, that is,

()

and obtained the following lemma.

Lemma 2.3 (see [12].)If the coefficients of g₁ and g₂ defined by (2.1) satisfy

()

where κ and μ are nonzero real numbers, then the cubic surface V(f) defined by

()

meets V(g_i) with G¹ continuity along V(g_i, h_i) (i = 1,2).

Now, we consider successively two classes of these nonsingular cubic blending surfaces:

(i)
the nonsingular cubic surfaces V(f) defined by (2.3), which are constructed under the above general assumptions, are called the general forms,
(ii)
if we add some extra restrictions to V(g₁), V(g₂), V(h₁), and V(h₂):
- (1)
  let V(g₁) and V(g₂) be quadratic surfaces defined by
  ()
  where {j, k} = {1,2, 3}∖{i} and r_i, x_i0, c_i, p_i, and l_i are nonzero real numbers;
- (2)
  let V(h₁) and V(h₂) be associated clipping planes defined by
  ()
  where h₁₀ and h₂₀ are nonzero real numbers,

then the nonsingular cubic surfaces V(f) defined by (2.3), which are constructed under this specific assumptions, are called the specific forms.

For these two classes of nonsingular cubic blending surfaces, our question is whether or not they have any good geometric property or any fast/exact parametrization algorithm.

3. Nonsingular Cubic Blending Surfaces: The Specific Forms

In this section, we mainly analyze some geometric information of the specific forms which is useful for the parametrization process and compute their uniform rational parametric representation.

3.1. Geometric Information

3.1.1. Lines on the Surfaces

According to the defining polynomial (2.3) and the results obtained in [13], we can easily find that V(h₁, h₂) is just one line on the surfaces.

Let h₃ = x₃. As showed in Section 2.1, we substitute the equation h₁ − λh₂ = 0 (λ is a parameter) of the pencil of planes into f(h₁, h₂, h₃) = 0 and obtain

()

of the residual conic 𝒞 after cancelling the factor h₂. It is easy to demonstrate the following proposition by checking each pair of V(g₁) and V(g₂) defined by (2.4).

Proposition 3.1. A_1,λ, …, A_4,λ in (3.1) satisfy that

(1)
A_1,λ, A_2,λ, A_3,λ, A_4,λ are polynomials of degree 1, 3, 2, 1 in λ, respectively,
(2)
A_2,λ = (λ² + 1)A_1,λ,
(3)
A_1,λ and A_4,λ differ by a nonzero constant.

We let

()

which is a polynomial of degree 4 in λ. Let b_i denote the coefficient of λⁱ in

, i = 0,1, …, 4, respectively. In other words,

The determinant of the Hessian of Q(h₂, h₃; λ) is

()

Since D(λ) factors into the product of A_1,λ and

, now it is possible to solve the equation D(λ) = 0 analytically by some fixed formula. So, we can find all the exact lines on the surfaces by repeating the same process on the new lines that have been found.

3.1.2. A Classification of the Surfaces

Let a + I · b be an arbitrary complex number, where a, b ∈ R and I denotes the imaginary unit. We formulate

()

The real and complex roots of the equation D(λ) = 0 correspond to the real and complex tritangent planes of the surfaces, respectively. In the complex tritangent planes, Q(h₂, h₃; λ) factors into two complex lines, that is,

()

where

denotes a complex root of D(λ) = 0. However, in the real tritangent planes, some of the factorizations of Q(h₂, h₃; λ) are real, and some are complex.

Segre in [14] classifies nonsingular cubic surfaces using Table 2, according to the number of real roots of D(λ) = 0 and whether or not the factorization of Q(h₂, h₃; λ) in the corresponding tritangent plane is real or complex.

Table 2. A classification of nonsingular cubic surfaces (derived from [14]).

Surface type	F₁	F₃	F₅	F₂	F₄	F₃
Number of real roots of D(λ) = 0	5	5	5	3	3	1
Number of real factorizations	5	3	1	3	1	1

Using Table 2 and Proposition 3.1, we can arrive at the following theorem.

Theorem 3.2. The specific forms of nonsingular cubic blending surfaces must be F₃, F₄, or F₅ surfaces.

Proof. Let λ₁ be the root of A_1,λ = 0. For all λ ∈ R∖{λ₁}, we have

()

The factorization of Q(h₂, h₃; λ) in a real tritangent plane is real if and only if there exists a λ^* ∈ R such that

()

Obviously, λ₁ is the only one which satisfies (3.7). Thus, the number of real factorizations in Table 2 is at most 1, and this completes the proof.

In what follows, we will give a complete solution for the surface type problem of the surfaces. The main tool we will use is the complete discrimination system for any polynomial with real coefficients proposed by Yang et al. in [15].

Lemma 3.3 (see [15].)Given a polynomial G(x) = a₀x^m + a₁x^m−1 + ⋯+a_m ∈ R[x], if the number of sign changes in the revised sign list with respect to G(x) is ν, then the number of pairs of the distinct conjugate imaginary roots of G(x) is ν. Moreover, if the number of nonvanishing members in the revised sign list with respect to G(x) is η, then the number of the distinct real roots is η − 2ν.

Applying Lemma 3.3 to defined in (3.2) and according to Table 2 and Theorem 3.2, we can eventually arrive at the following proposition.

Proposition 3.4. The discriminant sequence [D₁, D₂, D₃, D₄] of is of the form

()

where b_i is the coefficient of λⁱ in

, i = 0,1, …, 4, respectively.

(1)
If one of the following conditions holds:
()
then the specific forms of nonsingular cubic blending surfaces are F₃ surfaces.
(2)
If one of the following conditions holds:
()
then the specific forms of nonsingular cubic blending surfaces are F₄ surfaces.
(3)
If one of the following conditions holds:
()
then the specific forms of nonsingular cubic blending surfaces are F₅ surfaces.

The above symbol “∧” indicates logical conjunction, which means that A∧B holds if and only if both A and B hold simultaneously.

3.2. The Uniform Rational Parametric Representation

Since our results of this subsection are based on the algorithm proposed by Berry and Patterson in [5], we first outline the strategy of their algorithm.

Suppose that an F_i (i = 3,4) surface is given by f(x₁, x₂, x₃) = 0.

(1)
Find a line l on the surface.
(2)
Find a pair of complex conjugate tritangent planes through l, and denote them by m and m^*.
(3)
Compute the factorizations of the residual conic 𝒞 (mentioned in Section 2.1) in m and m^*, and denote them by m₁ · m₂ and , respectively.
(4)
Construct a 3 × 3 complex matrix of the form
()
such that k is a complex number; p is a real plane; .
(5)
Find a real matrix U which is equivalent to and satisfies det (U) = f.
(6)
Compute the Hilbert-Burch matrix H from the equation

()

A rational parametrization of the surface V(f) is

()

where H_i are the 3 × 3 submatrices of H by cancelling the ith column.

Now, using the notations and conclusions in Section 3.1, we apply the algorithm to the specific forms of type F₃ and F₄.

According to Theorem 3.2, we can only choose a pair of complex conjugate roots, denoted by α and α^*, of the equation D(λ) = 0. In the two corresponding complex tritangent planes, the factorizations of Q(h₂, h₃; α) and Q(h₂, h₃; α^*) must be the form (3.5). Thus, we can construct the uniform 3 × 3 complex matrix

of the form

()

where

()

Proposition 3.5. p₀, …, p₃, and k in are given by

()

Similarly, the uniform Hilbert-Burch matrix H can be constructed. We let Re(z) and Im (z) denote the real part and the imaginary part of the complex number z, respectively.

Proposition 3.6. The uniform Hilbert-Burch matrix H is of the form

()

where

()

Finally, the uniform rational parametric representation for the specific forms of type F₃ and F₄ is

()

where H_i are the 3 × 3 submatrices of (3.18) by cancelling the ith column.

4. Nonsingular Cubic Blending Surfaces: The General Forms

In this section, we will see that for the general forms, their parametrizations can also be computed using the symbolic computation. However, their parametrizations are too complicated to be written down in a uniform form.

Similarly as the analysis in Section 3, we can easily come to the following results. According to the defining polynomial (2.3),

(i)
V(h₁, h₂) is just one line on the surfaces,
(ii)
the equation of the residual conic 𝒞 is
()
where
()
(iii)
the determinant of the Hessian of Q(h₂, h₃; λ) is
()
where the second factor in the above product is some certain polynomial of degree 4 in λ. Thus we can solve the equation D(λ) = 0 analytically,
(iv)
at each root of D(λ) = 0, we discuss the sign of the minor
()
of (4.3). Then, we can determine the surface type and other geometric information using Table 2 and Table 1.

Because of the existence of the mixed term h₃h₂ in (4.1), the uniform representation of the factorization of (4.1) is very complicated, and so, we omit it. Finally, using the algorithm in [5], the rational parametrization of the general forms can be achieved without approximate calculation.

5. Example

Example 5.1. We are given a nonsingular cubic blending surface V(f) defined by the implicit equation

()

and we also know that V(f) meets the circular cone V(g_i) with G¹ continuity along V(g_i, h_i), i = 1,2, where

()

First, we let h₃ = x₃, and rewrite f(x₁, x₂, x₃) = 0 in the following form:

()

Geometric Information V(h₁, h₂) is a line on V(f). The equation of the residual conic 𝒞 is

()

The determinant of the Hessian of Q(h₂, h₃; λ) is

()

The discriminant sequence [D₁, D₂, D₃, D₄] of

satisfies that

()

According to Proposition 3.4, we know that V(f) is an F₃ surface and other information of V(f) from Table 1.

Rational Parametrization We choose a pair of complex conjugate roots of D(λ) = 0, I and −I. As mentioned in Section 3.2, we construct the complex matrix as

()

and the Hilbert-Burch matrix H as

()

Finally, the rational parametrization of V(f) is

()

Example 5.2. We are given a nonsingular cubic blending surface V(f) defined by the implicit equation

()

and we also know that V(f) meets the quadratic V(g_i) with G¹ continuity along V(g_i, h_i), i = 1,2, where

()

First, we let h₃ = x₃ and rewrite f(x₁, x₂, x₃) = 0 in the following form:

()

Geometric Information V(h₁, h₂) is a line on V(f). The equation of the residual conic 𝒞 is

()

The determinant of the Hessian of Q(h₂, h₃; λ) is

()

The discriminant sequence [D₁, D₂, D₃, D₄] of −(3λ⁴ + 10λ² + 3) satisfies that

()

According to Lemma 3.3 and the coefficients of

and

in Q(h₂, h₃; λ), we know that V(f) is an F₃ surface and other information of V(f) from Table 1.

Rational Parametrization We choose a pair of complex conjugate roots of D(λ) = 0, and . We can construct the complex matrix as

()

where

()

and the Hilbert-Burch matrix H as

()

Finally, the rational parametrization of V(f) is

()

where

()

6. Conclusions

This paper is concerned with nonsingular cubic blending surfaces. We mainly discuss geometric information and rational parametrization of them. Since their underlying geometric properties, the rational parametrization can be implemented using the symbolic computation, while for general nonsingular cubic surfaces, it has to resort to floating point numbers. In the future, we will focus on the analysis of singular cubic blending surfaces and nonsingular cubic blending surfaces of type F₅.

Acknowledgments

The authors express their deep gratitude to the referee for useful suggestions and comments which have improved the paper. This work is partly supported by National Science Foundation of China (Grant no. 60973155).

References

1 Henderson A., The Twenty-Seven Lines Upon the Cubic Surface, 1911, Hafner Publishing, New York, NY, USA, 0119139.
Google Scholar
2 Sederberg T. and Snively J., R. Martin, Parametrization of cubic algebraic surfaces, The Mathematics of Surfaces II, 1987, 11, Clarendon Press, Oxford, UK, 299–319, 927342, ZBL0102.37805.
Google Scholar
3 Bajaj C., Holt R., and Netravali A., Rational parameterizations of nonsingular real cubic-surfaces, ACM Transactions on Graphics. (1998) 17, no. 1, 1–31, https://doi.org/10.1145/269799.269800, ZBL0676.14001.
10.1145/269799.269800
Web of Science® Google Scholar
4 Polo-Blanco I. and Top J., A remark on parameterizing nonsingular cubic surfaces, Computer Aided Geometric Design. (2009) 26, no. 8, 842–849, https://doi.org/10.1016/j.cagd.2009.06.001, 2573062.
10.1016/j.cagd.2009.06.001
Web of Science® Google Scholar
5 Berry T. and Patterson R., Implicitization and parametrization of nonsingular cubic surfaces, Computer Aided Geometric Design. (2001) 18, no. 8, 723–738, https://doi.org/10.1016/S0167-8396(01)00048-6, 1857994, ZBL1205.65101.
10.1016/S0167-8396(01)00048-6
Web of Science® Google Scholar
6 Chen F., Shen L., and Deng J., Implicitization and parametrization of quadratic and cubic surfaces by μ-bases, Computing. (2007) 79, no. 2–4, 131–142, https://doi.org/10.1007/s00607-006-0192-0, 2295516, ZBL0983.68221.
10.1007/s00607-006-0192-0
Web of Science® Google Scholar
7 Wu T., Gao W., and Feng G., Blending of implicit algebraic surfaces, Proceedings of the ASCM, August 1995, Beijing, China, 125–131.
Google Scholar
8 Wu T. and Zhou Y., On blending of several quadratic algebraic surfaces, Computer Aided Geometric Design. (2000) 17, no. 8, 759–766, https://doi.org/10.1016/S0167-8396(00)00023-6, 1781086.
10.1016/S0167-8396(00)00023-6
Web of Science® Google Scholar
9 Lei N., Blending of two quadratic surfaces and Wu Wen-tsün′s formulae, Ph.D. thesis, 2002, Jilin University, Jilin, China, ZBL0948.68182.
Google Scholar
10 Wu T. and Cheng H., Basic lines, axes and geometric modeling on implicit algebraic surfaces, Journal of Computational and Applied Mathematics. (2006) 195, no. 1-2, 212–219, https://doi.org/10.1016/j.cam.2005.03.084, 2244637.
10.1016/j.cam.2005.03.084
Web of Science® Google Scholar
11 Sederberg T., Techniques for cubic algebraic surfaces II, IEEE Computer Graphics and Applications. (1990) 10, no. 5, 12–21, https://doi.org/10.1109/38.59032.
10.1109/38.59032
Google Scholar
12 Wu T., Lei N., and Cheng J., Wu Wen-tsün′s formulae for the blending of pipe surfaces, Northeastern Mathematical Journal. (2001) 17, no. 4, 383–386, 1968347.
Google Scholar
13 Li C., A software system on blending of pipe surfaces, M.S. thesis, 2000, Jilin University, Jilin, China, ZBL1031.65030.
Google Scholar
14 Segre B., The Non-Singular Cubic Surfaces, 1942, Oxford University Press, Oxford, UK, 0008171, ZBL1097.65040.
Google Scholar
15 Yang L., Zeng J., and Hou X., A complete discrimination system for polynomials, Science in China. (1996) E39, no. 6, 628–646, 1438754.
Google Scholar

All articles

Geometric Information and Rational Parametrization of Nonsingular Cubic Blending Surfaces

Abstract

1. Introduction

2. Notations and Preliminaries

2.1. Geometric Information of Nonsingular Cubic Surfaces

2.2. Nonsingular Cubic Blending Surfaces

3. Nonsingular Cubic Blending Surfaces: The Specific Forms

3.1. Geometric Information

3.1.1. Lines on the Surfaces

3.1.2. A Classification of the Surfaces

3.2. The Uniform Rational Parametric Representation

4. Nonsingular Cubic Blending Surfaces: The General Forms

5. Example

6. Conclusions

Acknowledgments

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Geometric Information and Rational Parametrization of Nonsingular Cubic Blending Surfaces

Abstract

1. Introduction

2. Notations and Preliminaries

2.1. Geometric Information of Nonsingular Cubic Surfaces

2.2. Nonsingular Cubic Blending Surfaces

3. Nonsingular Cubic Blending Surfaces: The Specific Forms

3.1. Geometric Information

3.1.1. Lines on the Surfaces

3.1.2. A Classification of the Surfaces

3.2. The Uniform Rational Parametric Representation

4. Nonsingular Cubic Blending Surfaces: The General Forms

5. Example

6. Conclusions

Acknowledgments

References

References

Related

Information