Geometric Information and Rational Parametrization of Nonsingular Cubic Blending Surfaces

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, 𝑥₁=𝑓₁𝑦₁,𝑦₂,𝑦₃𝑓₄𝑦₁,𝑦₂,𝑦₃,𝑥₂=𝑓₂𝑦₁,𝑦₂,𝑦₃𝑓₄𝑦₁,𝑦₂,𝑦₃,𝑥₃=𝑓₃𝑦₁,𝑦₂,𝑦₃𝑓₄𝑦₁,𝑦₂,𝑦₃, (1.1) 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 𝑓=𝑏₁𝑥₂−𝑑₂+𝑏₂𝑥₁−𝑑₁𝑥²₁+𝑥²₂+𝑥²₃−𝑟²−𝑥₁−𝑑₁𝑥₂−𝑑₂, (1.2) 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]).

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,𝑔_𝑖=𝑔_𝑖,21ℎ²₁+𝑔_𝑖,22ℎ²₂+𝑔_𝑖,23ℎ²₃+𝑔_𝑖,𝑚1ℎ₂ℎ₃+𝑔_𝑖,𝑚2ℎ₁ℎ₃+𝑔_𝑖,𝑚3ℎ₁ℎ₂+𝑔_𝑖,11ℎ₁+𝑔_𝑖,12ℎ₂+𝑔_𝑖,13ℎ₃+𝑔_𝑖,0(𝑖=1,2), (2.1) and obtained the following lemma.

Lemma 2.3 (see [12]). If the coefficients of 𝑔₁ and 𝑔₂ defined by (2.1) satisfy 𝑔_1,23𝑔_2,23=𝑔_1,13𝑔_2,13=𝑔_1,0𝑔_2,0𝑔=𝜅,_1,11−𝜅𝑔_2,11𝑔_1,12−𝜅𝑔_2,12=𝑔_1,𝑚2−𝜅𝑔_2,𝑚2𝑔_1,𝑚1−𝜅𝑔_2,𝑚1=𝜇, (2.2) where 𝜅 and 𝜇 are nonzero real numbers, then the cubic surface 𝑉(𝑓) defined by 𝑓=−𝜇ℎ₁+ℎ₂𝑔₁+𝑔_1,21−𝜅𝑔_2,21𝜇ℎ₁−ℎ₂+𝜇𝑔_1,𝑚3−𝜅𝑔_2,𝑚3ℎ₂+𝜇𝑔_1,𝑚2−𝜅𝑔_2,𝑚2ℎ₃+𝜇𝑔_1,11−𝜅𝑔_2,11ℎ²₁ (2.3) 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 𝑔_𝑖=𝑥²_𝑗+𝑥²_𝑘−𝑟²_𝑖,𝑥_𝑖−𝑥_𝑖0²+𝑥²_𝑗+𝑥²_𝑘−𝑟²_𝑖,𝑥²_𝑗+𝑥²_𝑘−𝑐²_𝑖𝑥_𝑖−𝑥_𝑖0²,𝑥²_𝑗+𝑥²_𝑘−2𝑝_𝑖𝑥_𝑖−𝑥_𝑖0,𝑥²_𝑗+𝑥²_𝑘−𝑐²_𝑖𝑥_𝑖−𝑥_𝑖0²−𝑙²_𝑖,𝑥²_𝑗+𝑥²_𝑘−𝑐²_𝑖𝑥_𝑖−𝑥_𝑖0²+𝑙²_𝑖,(𝑖=1,2), (2.4) 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 ℎ_𝑖=𝑥_𝑖+ℎ_𝑖0(𝑖=1,2), (2.5) 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 𝑄ℎ₂,ℎ₃;𝜆=𝐴_1,𝜆ℎ²₃+𝐴_2,𝜆ℎ²₂+𝐴_3,𝜆ℎ₂+𝐴_4,𝜆=0, (3.1) 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 letD(𝜆)𝐴∶=^2,𝜆12𝐴_3,𝜆12𝐴_3,𝜆𝐴_4,𝜆=𝐴_2,𝜆𝐴_4,𝜆−14𝐴²_3,𝜆, (3.2) which is a polynomial of degree 4 in 𝜆. Let 𝑏_𝑖 denote the coefficient of 𝜆^𝑖 in D(𝜆), 𝑖=0,1,…,4, respectively. In other words, D(𝜆)=𝑏₄𝜆⁴+𝑏₃𝜆³+𝑏₂𝜆²+𝑏₁𝜆+𝑏₀.

The determinant of the Hessian of 𝑄(ℎ₂,ℎ₃;𝜆) is D(𝜆)=𝐴_1,𝜆000𝐴_2,𝜆12𝐴_3,𝜆012𝐴_3,𝜆𝐴_4,𝜆=𝐴_1,𝜆D(𝜆). (3.3) Since D(𝜆) factors into the product of 𝐴_1,𝜆 and D(𝜆), 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 √√𝑎+I⋅𝑏=22𝑎²+𝑏²+𝑎+I⋅𝑎²+𝑏²√−𝑎,if𝑏>0or𝑏=0and𝑎<0,22𝑎²+𝑏²+𝑎−I⋅𝑎²+𝑏²−𝑎,if𝑏<0or𝑏=0and𝑎≥0. (3.4)

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, 𝑄ℎ₂,ℎ₃;𝜆=𝐴_2,𝜆ℎ₂+I⋅𝐴_1,𝜆ℎ₃+𝐴_3,𝜆2𝐴_2,𝜆𝐴_2,𝜆ℎ₂−I⋅𝐴_1,𝜆ℎ₃+𝐴_3,𝜆2𝐴_2,𝜆, (3.5) where 𝜆 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]).

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 𝐴_1,𝜆𝐴_2,𝜆=1+𝜆²𝐴²_1,𝜆>0. (3.6) 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 D𝜆^∗=0,𝐴_1,𝜆^∗𝐴_2,𝜆^∗≤0. (3.7) 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 D(𝜆) 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 D(𝜆) is of the form 𝐷₁𝐷=1,₂=−8𝑏₂𝑏₄+3𝑏²₃,𝐷₃=16𝑏₀𝑏₂−18𝑏²₁𝑏²₄+14𝑏₁𝑏₂𝑏₃−6𝑏₀𝑏²₃−4𝑏³₂𝑏₄+𝑏²₂𝑏²₃−3𝑏₁𝑏³₃,𝐷₄=256𝑏³₀𝑏³₄+144𝑏₀𝑏²₁𝑏₂−27𝑏⁴₁−192𝑏²₀𝑏₁𝑏₃−128𝑏²₀𝑏²₂𝑏²₄+18𝑏³₁𝑏₂𝑏₃−4𝑏²₁𝑏³₂+16𝑏₀𝑏⁴₂−80𝑏₀𝑏₁𝑏²₂𝑏₃−6𝑏₀𝑏²₁𝑏²₃+144𝑏²₀𝑏₂𝑏²₃𝑏₄−4𝑏₀𝑏³₂𝑏²₃+18𝑏₀𝑏₁𝑏₂𝑏³₃−27𝑏²₀𝑏⁴₃+𝑏²₁𝑏²₂𝑏²₃−4𝑏³₁𝑏³₃, (3.8) where 𝑏_𝑖 is the coefficient of 𝜆^𝑖 in D(𝜆), 𝑖=0,1,…,4, respectively.(1)If one of the following conditions holds: 𝐷₂<0∧𝐷₃<0∧𝐷₄>0;D₂≥0∧𝐷₃≤0∧𝐷₄𝐷>0;₂<0∧𝐷₃≥0;𝐷₂=0∧𝐷₃>0, (3.9) then the specific forms of nonsingular cubic blending surfaces are 𝐹₃ surfaces. (2)If one of the following conditions holds: 𝐷₂≤0∧𝐷₃<0∧𝐷₄≤0;𝐷₂=0∧𝐷₃=0∧𝐷₄𝐷<0;₂>0∧𝐷₃<0∧𝐷₄=0;𝐷₂>0∧𝐷₄<0, (3.10) then the specific forms of nonsingular cubic blending surfaces are 𝐹₄ surfaces. (3)If one of the following conditions holds: 𝐷₂>0∧𝐷₃>0∧𝐷₄≥0;𝐷₂≥0∧𝐷₃=0∧𝐷₄=0, (3.11) 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 𝑚^∗₁⋅𝑚^∗₂, respectively. (4)Construct a 3×3 complex matrix 𝑈 of the form 𝑚0𝑚₁0𝑚^∗𝑚^∗₁𝑘𝑚₂𝑘^∗𝑚^∗₂𝑝, (3.12) such that 𝑘 is a complex number; 𝑝 is a real plane; det(𝑈)=𝑓.(5)Find a real matrix 𝑈 which is equivalent to 𝑈 and satisfies det(𝑈)=𝑓.(6) Compute the Hilbert-Burch matrix 𝐻 from the equation𝑈𝑦₁,𝑦₂,𝑦₃^𝑇=𝐻𝑥₁,𝑥₂,𝑥₃,1^𝑇. (3.13) A rational parametrization of the surface 𝑉(𝑓) is 𝑥₁𝐻=−det₁det𝐻₄,𝑥₂=det𝐻₂det𝐻₄,𝑥₃𝐻=−det₃det𝐻₄, (3.14) 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 𝑈 of the form ℎ₁−𝛼ℎ₂0𝐴_2,𝛼ℎ₂+I⋅𝐴_1,𝛼ℎ₃+𝐴_3,𝛼2𝐴_2,𝛼0ℎ₁−𝛼^∗ℎ₂𝐴_2,𝛼∗ℎ₂−I⋅𝐴_1,𝛼∗ℎ₃+𝐴_3,𝛼∗2𝐴_2,𝛼∗𝑢₃₁𝑢₃₂𝑝₁ℎ₁+𝑝₂ℎ₂+𝑝₃ℎ₃+𝑝₀, (3.15) where 𝑢₃₁=𝑘𝐴_2,𝛼ℎ₂−I⋅𝐴_1,𝛼ℎ₃+𝐴_3,𝛼2𝐴_2,𝛼,𝑢₃₂=𝑘^∗𝐴_2,𝛼∗ℎ₂+I⋅𝐴_1,𝛼∗ℎ₃+𝐴_3,𝛼∗2𝐴_2,𝛼∗. (3.16)

Proposition 3.5. 𝑝₀,…,𝑝₃, and 𝑘 in 𝑈 are given by 𝑝₀=−𝜇𝜅𝑔_2,11,𝑝₁=−𝜇𝜅𝑔_2,21,𝑝₃=−𝜇𝜅𝑔_2,𝑚2,𝑝₂=−𝜇𝜅𝛼+𝛼^∗𝑔_2,21+𝜅𝑔_2,21−𝜇𝑔_2,𝑚3,1𝑘=𝐴_2,𝛼𝐴_3,𝛼∗−𝐴_2,𝛼∗𝐴_3,𝛼𝛼+𝛼^∗𝑝₀𝐴_2,𝛼∗−𝑝₂𝐴_3,𝛼∗+𝛼𝛼^∗𝑝₁𝐴_3,𝛼∗+𝑔_1,11−𝜇𝑔_1,12𝐴_2,𝛼∗−𝑔_1,𝑚3−𝜇𝑔_1,22𝐴_3,𝛼∗. (3.17)

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 2𝑦₁−2Re(𝛼)𝑦₁+Im(𝛼)𝑦₂+2Re𝐴_2,𝛼𝑦₃−2Im𝐴_1,𝛼𝑦₃𝐴Re^3,𝛼𝐴_2,𝛼𝑦₃12𝑦₂−Im(𝛼)𝑦₁−12Re(𝛼)𝑦₂+Im𝐴_2,𝛼𝑦₃Re𝐴_1,𝛼𝑦₃𝐴Im^3,𝛼𝐴_2,𝛼𝑦₃𝑝₁𝑦₃2Re𝑘𝐴_2,𝛼𝑦₁−Im𝑘𝐴_2,𝛼𝑦₂+𝑝₂𝑦₃𝑒₃₃𝑒₃₄, (3.18) where 𝑒₃₃=2Im𝑘𝐴_1,𝛼𝑦₁+Re𝑘𝐴_1,𝛼𝑦₂+𝑝₃𝑦₃,𝑒₃₄𝑘𝐴=Re^3,𝛼𝐴_2,𝛼𝑦₁−12𝑘𝐴Im^3,𝛼𝐴_2,𝛼𝑦₂+𝑝₀𝑦₃. (3.19)

Finally, the uniform rational parametric representation for the specific forms of type 𝐹₃ and 𝐹₄ is 𝑥₁𝐻=−det₁+ℎ₁₀det𝐻₄det𝐻₄,𝑥₂=det𝐻₂−ℎ₂₀det𝐻₄det𝐻₄,𝑥₃𝐻=−det₃det𝐻₄, (3.20) 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 𝑄ℎ₂,ℎ₃;𝜆=𝑎₁₁ℎ²₃+𝑎₁₂ℎ₃ℎ₂+𝑎₂₂ℎ²₂+𝑎₁₃ℎ₃+𝑎₂₃ℎ₂+𝑎₃₃=0, (4.1) where 𝑎₁₁=𝑔_1,23,𝑎1−𝜇𝜆₁₂=𝑔_1,𝑚11−𝜇𝜆+𝑔_1,𝑚2−𝜇𝜅𝑔_2,𝑚2𝜆𝑎𝜆,₂₂=1−𝜇𝜆𝜅𝑔_2,21𝜆²+𝑔_1,𝑚3𝜆+𝑔_1,22+𝜇𝑔_1,𝑚3−𝜅𝑔_2,𝑚3𝜆²,𝑎₁₃=𝑔_1,13,𝑎1−𝜇𝜆₂₃=𝑔_1,121−𝜇𝜆+𝑔_1,11−𝜇𝜅𝑔_2,11𝜆𝑎𝜆,₃₃=1−𝜇𝜆𝑔_1,0, (4.2)(iii)the determinant of the Hessian of 𝑄(ℎ₂,ℎ₃;𝜆) is D(𝜆)=𝑎₁₁12𝑎₁₂12𝑎₁₃12𝑎₁₂𝑎₂₂12𝑎₂₃12𝑎₁₃12𝑎₂₃𝑎₃₃=1−𝜇𝜆𝑇₄𝜆⁴+⋯, (4.3) 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 𝑎₁₁12𝑎₁₂12𝑎₁₂𝑎₂₂, (4.4) 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 𝑓𝑥₁,𝑥₂,𝑥₃=𝑥³₁+𝑥²₁𝑥₂+𝑥₁𝑥²₂+𝑥₁𝑥²₃+𝑥³₂+𝑥₂𝑥²₃−6𝑥²₁−12𝑥₁𝑥₂−6𝑥²₂−6𝑥²₃+21𝑥₁+21𝑥₂−18=0, (5.1) 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 𝑔₁=𝑥²₂+𝑥²₃−32𝑥₁−1²,ℎ₁=𝑥₁𝑔−3,₂=𝑥²₁+𝑥²₃−32𝑥₂−1²,ℎ₂=𝑥₂−3. (5.2)
First, we let ℎ₃=𝑥₃, and rewrite 𝑓(𝑥₁,𝑥₂,𝑥₃)=0 in the following form: 𝑓ℎ₁,ℎ₂,ℎ₃=ℎ₁+ℎ₂ℎ₂+3²+ℎ²₃−32ℎ₁+2²+52ℎ₁+52ℎ₂+12ℎ²₁=0. (5.3)

Geometric Information
𝑉(ℎ₁,ℎ₂) is a line on 𝑉(𝑓). The equation of the residual conic 𝒞 is 𝑄ℎ₂,ℎ₃;𝜆=(𝜆+1)ℎ²₃+(𝜆+1)𝜆²+1ℎ²₂+6𝜆²+1ℎ₂+3(𝜆+1)=0. (5.4) The determinant of the Hessian of 𝑄(ℎ₂,ℎ₃;𝜆) is D(𝜆)=(𝜆+1)D(𝜆)=−6(𝜆+1)𝜆²+1𝜆²−𝜆+1. (5.5) The discriminant sequence [𝐷₁,𝐷₂,𝐷₃,𝐷₄] of D(𝜆) satisfies that 𝐷₁=1,𝐷₂=−468<0,𝐷₃=6480>0,𝐷₄=559872>0. (5.6) 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 𝑈 as ℎ₁−I⋅ℎ₂0(1+I)ℎ₃+√3⋅I0ℎ₁+I⋅ℎ₂(1−I)ℎ₃−√13⋅I2⋅I⋅ℎ₃−√−13⋅I2⋅I⋅ℎ₃+√3⋅Iℎ₁+ℎ₂+6, (5.7) and the Hilbert-Burch matrix 𝐻 as 2𝑦₁𝑦₂2𝑦₃−2√3𝑦₃12𝑦₂−𝑦₁𝑦₃√3𝑦₃𝑦₃𝑦₃−12𝑦₂√3𝑦₁+6𝑦₃. (5.8) Finally, the rational parametrization of 𝑉(𝑓) is 𝑥₁=3𝑦²₁𝑦₂−2√3𝑦²₁𝑦₃−12𝑦₁𝑦²₃+(3/4)𝑦³₂−√3/2𝑦²₂𝑦₃−4√3𝑦³₃𝑦²₁𝑦₂+(1/4)𝑦³₂+2𝑦₂𝑦²₃,𝑥₂=3𝑦²₁𝑦₂+2√3𝑦²₁𝑦₃+12𝑦₁𝑦²₃+(3/4)𝑦³₂+√3/2𝑦²₂𝑦₃+4√3𝑦³₃𝑦²₁𝑦₂+(1/4)𝑦³₂+2𝑦₂𝑦²₃,𝑥₃=2√3𝑦³₁+12𝑦²₁𝑦₃+√3/2𝑦₁𝑦²₂+4√3𝑦₁𝑦²₃+3𝑦²₂𝑦₃𝑦²₁𝑦₂+(1/4)𝑦³₂+2𝑦₂𝑦²₃. (5.9)

Example 5.2. We are given a nonsingular cubic blending surface 𝑉(𝑓) defined by the implicit equation 𝑓𝑥₁,𝑥₂,𝑥₃=𝑥³₁+4𝑥²₁𝑥₂+4𝑥²₁𝑥₃+6𝑥₁𝑥²₂+12𝑥₁𝑥₂𝑥₃+7𝑥₁𝑥²₃+4𝑥³₂+12𝑥²₂𝑥₃+14𝑥₂𝑥²₃+6𝑥³₃−2𝑥²₁−8𝑥₁𝑥₂−6𝑥₁𝑥₃−8𝑥²₂−12𝑥₂𝑥₃−8𝑥²₃−𝑥₁−2𝑥₂−10𝑥₃+4=0, (5.10) 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 𝑔₁=𝑥²₁+2𝑥₁𝑥₂+2𝑥₁𝑥₃+2𝑥²₂+4𝑥₂𝑥₃+3𝑥²₃−6𝑥₁−4𝑥₂−2𝑥₃𝑔+7,₂=𝑥²₁+2𝑥₁𝑥₂+2𝑥₁𝑥₃+2𝑥²₂+4𝑥₂𝑥₃+3𝑥²₃+2𝑥₁−4𝑥₂−2𝑥₃ℎ+7,₁=𝑥₁+𝑥₂+𝑥₃−2,ℎ₂=𝑥₂+𝑥₃−2. (5.11) First, we let ℎ₃=𝑥₃ and rewrite 𝑓(𝑥₁,𝑥₂,𝑥₃)=0 in the following form: 𝑓ℎ₁,ℎ₂,ℎ₃=ℎ₁+ℎ₂ℎ₂+3²+ℎ₃+1²+ℎ₁−1²−4+8ℎ²₁=0. (5.12)

Geometric Information
𝑉(ℎ₁,ℎ₂) is a line on 𝑉(𝑓). The equation of the residual conic 𝒞 is 𝑄ℎ₂,ℎ₃;𝜆=(𝜆+1)ℎ²₃+(𝜆+1)𝜆²+1ℎ²₂+2(𝜆+1)ℎ₃+6𝜆²+4𝜆+6ℎ₂+7(𝜆+1)=0. (5.13) The determinant of the Hessian of 𝑄(ℎ₂,ℎ₃;𝜆) is D(𝜆)=0()𝜆𝜆+10𝜆+1𝜆+1²+13𝜆²+2𝜆+3𝜆+13𝜆²()(+2𝜆+37𝜆+1=−𝜆+1)3𝜆⁴+10𝜆²+3. (5.14) The discriminant sequence [𝐷₁,𝐷₂,𝐷₃,𝐷₄] of −(3𝜆⁴+10𝜆²+3) satisfies that 𝐷₁=1,𝐷₂=−240<0,𝐷₃=−7680<0,𝐷₄=589824>0. (5.15) According to Lemma 3.3 and the coefficients of ℎ²₃ and ℎ²₂ 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, √3⋅I and −√3⋅I. We can construct the complex matrix 𝑈 as ℎ₁−√3⋅I⋅ℎ₂01+√3⋅Iℎ₃+√2ℎ₂+1−√6⋅I0ℎ₁+√3⋅I⋅ℎ₂1−√3⋅Iℎ₃+√2ℎ₂+1+√6⋅I𝑢₃₁𝑢₃₂ℎ₁+ℎ₂+6, (5.16) where 𝑢₃₁=√36⋅I⋅ℎ₃−√2ℎ₂+1+√6⋅I,𝑢₃₂√=−36⋅I⋅ℎ₃−√2ℎ₂+1−√6⋅I, (5.17) and the Hilbert-Burch matrix 𝐻 as 2𝑦₁√3𝑦₂+2√2𝑦₃2𝑦₃2+6√2𝑦₃12𝑦₂−√3𝑦₁+√6𝑦₃√3𝑦₃1−√6𝑦₃𝑦₃√66𝑦₂+𝑦₃−√36𝑦₂−√2𝑦₁−√36𝑦₂+6𝑦₃. (5.18) Finally, the rational parametrization of 𝑉(𝑓) is 𝑥₁∶𝑥₂∶𝑥₃=𝑌₁𝑌₄∶𝑌₂𝑌₄∶𝑌₃𝑌₄, (5.19) where 𝑌₁=4√6𝑦²₁𝑦₃−24√3𝑦₁𝑦²₃+√6𝑦²₂𝑦₃+4𝑦₂𝑦²₃+16√6𝑦³₃,𝑌₂=2√6𝑦³₁+3𝑦²₁𝑦₂−2√6+16√3𝑦²₁𝑦₃+√62𝑦₁𝑦²₂+4−6√2𝑦₁𝑦₂𝑦₃+12√3+20√6𝑦₁𝑦²₃+34𝑦³₂+√62−2√3𝑦²₂𝑦₃+6−6√2𝑦₂𝑦²₃−8√6+16√3𝑦³₃,𝑌₃=−2√6𝑦³₁−𝑦²₁𝑦₂+16√3𝑦²₁𝑦₃−√62𝑦₁𝑦²₂+2√2−4𝑦₁𝑦₂𝑦₃−20√6𝑦₁𝑦²₃−14𝑦³₂+2√√3−63𝑦²₂𝑦₃+6√2−4𝑦₂𝑦²₃+16√3𝑦³₃,𝑌₄=𝑦²₁𝑦₂−2√2𝑦₁𝑦₂𝑦₃+14𝑦³₂+√63𝑦²₂𝑦₃+4𝑦₂𝑦²₃. (5.20)

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).