Explicit Continuity Conditions for G 1 Connection of S- λ Curves and Surfaces

: The S- λ model is one of the most useful tools for shape designs and geometric representations in computer-aided geometric design (CAGD), which is due to its good geometric properties such as symmetry, shape adjustable property. With the aim to solve the problem that complex S- λ curves and surfaces cannot be constructed by a single curve and surface, the explicit continuity conditions for G 1 connection of S- λ curves and surfaces are investigated in this paper. On the basis of linear independence and terminal properties of S- λ basis functions, the conditions of G 1 geometric continuity between two adjacent S- λ curves and surfaces are proposed, respectively. Modeling examples imply that the continuity conditions proposed in this paper are easy and e ﬀ ective, which indicate that the S- λ curves and surfaces can be used as a powerful supplement of complex curves and surfaces design in computer aided design / computer aided manufacturing (CAD / CAM)


Introduction
In the design of complex curves and surfaces, the basis functions of geometric models determine the properties of the curves and surfaces constructed. The basis functions of common models in CAGD include the Bernstein basis function, the Poisson basis function, the negative Bernstein basis function, the B-spline basis function and so on, which are closely related to the probability distribution-especially the discrete probability distribution [1,2]. For example, the Bernstein basis function contained in the Bézier model is taken from binomial distribution, and the B-spline basis function used in the B-spline model is closely related to some stochastic processes [3]. The negative Bernstein basis function which was proposed by Goldman [4] corresponds to the negative binomial distribution, and the Poisson basis function advanced by Goldman and Morin [5] corresponds to the Poisson distribution. The S-λ basis functions proposed by [6,7] are a kind of discrete probability modeling defined by generating function and transformation factor combined with probability convolution. By taking different generating functions and transformation factors, different S-λ basis functions can be obtained, such as the Bernstein basis function, the Poisson basis function, the negative Bernstein basis function, etc. Therefore, we can study these well-known basis functions mentioned above in a unified way by S-λ basis function. Furthermore, S-λ curves and surfaces include Bézier curves and surfaces, Poisson curves and surfaces, rational Bézier curves and surfaces as well as many other curves and surfaces. In conclusion, the studies

Univariate S-λ Basis Functions and S-λ Curve
For any given positive integer m, let A j m j=0 be a given sequence of positive numbers. S(x) = m j=0 A j x j is the generating function with radius of convergence R, which satisfies S(0) = 1. Let λ(x) be Mathematics 2020, 8,1359 3 of 18 a continuous strictly monotone increasing function satisfying λ(0) = 0 and mapping [0, R 1 ) onto [0, R), where R and R 1 may be +∞. Furthermore, λ(x) is called the transformation factor. For x ∈ [0, R 1 ), let Then we have P 1, j (x) ∈ C[0, R 1 ) as well as m j=0 P 1, j (x) = 1. According to the theory above, we give the following definition of S-λ basis functions. Definition 1. Let random variable ξ be an integer valued and satisfy . Its generating function is Moreover, let {ξ i } ∞ i=1 be a sequence of independent random variables with the same distribution as ξ. Let η n = n i=1 ξ i , according to the convolution formula of probability, we have , j = 0, 1, 2, . . . , mn.
It can be proved that S-λ basis defined by (5) possess many properties similar to those of Bernstein basis, such as non-negativity, partition of unity, linear independence, degree elevation, etc.

Definition 2.
For any given S-λ basis functions P n,j (x) mn j=0 defined by (5), the S-λ curves can be defined as follows [6]: where V j mn j=0 are control points of S-λ curves.
Similarly, the S-λ curves also possess many important properties, such as terminal properties, convex hull, interpolation and variation diminishing.

Tensor Product S-λ Basis Functions and S-λ Surface
Denote < i, j > as a set of two-dimensional indicators. Let A m = A , respectively.
That is, sequence T <m,n> is the two-dimensional convolution of sequence T with respect to index < i, j >.
Obviously, if the generating functions of nonnegative sequences A and B are F(s) and G(t), respectively, then the binary generating functions of sequences T and T <m,n> are as below: If PF m,i (s) are the S-λ basis functions determined by the generating function F(s) and the transformation factor λ 1 (s), PG n,j (t) are the S-λ basis functions determined by the generating function G(t) and the transformation factor λ 2 (t), then their tensor product S-λ basis function can be defined by P <m,n>,<i,j> (s, t) = PF m,i (s)PG n,j (t), 0 ≤ i ≤ ml 1 ; 0 ≤ j ≤ nl 2 .
As we have done in the univariate S-λ basis function, according to the definition of convolution, we can define the tensor product S-λ basis functions by giving the binary generating function and the transformation factor directly. From the marks given above, we have the following definitions. Definition 3. Let S(s, t) be the binary generating function satisfying S(0, 0) ≡ 1 , λ 1 (s) and λ 2 (t) be two strictly monotone continuous functions on [0, R 1 ) and [0, R 2 ), respectively. Then the tensor product S-λ basis function is defined as follows [7] P <m,n>,<i,j> (s, t) = It is obvious that P <m,n>,<i,j> (s, t) = PF m,i (s)PG n,j (t).
Tensor product S-λ basis function inherits many properties of univariate S-λ basis function, which includes nonnegativity, Partition of unity, interpolation, linear independence, etc. By using the tensor product S-λ basis function defined by (10), we can define tensor product S-λ surface as below. , the tensor product S-λ surface patch can be defined as below [7] C(s, t) = ml 1 i=0 nl 2 j=0 P <m,n>,<i,j> (s, t)V <i,j> , where P <m,n>,<i,j> (s, t) are the tensor product S-λ basis functions. From the properties of tensor product S-λ basis function, we can obtain that tensor product S-λ surface also has most of the geometric properties of the S-λ curve, such as affine invariance, convex hull property, etc., and the boundary curve of S-λ surface is a S-λ curve.

Geometric Continuity Conditions for S-λ Curves and Surfaces
In the geometric design of curve and surface modeling, single curve or surface is difficult to describe complex geometric models, so it is often necessary to give multiple curves or surfaces, and then according to certain continuity conditions, these curve segments or surface patches are spliced into a complete curve or surface patch. There are two ways to measure the smooth continuity of curves and surfaces, one is parameter continuity and the other is geometric continuity. Here, we will discuss G 1 and C 1 continuity conditions for S-λ curves and surfaces. These smooth continuity conditions will be discussed one-by-one below.

The G 1 and C 1 Smooth Continuity for S-λ Curves
Given a S-λ curve C(x) = mn j=0 P n,j (x)V j , where the basis functions are P n,j (x) = [S(λ(x))] n , ( j = 0, 1, · · · , mn). In addition, the generating function and transformation factor of S-λ curve are , respectively. Let's solve the derivative of the S-λ curve at the initial endpoint first, that is According to the definition of derivative when j 0, we have when j = 0, we obtain Substituting (15) and (14) into (12), we can obtain and dλ(x) dx = 1 (1−x) 2 , so the derivative of the curve at the initial endpoint can be obtained as below Now, let's solve the derivative of the S-λ curve at the another endpoint, that is to say On the basis of definition of derivative, we have However so we obtain When j = mn, and Consequently, When j = mn − 1, That is Owing to so we can obtain the conclusion Similarly, when j ≤ mn − 2, notice that the highest power of molecule λ(x) in P n,j Substituting (24) and (29) as well as (30) into (18), the derivative of the S-λ curve at the another endpoint can be represented as follows Hence, the following theorems can be obtained based on the theory above.

Theorem 1.
Given two S-λ curves P(x) and Q(x) with the control points P 0 , P 1 , . . . , P ml 1 and B j t j be the generating functions of them. Moreover, transformation factor of both is λ( . When two curves meet the following conditions then two S-λ curves reach G 1 smooth continuity at the common joint. In particular, when α = 1 in Theorem 1, the C 1 smooth condition of two S-λ curves can be obtained as below. Theorem 2. When two S-λ curves P(x) and Q(x) satisfy the conditions then two S-λ curves reach C 1 smooth continuity at the common joint.

The G 1 and C 1 Smooth Continuity for S-λ Surfaces
Given the generating functions F(s) = so the basis functions of tensor product S-λ surface are obtained as follows On the basis of basis functions above, two S-λ surfaces C 1 and C 2 are defined as below: where U <i,j> and V <i,j> are the control points of the corresponding surfaces.

Lemma 1.
If two adjacent S-λ surfaces C 1 and C 2 need to reach G 1 smooth continuity in the u direction, they are required to meet the following conditions The condition (37) means that two surface patches have a common joint, and the condition (38) requires that two surface patches have a common tangent plane at the common joint. Let's consider the condition (37) at first. As we all know, According to (37), we have Owing to the linearly independence of basis functions, it must exist In addition, (38) can be simplified as follows: where f is a positive real number.
The left side of the Equation (43) can be calculated as follows: When j = nl 2 , we have When j = nl 2 − 1, we obtain If j ≤ nl 2 − 2, it is easy to check that From the above calculation, we can obtain the conclusion as below Now let's calculate the right side of the equation (43). As we know, If j ≥ 2, it is obvious that Hence, by the calculation above, we can get Based on the (43), (49) and (54), we have Moreover, the basis function is linearly independent, so we have where f is a positive real number.
From (42) and (56), the sufficient conditions for two adjacent S-λ surfaces reach G 1 continuity in the u direction can be obtained, that is Theorem 3.

Theorem 3.
Given two adjacent S-λ surface patches C 1 (s, t) = ml 1 i=0 nl 2 j=0 P <m,n>,<i,j> (s, t)U <i,j> as well as C 2 (s, t) = ml 1 i=0 kl 3 j=0 P <m,k>,<i,j> (s, t)V <i,j> with the basis functions P <m,n>,<i,j> (s, t) = PF m,i (s)PG n,j (t) and P <m,k>,<i, j> (s, t) = PF m,i (s)PQ k,j (t), respectively. Here, and U <i,j> and V <i,j> are the control points of the corresponding surfaces. When two S-λ surfaces C 1 and C 2 satisfy the following conditions the two surface patches reach G 1 continuity in the u direction, where f is a positive real number.
Similarly, the following two theorems can be obtained as well.

Theorem 4.
Given two adjacent S-λ surface patches C 1 (s, t) = ml 1 i=0 nl 2 j=0 P <m,n>,<i,j> (s, t)U <i,j> as well as P <k,n>,<i,j> (s, t)V <i,j> with the basis functions P <m,n>,<i,j> (s, t) = PF m,i (s)PG n,j (t) and P <k,n>,<i,j> (s, t) = PQ k,i (s)PG n,j (t), respectively. Here, and U <i,j> and V <i,j> are the control points of the corresponding surfaces. When C 1 and C 2 satisfy the following conditions the two surface patches reach G 1 continuity in the v direction, where f is a positive real number.

Theorem 5.
Given two adjacent S-λ surface patches C 1 (s, t) = ml 1 i=0 nl 2 j=0 P <m,n>,<i,j> (s, t)U <i,j> as well as C 2 (s, t) = kl 3 i=0 ml 1 j=0 P <k,m>,<i,j> (s, t)V <i,j> with the basis functions P <m,n>,<i,j> (s, t) = PF m,i (s)PG n,j (t) and P <k,m>,<i, j> (s, t) = PQ k,i (s)PF m,j (t), respectively. Here, where s × t ∈ [0, 1) × [0, 1) , U <i,j> and V <i,j> are the control points of the corresponding surfaces. When C 1 and C 2 satisfy the following conditions: the two surface patches reach G 1 continuity in the u and v direction, where f is a positive real number. When f = 1 in Theorem 3 to Theorem 5, the two S-λ surfaces reach C 1 continuity in the splicing direction.
the two surface patches reach G 1 continuity in the u and v direction, where f is a positive real number.
When 1 = f in Theorem 3 to Theorem 5, the two S-λ surfaces reach C 1 continuity in the splicing direction.

Conclusions
In this paper, we introduced the S-λ curves and surfaces and derived the geometric conditions for G 1 smooth continuity between two adjacent S-λ curves and surfaces, respectively. In addition-in terms of the proposed continuity conditions-some modeling examples are provided to verify the effectiveness of the method. In summary, the research in this paper not only provides a unified method for dealing with common geometric models (such as the Bézier model, the B-spline model and the Poisson model), but also provides a practical technology with distinctive characteristics for the CAGD system. The advantages of the S-λ model can be summarized as follows: (a) The G 1 smooth continuity conditions for S-λ curves and surfaces proposed in this paper extend the conclusions of S-λ model constructed by Fan and Zeng [6,7]; (b) For any composite S-λ curves and surfaces which satisfy G 1 or C 1 geometric continuity, its local and global shape can be adjusted flexibly by modifying the shape parameters without changing the control points;