The Generalized H-B é zier Model: Geometric Continuity Conditions and Applications to Curve and Surface Modeling

: The local controlled generalized H-B é zier model is one of the most useful tools for shape designs and geometric representations in computer-aided geometric design (CAGD), which is owed to its good geometric properties, e.g., symmetry and shape adjustable property. In this paper, some geometric continuity conditions for the generalized cubic H-B é zier model are studied for the purpose of constructing shape-controlled complex curves and surfaces in engineering. Firstly, based on the linear independence of generalized H-B é zier basis functions (GHBF), the conditions of ﬁrst-order and second-order geometric continuity (namely, G 1 and G 2 continuity) between two adjacent generalized cubic H-B é zier curves are proposed. Furthermore, following analysis of the terminal properties of GHBF, the conditions of G 1 geometric continuity between two adjacent generalized H-B é zier surfaces are derived and then simpliﬁed by choosing appropriate shape parameters. Finally, two operable procedures of smooth continuity for the generalized H-B é zier model are devised. Modeling examples show that the smooth continuity technology of the generalized H-B é zier model can improve the e ﬃ ciency of computer design for complex curve and surface models.


Introduction
Parametric curves and surfaces are the main tools used to describe the geometric shape of products in computer-aided geometric design (CAGD), and their related theories and technologies are also the bases and core contents of the whole CAGD [1,2].Traditional Bézier curves and surfaces are parametric curves and surfaces constructed from Bernstein basis functions.They are not only an important method for representing complex curves and surfaces in the field of CAGD, but also a powerful tool for shape designs and geometric representations of various products [3,4].However, the Bézier model has some limitations: (i) the shape of the Bézier model is only determined by its control points; and (ii) the Bézier model cannot accurately represent conic curves and surfaces [5].To overcome these limitations, the shapes of the rational Bézier model and non-uniform rational B-spline (NURBS) with fixed control points can be modified by altering the weight factor [1].The rational Bézier model and NURBS can accurately express circles and ellipses, but they cannot precisely describe transcendental curves such as hyperbolic curves, catenary curves, and so on [3].In order to preserve the good properties of the classical Bézier model and enhance its shape adjustability, scholars have recently constructed many curves and surfaces with shape parameters .
In view of the remarkable advantages of the H-Bézier model, scholars have done much research on the related theories and applications of the H-Bézier model in recent years.Wang and Yang [11] presented a characterization diagram of the planar cubic H-Bézier model and studied the existence conditions of singularities and inflection points.Qian and Tang [12] investigated the sufficient and necessary conditions for continuity of curvature and tangency between H-Bézier curves of order 4. Fan and Wang [13] presented the matrix representation of the H-Bézier basis and the H-B-spline basis using recursion.Wu [14] analyzed the geometric characteristics of the cubic H-Bézier model, e.g., convexity, singularities, and inflection points.Zhao and Wang [15] proposed a geometric characterization of the H-Bézier model and then compared the singularities of the Bézier, C-Bézier, and H-Bézier models.Wang et al. [16] proposed an effective algorithm of approximate reduction for H-Bézier curves based on generalized inverse matrix theory.Qin et al. [17] constructed the PHH-Bézier curve and proposed sufficient and necessary conditions for a cubic H-Bézier curve to be a PH curve.In [18], Lee and Ahn obtained the limit curves of nth-degree H-Bézier curves as α → ∞ .Cao et al. [19] proposed two algorithms for the approximation of the respective offset curves of H-Bézier and H-B-spline spline curves.Huang and Wang [20] presented a novel orthogonal basis in hyperbolic hybrid polynomial space which retains prominent characteristics of the H-Bézier basis.Zhang et al. [21] proposed a novel method for unifying C-curves and H-curves by extending the calculation to complex numbers.As is well known, the global shape of the H-Bézier model can be altered without modifying the positions of control points.However, the H-Bézier model in [9][10][11][12][13][14][15][16][17][18][19][20][21] has no local shape adjustability due to the lack of local shape parameters.In [22], Hu et al. presented a novel local controlled H-Bézier model called the generalized H-Bézier model.The global and local shapes of the new model can be adjusted by changing its local shape parameters while the control points are maintained.Compared with the H-Bézier model, the generalized H-Bézier model in [22] has greater advantages and plays a key role in describing complex curves and surfaces.In order to handle the problem of not being able to construct complex curves and surfaces using a single generalized H-Bézier model, in this paper we propose the respective geometric continuity conditions between two adjacent generalized H-Bézier curves and surfaces.
The rest of the paper is organized as follows.The definition and properties of the generalized H-Bézier model are given in Section 2. In Section 3, we propose the geometric continuity conditions for the generalized H-Bézier model and give some examples.Two practical applications are discussed in Section 4, and finally, conclusions are given in Section 5.
Figure 1 shows a graphical representation of the basis functions h i,3 (t) (i = 0, 1, 2, 3) in (3) with variation of the shape parameters.One can see from Figure 1 that modification of the three shape parameters has a distinct effect on the basis functions h i,3 (t) (i = 0, 1, 2, 3).
Theorem 2. The curves H(t; Ω) in (5) have the following terminal properties: Proof.According to the terminal properties in (4), as well as the definition of the generalized cubic H-Bézier curves, we can deduce the terminal properties (6) of the curves H(t; Ω), thus proving Theorem 2.

Shape Control of the Generalized Cubic H-Bézier Curves
For t ∈ [0, 1], we rewrite (5) as follows: in which where f i (t; α i )(i =  For a given t ∈ (0, 1) and a fixed control polygon, the expression of the curves H( t; Ω) describes the movement locus of a fixed point H( t) when one of the shape parameters, α i (i = 1, 2, 3), changes.
Afterwards, we obtain the partial derivative of (7) with respect to α i as follows: which signifies that the trace curve H( t; Ω) of a fixed point H( t) is a straight line only when the shape parameter α i changes (as shown in Figure 3).

Definition of Generalized Bicubic H-Bézier Surfaces
Definition 3. Given a set of control points The tensor products of generalized bicubic H-Bézier surfaces have properties similar to those of the tensor products of classical bicubic H-Bézier ones.By keeping the control mesh fixed, the shape of the generalized bicubic H-Bézier surface can also be modified by altering the shape parameters.Figures 4 and 5 show the different behaviors of surfaces ) , ; , ( by alteration of the shape parameters in their domain.Obviously, for a fixed control polygon and a given t ∈ (0, 1), the value change of one shape parameter α i (i = 1, 2, 3) will result in a linear change of a point H( t) on the curves.For a fixed t ∈ (0, 1), according to (8), we know that We can see from ( 10) that f 1 ( t; α 1 ) is an increasing function with α 1 (see Figure 2a), which means that the generalized cubic H-Bézier curve will move along the same direction as P 1 − P 0 with increasing α 1 , and vice versa.Analogously, we deduce that the shape parameter α 3 has a similar effect on the edge P 2 − P 3 on the basis of the monotonicity principle of f 3 (t; α 3 ) (as shown in Figure 2c).Notice, however, that f 2 (t; α 2 ) augments or diminishes with increasing α 2 for any fixed t ∈ (0, 0.5) or t ∈ (0.5, 1), respectively (as shown in Figure 2b), which signifies that the curves H(t; Ω) defined on intervals (0, 0.5) and (0.5, 1) move in the same and opposite direction as the edge P 2 − P 1 as α 2 increases, respectively.From the discussions above, the effect laws of the shape parameters α i (i = 1, 2, 3) on generalized cubic H-Bézier curves are obtained under the circumstance of the control points of the curves remaining unchanged.
(a) For fixed shape parameters α i (i = 2, 3), the curves H(t; Ω) will gradually approach or go far away from the control point P 1 when increasing or decreasing the value of shape parameter α 1 .
That means that the shape parameter α 1 is mainly used to adjust the local shape of the curves in the vicinity of control point P 1 (as shown in Figure 3a).(b) For fixed shape parameters α 1 and α 3 , the curves H(t; Ω) will simultaneously move away from the control points P 1 and P 2 with increasing value of the shape parameter α 2 (as shown in Figure 3b).What calls for special attention is that the location of point H(0.5;Ω) will not change as α 2 increases.(c) For fixed shape parameters α 1 and α 2 , the curves H(t; Ω) will get nearer to or farther away from control point P 3 when increasing or decreasing the value of shape parameter α 3 .That means that the shape parameter α 3 is mainly utilized to control the local shape of the curves in the vicinity of control point P 3 (as shown in Figure 3c).
Besides this, the curves H(t; Ω) gradually approach or move away from their control polygon spanned by P i (i = 0, 1, 2, 3) when increasing or decreasing the values of shape parameters α 1 and α 3 simultaneously (as shown in Figure 3d).In conclusion, we can see expressly that the shape parameters α i (i = 1, 2, 3) can act as tension parameters, and their influence rules on the curves H(t; Ω) are perspicuous (see Figure 3).
Figure 3 displays the influences of the three shape parameters on the cubic generalized H-Bézier curves when their control points are fixed.Figure 3a-c shows the curves with two fixed shape parameters while changing α 1 , α 2 , α 3 , respectively.Figure 3d shows the curves while altering shape parameters α 1 and α 3 simultaneously.The influence rules of the shape parameters on the curves H(t; Ω) can be seen distinctly in Figure 3.Note that, in Figure 3, the values in parentheses show the changes of the shape parameters in the corresponding position, with the values from left to right in parentheses corresponding to the H-Bézier curves from the black dotted curve to the green solid curve in each graph; the red asterisks marked on the different generalized cubic H-Bézier curves in each figure represent the points with the same parameter t, so that the straight lines connecting these points are the motion trajectory of the points corresponding to t when altering the shape parameters.
The tensor products of generalized bicubic H-Bézier surfaces have properties similar to those of the tensor products of classical bicubic H-Bézier ones.By keeping the control mesh fixed, the shape of the generalized bicubic H-Bézier surface can also be modified by altering the shape parameters.Figures 4 and 5 show the different behaviors of surfaces S(u, v; Ω α , Ω β ) by alteration of the shape parameters in their domain.

Geometric Continuity Conditions for the Generalized Cubic H-Bézier Model
In the design of curve and surface modeling, we often encounter some complex curves and surfaces that cannot be represented by a single curve and surface, so the technology of smooth splicing between multiple adjacent curves and surfaces seems to be very important.According to the smooth continuity conditions of parametric curves and surfaces, we will discuss the G 1 and G 2 continuity conditions for generalized cubic H-Bézier curves and G 1 continuity conditions for

Geometric Continuity Conditions for the Generalized Cubic H-Bézier Model
In the design of curve and surface modeling, we often encounter some complex curves and surfaces that cannot be represented by a single curve and surface, so the technology of smooth splicing between multiple adjacent curves and surfaces seems to be very important.According to the smooth continuity conditions of parametric curves and surfaces, we will discuss the G 1 and G 2 continuity conditions for generalized cubic H-Bézier curves and G 1 continuity conditions for generalized bicubic H-Bézier surfaces.Since the smooth continuity of surfaces has directionality, there are three different ways for surfaces to achieve smooth continuity at the joint, which are the continuity of the two surfaces in the u direction, the continuity of one surface in the u direction and the other in the v direction, and the continuity of the two surfaces in the v direction.These smooth continuity conditions will be discussed one by one below.

G 1 and G 2 Smooth Continuity for Generalized Cubic H-Bézier Curves
For the convenience of our discussion, it is assumed that the expressions of the two generalized cubic H-Bézier curves to be spliced are as follows: where and H 2 (t, Ω 2 ), respectively.
Theorem 3.For the two adjacent generalized cubic H-Bézier curves defined by ( 12), the necessary and sufficient conditions of G 1 smooth continuity at the joint are as follows: where λ > 0 is an arbitrary constant.
Proof.In order to make two adjacent curves H 1 (t; Ω 1 ) and H 2 (t; Ω 2 ) reach G 1 smooth continuity at the joint, they should satisfy G 0 continuity in the first place, which means that the two splicing curves should be connected from end to end: At the same time, G 1 continuity of the two curves needs to have the same tangent direction at the splicing point [23,24]; that is, where λ > 0 is an arbitrary constant.
Proof.If the two adjacent generalized cubic H-Bézier curves H 1 (t; Ω 1 ) and H 2 (t; Ω 2 ) achieve G 2 continuity at the joint, they are required to achieve G 1 continuity first (as shown in ( 13)).Furthermore, G 2 continuity of the two adjacent curves also needs to be satisfied [23][24][25]: where λ > 0, µ is an arbitrary constant.
In terms of the terminal properties (6) of generalized cubic H-Bézier curves, we have Finally, the third equation of ( 18) can be obtained by combining (13) and (20).Thus, Theorem 4 is proved.Specifically, Equation ( 18) is the necessary and sufficient condition for C 2 smooth continuity of two adjacent generalized cubic H-Bézier curves when λ = 1, µ = 0. Remark 3.While keeping the continuity conditions of G 1 and G 2 unchanged, we can adjust the global and local shapes of composite generalized cubic H-Bézier curves by modifying the values of shape parameters, which is the advantage of splicing generalized cubic H-Bézier curves.
According to the two smooth continuity conditions of generalized cubic H-Bézier curves discussed above, we take G 2 continuity as an example to give a specific splicing algorithm for two adjacent curves.The procedure is shown in Algorithm 1. for [3]; } Plot the initial generalized cubic H-Bézier curve according to Hx and Hy; Compute the first three control points of the second curve according to (18) and store them in Qx[i] and Qy[i] (i = 0,1,2); Give the last control point Qx [3],Qy [3]; [3]; } Plot the generalized cubic H-Bézier curve that needs to reach G 2 smooth continuity according to HHx and HHy; } Remark 4. According to Algorithm 1, composite generalized cubic H-Bézier curves with G 2 continuity can be easily obtained.Similarly, when "the first three control points of the second curve are calculated according to (18)" and the next step "give the last control points" in Algorithm 1 are replaced by "the first two control points of the second curve are calculated in terms of ( 13)" and "give the remaining two control points", generalized cubic H-Bézier curves with G 1 continuity can be obtained by Algorithm 1.
Figures 6 and 7 show examples of shape adjustment of the composite generalized cubic H-Bézier curves with G 1 and G 2 continuity, respectively, which can be obtained by Algorithm 1.It can be clearly seen from the figure that the generalized cubic H-Bézier curve has superior shape adjustability, which allows broader practical applications of the generalized cubic H-Bézier curve.
(18)" and the next step "give the last control points" in Algorithm 1 are replaced by "the first two control points of the second curve are calculated in terms of ( 13)" and "give the remaining two control points", generalized cubic H-Bézier curves with G 1 continuity can be obtained by Algorithm 1.

G 1 Smooth Continuity for Generalized Cubic H-Bézier Surfaces
For the convenience of this discussion, the splicing of two generalized bicubic H-Bézier surface patches is considered here.Suppose that the two generalized bicubic H-Bézier surface patches to be spliced are defined as follows:

G 1 Smooth Continuity for Generalized Cubic H-Bézier Surfaces
For the convenience of this discussion, the splicing of two generalized bicubic H-Bézier surface patches is considered here.Suppose that the two generalized bicubic H-Bézier surface patches to be spliced are defined as follows: where 2) represent the control mesh points of the kth surface patch.Theorem 5.If the shape parameters and control mesh points of two generalized bicubic H-Bézier surface patches (21) satisfy then the u direction of S 1 and the u direction of S 2 reach G 1 smooth continuity at the common boundary, where γ > 0 is a real constant.
Proof.In order to make two adjacent generalized bicubic H-Bézier surfaces S 1 and S 2 reach G 1 smooth continuity in the u direction, they should satisfy G 0 continuity first, which means that Based on the boundary properties of the generalized bicubic H-Bézier surface, we can obtain Due to the linear independence of generalized cubic H-Bézier basis functions, and comparing the coefficients of the two sides of (24), Equation ( 24) can be further simplified and the following obtained: Also, the two H-Bézier surface patches need to satisfy normal vector direction continuity at the common boundary [26], i.e., where γ(u) > 0 is the scaling factor between their normal vectors.In order to facilitate the calculation, (26) can be simplified by the Faux method as follows [26]: where γ > 0 is a real constant; (27) means that the directions of cross-boundary tangents of the two surfaces are continuous.After simple calculation, (27) can be simplified to Due to the linear independence of generalized cubic H-Bézier basis functions, Equation (28) can be simplified by comparing coefficients as follows: By combining the conclusions in ( 25), (29) can be reduced to Therefore, ( 25) and (30) constitute G 1 smooth continuity conditions of two generalized bicubic H-Bézier surface patches in the u direction.Theorem 5 is thus proved.Theorem 6.If the shape parameters and control mesh points of two generalized bicubic H-Bézier surface patches (21) satisfy then the u direction of S 1 and the v direction of S 2 reach G 1 smooth continuity at the common boundary, where γ > 0 is a real constant.
Proof.In order to make two adjacent generalized bicubic H-Bézier surfaces S 1 and S 2 reach G 1 smooth continuity in the u and v directions, they should satisfy G 0 continuity first, which means Based on the boundary properties of the generalized bicubic H-Bézier surface, we can obtain Then, the two H-Bézier surface patches need to satisfy continuity of normal vectors in the u and v directions at the common boundary, i.e., [24,26] where γ > 0 is a real constant.
After simple calculation, (34) can be simplified to Due to the linear independence of generalized cubic H-Bézier basis functions, (35) can be simplified by comparing coefficients as follows: Furthermore, by combining the conclusions in (33), (36) can be reduced to Therefore, (33) and (37) constitute the G 1 smooth continuity conditions of two generalized bicubic H-Bézier surface patches in the u and v directions.Theorem 6 is thus proved.
Analogously, for the G 1 smooth continuity conditions of two generalized bicubic H-Bézier surface patches in the v direction, it can be easily proved that the following theorem holds.Theorem 7. If the shape parameters and control mesh points of two generalized bicubic H-Bézier surface patches (17) satisfy then the v direction of S 1 and the v direction of S 2 reach G 1 smooth continuity at the common boundary, where γ > 0 is a real constant.
From the above discussion, it can be seen that in order to obtain a generalized bicubic H-Bézier surface that satisfies the G 1 continuity condition, we are only required to calculate the first two rows of control points of the second generalized bicubic H-Bézier surface according to the corresponding conditions; then the last two rows of control points can be given.Here, we devise a specific operable procedure for generating composite generalized bicubic H-Bézier surfaces with G 1 smooth continuity in the u direction.The procedure is shown in Algorithm 2.
Figure 8 shows a modeling example to illustrate G 1 smooth continuity in the u direction between two generalized bicubic H-Bézier surfaces and modifies the shape of the composite surface by changing shape parameters without changing the continuity conditions.In Figure 8, the green mesh surfaces represent the original surface S 1 (u, v; Ω α , Ω β ), the blue polylines are its control mesh, and the circles on the polylines represent control points (for visual effect, only four corners of the surfaces are shown here; the same below).The yellow mesh surfaces are the constructed surfaces S 2 (u, v; Ωα , Ωβ ) which satisfy the G 1 smooth continuity conditions in the u direction, that is, Equation (22).The red polylines are the control meshes of the surfaces S 2 (u, v; Ωα , Ωβ ), and the circles on the polylines denote their control points.In Figure 8a, the shape parameters of surfaces S 1 (u, v; Ω α , Ω β ) and S 2 (u, v; Ωα , Ωβ ) are Ω α = Ωα = {1, 2, 1}, γ = 0.9. Figure 8b was obtained by modifying the value of the partial shape parameters in Ω β , Ωβ (specific modifications can be seen in Figure 8) on the basis of Figure 8a.
Mathematics 2020, 8, x FOR PEER REVIEW 19 of 24 Figure 8 shows a modeling example to illustrate G 1 smooth continuity in the u direction between two generalized bicubic H-Bézier surfaces and modifies the shape of the composite surface by changing shape parameters without changing the continuity conditions.In Figure 8, the green mesh surfaces represent the original surface ) , ; , ( , the blue polylines are its control mesh, and the circles on the polylines represent control points (for visual effect, only four corners of the surfaces are shown here; the same below).The yellow mesh surfaces are the constructed surfaces (specific modifications can be seen in Figure 8) on the basis of Figure 8a. Figure 9 shows a modeling example to demonstrate G 1 smooth continuity in the u and v directions between two generalized bicubic H-Bézier surfaces and modifies the shape of the composite surface by changing shape parameters without changing the continuity conditions.In Figure 9, the green mesh surfaces represent the original surface (specific modifications can be seen in Figure 9) so as to adjust the shape of the composite surfaces.Figure 9 shows a modeling example to demonstrate G 1 smooth continuity in the u and v directions between two generalized bicubic H-Bézier surfaces and modifies the shape of the composite surface by changing shape parameters without changing the continuity conditions.In Figure 9, the green mesh surfaces represent the original surface S 1 (u, v; Ω α , Ω β ) and the yellow mesh surfaces are the constructed surfaces S 2 (u, v; Ωα , Ωβ ) which satisfy the G 1 smooth continuity conditions in the u and v directions, that is, Equation (31).In Figure 9a, the shape parameters of surfaces S 1 (u, v; Ω α , Ω β ) and S 2 (u, v; Ωα , Ωβ ) are Ω α = Ωβ = {1, 5, 1}, γ = 0.5.In Figure 9b, we take Ω α = Ωβ = {5, 2, 5} and modify some shape parameter values in Ω β , Ωα (specific modifications can be seen in Figure 9) so as to adjust the shape of the composite surfaces.
Figure 10 shows a modeling example to display G 1 smooth continuity in the v direction between two generalized bicubic H-Bézier surfaces and modifies the shape of the composite surface by changing shape parameters without changing the continuity conditions.In Figure 10, the green mesh surfaces represent the initial surface S 1 (u, v; Ω α , Ω β ) and the yellow mesh surfaces are the constructed surfaces S 2 (u, v; Ωα , Ωβ ) which satisfy the G 1 smooth continuity conditions in the v direction, that is, Equation (38).In Figure 10a, the shape parameters of surfaces S 1 (u, v; Ω α , Ω β ) and S 2 (u, v; Ωα , Ωβ ) are Ω β = Ωβ = {1, 2, 1}, γ = 0.9. Figure 10b is obtained by modifying the value of the partial shape parameters in Ω α , Ωα on the basis of Figure 10a.From Figure 8 to Figure 10, it can be seen that when the continuity condition of the surface remains unchanged, modification of the local or global shape of by using the method proposed in this paper, various types of projection contour curve schemes of the car headlight can be obtained quickly.Then we can project the contour curves onto the surface of the car front face, and different styles of car headlight modeling schemes (as shown in Figure 12b) can be obtained using surface trimming technology.The appearance design of a ship hull using generalized H-Bézier surfaces is displayed in Figure 13.In Figure 13, the body of this ship hull is composed of four generalized H-Bézier surfaces with G 1 smooth continuity.The modeling examples provided show that our methods are effective and provide an alternative for constructing complex curves and surfaces in industry.generalized H-Bézier curves.In Figure 11, the contour curves of the car headlight are composed of eight generalized H-Bézier curves with G 1 smooth continuity.As can be seen from Figure 11, designers can use control points to delineate the initial shape of the contour curves of the car headlight, and then the shape of the initial contour curves can be adjusted slightly by modifying the local shape parameters, so as to quickly obtain satisfactory contour curves in car headlight modeling.In automobile modeling design, the car front face is undoubtedly the most important part of the car body, and the contour curves of the car headlight are an important visual feature in car front face modeling.As shown in Figure 12a, by using the method proposed in this paper, various types of projection contour curve schemes of the car headlight can be obtained quickly.Then we can project the contour curves onto the surface of the car front face, and different styles of car headlight modeling schemes (as shown in Figure 12b) can be obtained using surface trimming technology.The appearance design of a ship hull using generalized H-Bézier surfaces is displayed in Figure 13.In Figure 13, the body of this ship hull is composed of four generalized H-Bézier surfaces with G 1 smooth continuity.The modeling examples provided show that our methods are effective and provide an alternative for constructing complex curves and surfaces in industry.eight generalized H-Bézier curves with G 1 smooth continuity.As can be seen from Figure 11, designers can use control points to delineate the initial shape of the contour curves of the car headlight, and then the shape of the initial contour curves can be adjusted slightly by modifying the local shape parameters, so as to quickly obtain satisfactory contour curves in car headlight modeling.In automobile modeling design, the car front face is undoubtedly the most important part of the car body, and the contour curves of the car headlight are an important visual feature in car front face modeling.As shown in Figure 12a, by using the method proposed in this paper, various types of projection contour curve schemes of the car headlight can be obtained quickly.Then we can project the contour curves onto the surface of the car front face, and different styles of car headlight modeling schemes (as shown in Figure 12b) can be obtained using surface trimming technology.The appearance design of a ship hull using generalized H-Bézier surfaces is displayed in Figure 13.In Figure 13, the body of this ship hull is composed of four generalized H-Bézier surfaces with G 1 smooth continuity.The modeling examples provided show that our methods are effective and provide an alternative for constructing complex curves and surfaces in industry.

Conclusions
In this paper, we investigated some properties of the generalized H-Bézier model and derived the geometric conditions for G 1 and G 2 smooth continuity between two adjacent generalized H-Bézier curves and for G 1 smooth continuity between two adjacent generalized H-Bézier surfaces.

Figure 2 .
Figure 2. The translational speed functions with different parameters t.

Figure 3 .
Figure 3.The influences of shape parameters on generalized cubic H-Bézier curves.

Figure 3 .
Figure 3.The influences of shape parameters on generalized cubic H-Bézier curves.

Figure 4 .
Figure 4.The effects of 1 α on the generalized bicubic H-Bézier surface.

Figure 4 .
Figure 4.The effects of α 1 on the generalized bicubic H-Bézier surface.

Figure 4 .
Figure 4.The effects of 1 α on the generalized bicubic H-Bézier surface.

Figure 5 .
Figure 5.The effects of 3 β on the generalized bicubic H-Bézier surface.

Figure 5 .
Figure 5.The effects of β 3 on the generalized bicubic H-Bézier surface.

Figures 6 Figure 6 .Figure 6 .Figure 7 .
Figures6 and 7show examples of shape adjustment of the composite generalized cubic H-Bézier curves with G 1 and G 2 continuity, respectively, which can be obtained by Algorithm 1.It can be clearly seen from the figure that the generalized cubic H-Bézier curve has superior shape adjustability, which allows broader practical applications of the generalized cubic H-Bézier curve.

Figure 7 .
Figure 7. Shape adjustment of the generalized cubic H-Bézier curves with G 2 continuity.
Figure 8b was obtained by modifying the value of the partial shape parameters in β β Ω Ω ,

Figure 8 .
Figure 8. Generalized bicubic H-Bézier surfaces with G 1 continuity in the u direction.
satisfy the G 1 smooth continuity conditions in the u and v directions, that is, Equation (31).In Figure9a,

Figure 8 .
Figure 8. Generalized bicubic H-Bézier surfaces with G 1 continuity in the u direction.

Figure 11 .Figure 12 .
Figure 11.Eight generalized H-Bézier curves used to define the contour curves of a car headlight.

Figure 11 .
Figure 11.Eight generalized H-Bézier curves used to define the contour curves of a car headlight.

Figure 11 .Figure 12 .
Figure 11.Eight generalized H-Bézier curves used to define the contour curves of a car headlight.

Figure 12 .
Figure 12.Designing a car headlight using projection and pruning techniques.

Figure 13 .
Figure 13.Designing the surface model of a ship hull.

Figure 13 .
Figure 13.Designing the surface model of a ship hull.