Extending cubic uniform B-splines by unified trigonometric and hyperbolic basis

doi:10.1016/j.gmod.2004.06.001

Graphical Models

Volume 67, Issue 2, March 2005, Pages 100-119

https://doi.org/10.1016/j.gmod.2004.06.001 Get rights and content

Abstract

In this paper, the trigonometric basis {sin t, cos t, t, 1} and the hyperbolic basis {sinh t, cosh t, t, 1} are unified by a shape parameter C (0 ≤ C < ∝). It yields the Functional B-splines (FB-splines) and its corresponding Subdivision B-splines (SB-splines). As well, a geometric proof of curvature continuity for SB-splines is provided. FB-splines and SB-splines inherited nearly all properties of B-splines, including the C² continuity, and can represent elliptic and hyperbolic arcs exactly. They are adjustable, and each control point b_i can have its unique shape parameter C_i. As C_i increases from 0 to ∝, the corresponding breakpoint of b_i on the curve is moved to the location of b_i, and the curvature of this breakpoint is increased from 0 to ∝ too. For a set of control points and their shape parameters, SB-spline and FB-spline have the same position, tangent, and curvature on each breakpoint. If two adjacent control points in the set have identical parameters, their SB-spline and FB-spline segments overlap. However, in general cases, FB-splines have no simple subdivision equation, and SB-splines have no common evaluation function. Furthermore, FB-splines and SB-splines can generate shape adjustable surfaces. They can represent the quadric surfaces precisely for engineering applications. However, the exact proof of C² continuity for the general SB-spline surfaces has not been obtained yet.

Introduction

Using the trigonometric basis {sin t, cos t, t, 1} instead of {t³, t², t, 1} in cubic uniform B-splines, we obtained the CB-splines [1], [2], [3]. Similarly, using the hyperbolic basis {sinh t, cosh t, t, 1} instead of {t³, t², t, 1} in cubic uniform B-splines, we can construct a curve family too. This has been studied as exponential B-splines [4], [5], [6], or splines in tension [7]. Just for convenience, we call them HB-splines here in this paper.

CB-splines and HB-splines are the same in structure and their shapes are adjustable. However, after comparing CB-splines and HB-splines, we found that a CB-spline is located on one side of the B-spline (the area with square dot in Fig. 1, Fig. 2), and it can represent the elliptic arc exactly. However, the HB-spline is located on the other side of the B-spline (the area with round dot in Fig. 1, Fig. 2), and it can represent the hyperbolic arc exactly. Therefore, we think that may be they are two different parts of a complete spline curve families. If we can unify them, the adjustable scope of the new curve family must be more complete.

Recently, Morin et al. [8] developed a subdivision rule that can connect two subdivision segments based on these bases in C² continuity. Their work enlightened us that the trigonometric basis {sin t, cos t, t, 1} and the hyperbolic basis {sinh t, cosh t, t, 1} can be unified by a parameter C. Then we obtained the Functional B-splines (FB-splines) and their corresponding Subdivision B-splines (SB-splines). FB-splines and SB-splines are generated by the unified trigonometric and hyperbolic basis. They include all the curves that are generated by each basis, and in both bases. Therefore, they can cross the B-splines and reach the both sides of B-splines.

Now we begin to introduce them from the CB-splines that use the basis {sin t, cos t, t, 1}.

Section snippets

Introduction of CB-Splines

Using the basis {sin t, cos t, t, 1} instead of {t³, t², t, 1} in cubic uniform B-spline, we have the CB-splines [2], [3] (see Fig. 1, where C is cos(α/2)):

(1) Let b₀, b₁, b₂, …, b_n, b_n + 1 be n + 2 ≥ 4 given control points. For an arbitrarily selected real value of α, 0 ≤ α ≤ π the CB-spline with parameter α can be calculated in a matrix form: $p_{i} (t) = \frac{1}{2 α (1 - C_{α})} [\sin t \cos t t 1] [\begin{matrix} C_{α} & - (1 + 2 C_{α}) & 2 + C_{α} & - 1 \\ - S_{α} & 2 S_{α} & - S_{α} & 0 \\ - 1 & 1 + 2 C_{α} & - (1 + 2 C_{α}) & 1 \\ α & - 2 α C_{α} & α & 0 \end{matrix}] [\begin{matrix} b_{i - 1} \\ b_{i} \\ b_{i + 1} \\ b_{i + 2} \end{matrix}],$ where C_α = cos α, S_α = sin α, 0 ≤ t ≤ α, i = 1, 2, …, n − 1.

In the following

Functional B-splines

However, the CB-spline curve family is incomplete. CB-splines are located only on one side of the B-splines, and their curvature on each breakpoint cannot be bigger than the B-splines.

Enlightened by [8], an interesting result is found: using sinh and cosh instead of sin and cos, respectively, and letting α be in 0 ≤ α < ∝, all above Eqs. (1), (2), (3a), (3b), (3c), (4) and properties are still true! The same methods used in [2], [3] can prove them all.

As mentioned above these curves generated by

Subdivision B-splines

As mentioned above, FB-splines have the subdivision Eq. (7) only if all control points in a curve share one parameter C. To generate C² continuity subdivision curve for the non-uniform parameter cases, Morin et al. [8] have succeeded in connecting two subdivision segments with different uniform parameters, by extending the subdivision equations for the connection. Here, we consider that each shape parameter belongs to its control point, not the whole segment, and want to construct the

The proof of C² continuity for SB-splines

Suppose F, A, B, and E are four control points of a SB-spline segment (see Fig. 5). They correspond to the above b_i − 1, b_i, b_i + 1, and b_i + 2 (i = 1, or 2, …, or n − 1). We prove that SB-spline segment has curvature right (or left) continuity in the limit point of A. For easy understanding, the proof is just given in 2D cases here, but it can be extended to the 3D cases easily.

Assume C_a and C_b are the parameters of points A and B; M and N are the midpoints of FB and AE, respectively, in Fig. 5. We set

FB-spline surfaces and SB-spline surfaces

Obviously, FB-splines and SB-splines can construct the shape adjustable surfaces of revolution in C² continuity conveniently.

Furthermore, the FB-splines can generate the tensor product surfaces with C² continuity easily. The FB-spline surfaces can represent the quadric surface precisely, and their shapes can be modified by the parameters too. Fig. 6 shows an example of the FB-spline surfaces, where different shapes are generated by the same control mesh, but with the different parameters. The

Conclusions

The unified trigonometric and hyperbolic basis is just a basis {sin t, cos t, t, 1} or {sinh t, cosh t, t, 1} that is switched by the shape parameter C (0 ≤ C ≤ 1 or 1 < C < ∝) smoothly. Using this unified basis, the cubic uniform B-splines are extended to FB-splines and SB-splines for functional calculation and subdivision processes, respectively.

For a set of control points and their shape parameters, SB-spline and FB-spline have the same position, tangent, and curvature on each breakpoint. If two

Acknowledgments

We would like to give our deep thanks to Prof. H. Nowacki for his very important guidance and suggestions, to Prof. Yang, to Mingchao for his careful revision in mathematics, and to the reviewers for their helpful comments and suggestions.

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^⋆: This work is supported by the National Natural Science Foundation of China (60073025) and TU-Berlin Germany.

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Extending cubic uniform B-splines by unified trigonometric and hyperbolic basis⋆

Abstract

Introduction

Section snippets

Introduction of CB-Splines

Functional B-splines

Subdivision B-splines

The proof of C2 continuity for SB-splines

FB-spline surfaces and SB-spline surfaces

Conclusions

Acknowledgments

Comput. Aided Geometric Design

Comput. Aided Geometric Design

Comput. Aided Geometric Design

Comput. Aided Geometric Design

The proof of C² continuity for SB-splines