# Cubic Trigonometric Nonuniform Spline Curves and Surfaces

Authors: Lanlan Yan
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

A class of cubic trigonometric nonuniform spline basis functions with a local shape parameter is constructed. Their totally positive property is proved. The associated spline curves inherit most properties of usual polynomialB-spline curves and enjoy some other advantageous properties for engineering design. They have C
2 continuity at single knots. For equidistant knots, they have C
3 continuity and C
5 continuity for particular choice of shape parameter. They can express freeform curves as well as ellipses. The associated spline surfaces can exactly represent the surfaces of revolution. Thus the curve and surface representation scheme unifies the representation of freeform shape and some analytical shapes, which is popular in engineering.


---

## Body

## 1. Introduction

As a unified mathematical model with many desirable properties,B-splines are widely applied to the modeling of freeform curves and surfaces. However, there are several limitations of the B-spline model, which restrict its applications. For example, once the knot vectors are specified, the positions of B-spline curves are relatively fixed to their control points. On the other hand, B-spline curves fail to represent conic curves except the parabolas, as well as some transcendental curves such as the helix and the catenary, which are often used in engineering.Although the nonuniform rationalB-spline (NURBS) can overcome the first shortcoming of B-splines to a certain extent, it fails to model transcendental curves. The NURBS model has several other potential limitations due to the relative complexity of rational basis functions. For example, rational form may be unstable, and derivatives and integrals are hard to compute. Consequently, in order to overcome the drawbacks of B-splines, it is necessary to explore new models.To enhance the flexibility ofB-spline models, some researchers have suggested many types of curves with shape parameters incorporated into the basis functions. For instance, Xu and Wang [1] proposed three kinds of extensions of cubic uniform B-spline. The advantage of the extensions is that they have shape parameters, which can be used to adjust the shape of the curves without shifting the control points. Costantini et al. [2] presented a method for the construction of cubic like B-splines with multiple knots. The proposed B-splines are equipped with tension parameters, associated to the knots, which permit a modification of their shape. Han [3] constructed a kind of piecewise quartic polynomial curves with a local shape parameter. Han [4] defined a class of piecewise quartic spline curves with three local shape parameters. Hu et al. [5] constructed a kind of B-spline curves with two local shape parameters. To expand the scope of shape representation of B-spline models, some researchers have suggested many types of curves defined on nonpolynomial space. For instance, Han presented the quadratic [6] and cubic [7] trigonometric polynomial curves with a shape parameter. Han and Zhu [8] defined trigonometric B-spline curves with three local shape parameters and a global shape parameter. Dube and Sharma [9] constructed the cubic trigonometric polynomial B-spline curves with a shape parameter. Wang et al. [10] presented nonuniform algebraic-trigonometric B-splines. The curves in [6–10] not only enjoy adjustable shape, but also can exactly represent ellipses. Yan and Liang [11] constructed a class of algebraic-trigonometric blending spline curves with two shape parameters. Except for shape adjustability, the curves admit exact representations for several remarkable curves. Lü et al. [12] defined uniform algebraic-hyperbolic blending B-spline curves, which possess adjustable shape and can represent exactly the hyperbola and the catenary.Totally positive property is one of the important properties of basis functions. Curves defined by totally positive basis must have variation diminishing and convexity preserving property. Goodman and Said [13] proved that the generalized Ball basis given in [14] is normalized totally positive and hence it possesses the same kind of shape preserving properties as the Bernstein basis [15]. Han and Zhu [16] proved that the cubic trigonometric Bézier basis given in [17] forms an optimal normalized totally positive basis. Zhu et al. [18] constructed four Bernstein-like basis functions, which form an optimal normalized totally positive basis. Based on the Bernstein-like basis, a class of totally positive B-spline-like basis functions is constructed. The associated B-spline curves have C
2 continuity at single knots and can be C
2
∩
F
C
k
+
3 (k
∈
Z
+) continuous for particular choice of shape parameters.The curves given in [1–10, 12, 18] have adjustable shape. In addition, the curves given in [6–10, 12] can represent exactly some conic curves and transcendental curves. However, whether the blending functions in [1, 3–10, 12] have total positivity is unknown, so whether the associated curves have variation diminishing is unknown. The curves in [2, 18] have variation diminishing, but they cannot represent conic curve or transcendental curve. The purpose of this paper is to define a kind of B-spline-like curves and surfaces, which have adjustable shape and can represent some elementary analytic curves and surfaces, and the curves have variation diminishing thus having a good shape control.The research topics of this paper and some existing documents, such as [19, 20], are similar. There are also some differences, though. By extending the global parameter to local parameter, Wu and Chen [19] presented cubic nonuniform trigonometric polynomial curves with multiple shape parameters. The curves are C
2 continuity for a nonuniform knot vector and C
3 continuity for a uniform knot vector, respectively. Han [20] presented quadratic trigonometric polynomial curves with local basis. The curves have C
2 continuity with a nonuniform knot vector and any value of the bias. Compared to [19, 20], novelty of this paper is listed as follows. First, it discusses the total positivity of the basis functions. This property makes the corresponding curves have variation diminishing property, which is one of the important properties of dominant Bézier curves and B-spline curves. Second, it provides the representation method of surface of revolution. In surface modeling, the construction of the rotational surface is a common problem. Third, it provides a class of higher order continuous curves, which can meet most of the needs in engineering.The rest of the paper is organized as follows. Section2 gives the definition and properties of the basis functions. Section 3 defines the associated curves and gives the representation of the ellipses and parabolas. Section 4 defines the associated surfaces and gives the representation of the rotating surfaces. Section 5 concludes the paper.


## 2. Basis Functions

### 2.1. The Construction of the Basis Functions

In [7], cubic trigonometric splines are presented for a nonuniform knot vector. These splines are used to define trigonometric spline curves. As special cases, the author also introduces a class of cubic trigonometric polynomial basis functions used to construct trigonometric Bézier curves. The original expression of the basis functions is as follows:(1)
B
i
-
3
u
=
1
-
sin
⁡
t
2
1
-
λ
sin
⁡
t
,
B
i
-
2
u
=
sin
⁡
t
1
-
sin
⁡
t
2
2
+
λ
1
-
sin
⁡
t
,
B
i
-
1
u
=
cos
⁡
t
1
-
cos
⁡
t
2
2
+
λ
1
-
cos
⁡
t
,
B
i
u
=
1
-
cos
⁡
t
2
1
-
λ
cos
⁡
t
.This set of basis functions contains only one shape parameter. In [17], it was further extended to possess two shape parameters.By changing theλ in the last two functions in (1) to μ, a class of cubic trigonometric Bézier basis functions with two shape parameters is defined in [17] as follows. Let λ
,
μ
∈
[
-
2,1
], for t
∈
[
0
,
π
/
2
]; the following four functions are defined to be the cubic trigonometric Bézier basis functions (T-Bézier basis for short), with two shape parameters λ and μ:(2)
T
0
t
;
λ
=
1
-
sin
⁡
t
2
1
-
λ
sin
⁡
t
,
T
1
t
;
λ
=
sin
⁡
t
1
-
sin
⁡
t
2
+
λ
-
λ
sin
⁡
t
,
T
2
t
;
μ
=
cos
⁡
t
1
-
cos
⁡
t
2
+
μ
-
μ
cos
⁡
t
,
T
3
t
;
μ
=
1
-
cos
⁡
t
2
1
-
μ
cos
⁡
t
.In [16], the T-Bézier basis was proved to be the optimal normalized totally positive basis of the space T
λ
,
μ
:
=
S
p
a
n
{
1
,
sin
2
⁡
t
,
(
1
-
sin
⁡
t
)
2
(
1
-
λ
sin
⁡
t
)
,
(
1
-
cos
⁡
t
)
2
(
1
-
μ
cos
⁡
t
)
} for λ
,
μ
∈
(
-
2,1
]. Hence the corresponding cubic trigonometric Bézier curves are suited for conformal curve design. However, the Bézier curve is a single curve segment. When using Bézier curve to describe complex shapes, the problem of joining curve segments smoothly needs to be solved. The Bézier form is the special case of B-spline. B-spline curves consist of many polynomial pieces, offering much more versatility than Bézier curves. Considering the B-spline is more suitable for expressing complex curve and surface, this paper will discuss more generally the B-spline form.Next we will construct a kind of cubic trigonometric spline basis function based on theT-Bézier basis.Given knotsu
0
<
u
1
<
⋯
<
u
n
+
4, we refer to U
=
(
u
0
,
u
1
,
…
,
u
n
+
4
) as a knot vector. Let h
j
=
u
j
+
1
-
u
j, and t
j
(
u
)
=
(
π
/
2
)
(
u
-
u
j
/
h
j
), j
=
0,1
,
…
,
n
+
3; we want to construct the associated spline basis functions as follows:(3)
B
i
u
=
B
i
,
0
t
i
,
u
∈
u
i
,
u
i
+
1
,
B
i
,
1
t
i
+
1
,
u
∈
u
i
+
1
,
u
i
+
2
,
B
i
,
2
t
i
+
2
,
u
∈
u
i
+
2
,
u
i
+
3
,
B
i
,
3
t
i
+
3
,
u
∈
u
i
+
3
,
u
i
+
4
,
0
,
u
∉
u
i
,
u
i
+
4for i
=
0,1
,
…
,
n. Here (4)
B
i
,
0
t
=
d
i
T
3
t
;
λ
i
+
1
,
B
i
,
1
t
=
c
i
+
1,0
T
0
t
;
λ
i
+
1
+
c
i
+
1,1
T
1
t
;
λ
i
+
1
+
c
i
+
1,2
T
2
t
;
λ
i
+
2
+
c
i
+
1,3
T
3
t
;
λ
i
+
2
,
B
i
,
2
t
=
b
i
+
2,0
T
0
t
;
λ
i
+
2
+
b
i
+
2,1
T
1
t
;
λ
i
+
2
+
b
i
+
2,2
T
2
t
;
λ
i
+
3
+
b
i
+
2,3
T
3
t
;
λ
i
+
3
,
B
i
,
3
t
=
a
i
+
3
T
0
t
;
λ
i
+
3
,in which T
j
(
t
;
λ
) for j
=
0,1
,
2,3 are the T-Bézier basis given in (2) and a
i
+
3
,
b
i
+
2
,
j
,
c
i
+
1
,
j
,
d
i for j
=
0,1
,
2,3 are undetermined coefficients.To determine the coefficient values, we impose two conditions onB
i
(
u
), which have C
2 continuity at each knot and form a partition of unity on the interval [
u
3
,
u
n
+
1
]. Then, we can compute the coefficients as follows:(5)
a
i
=
β
i
-
1
h
i
2
α
i
γ
i
,
b
i
,
0
=
β
i
γ
i
+
1
φ
i
-
1
h
i
+
γ
i
ϕ
i
+
1
h
i
-
1
α
i
γ
i
γ
i
+
1
,
b
i
,
1
=
β
i
ϕ
i
+
1
γ
i
+
1
,
b
i
,
2
=
β
i
h
i
+
1
γ
i
+
1
,
b
i
,
3
=
a
i
+
1
,
d
i
=
β
i
+
2
h
i
2
α
i
+
1
γ
i
+
2
,
c
i
,
3
=
β
i
+
1
γ
i
+
2
φ
i
h
i
+
1
+
γ
i
+
1
ϕ
i
+
2
h
i
α
i
+
1
γ
i
+
1
γ
i
+
2
,
c
i
,
2
=
β
i
+
1
φ
i
γ
i
+
1
,
c
i
,
1
=
β
i
+
1
h
i
-
1
γ
i
+
1
,
c
i
,
0
=
d
i
-
1
,where α
i
=
h
i
-
1
+
h
i, β
i
=
1
+
2
λ
i, φ
i
=
h
i
-
1
+
β
i
h
i, and ϕ
i
=
β
i
h
i
-
1
+
h
i, γ
i
=
β
i
h
i
-
2
+
β
i
-
1
β
i
h
i
-
1
+
β
i
-
1
h
i.Foru
∈
[
u
i
,
u
i
+
1
), i
=
3,4
,
…
,
n, we have B
j
(
u
)
=
0 for j
≠
i
-
3
,
i
-
2
,
i
-
1
,
i, and (6)
B
i
-
3
u
=
B
i
-
3,3
t
i
,
B
i
-
2
u
=
B
i
-
2,2
t
i
,
B
i
-
1
u
=
B
i
-
1,1
t
i
,
B
i
u
=
B
i
,
0
t
i
.We can easily check that(7)
B
i
-
3
u
,
B
i
-
2
u
,
B
i
-
1
u
,
B
i
u
=
T
0
t
i
,
λ
i
,
T
1
t
i
,
λ
i
,
T
2
t
i
,
λ
i
+
1
,
T
3
t
i
,
λ
i
+
1
H
i
,where (8)
H
i
=
a
i
b
i
,
0
c
i
,
0
0
0
b
i
,
1
c
i
,
1
0
0
b
i
,
2
c
i
,
2
0
0
b
i
,
3
c
i
,
3
d
i
.Theorem 1.
Whenλ
i
-
1
,
λ
i
,
λ
i
+
1
,
λ
i
+
2
∈
(
-
1
/
2
,
1
], H
i is a nonsingular stochastic and totally positive matrix.Proof.
Whenλ
i
-
1
,
λ
i
,
λ
i
+
1
,
λ
i
+
2
∈
(
-
1
/
2
,
1
], it is obvious that a
i
,
b
i
,
j
,
c
i
,
j
,
d
i are all positive for j
=
0,1
,
2,3. In addition, from (5) we can verify that a
i
+
b
i
,
0
+
c
i
,
0
=
b
i
,
1
+
c
i
,
1
=
b
i
,
2
+
c
i
,
2
=
b
i
,
3
+
c
i
,
3
+
d
i
=
1. This means that H
i is stochastic. With direct computation we have(9)
b
i
,
0
c
i
,
0
b
i
,
1
c
i
,
1
=
η
i
φ
i
-
1
h
i
-
1
α
i
γ
i
γ
i
+
1
>
0
,
b
i
,
0
c
i
,
0
b
i
,
2
c
i
,
2
=
η
i
γ
i
+
1
φ
i
-
1
φ
i
+
γ
i
δ
i
h
i
-
1
α
i
γ
i
γ
i
+
1
2
>
0
,
b
i
,
0
c
i
,
0
b
i
,
3
c
i
,
3
=
η
i
γ
i
+
1
γ
i
+
2
φ
i
-
1
φ
i
h
i
+
1
+
γ
i
+
1
2
φ
i
-
1
ϕ
i
+
2
h
i
+
γ
i
γ
i
+
1
ϕ
i
+
1
ϕ
i
+
2
h
i
-
1
+
γ
i
γ
i
+
2
δ
i
h
i
-
1
h
i
+
1
α
i
α
i
+
1
γ
i
γ
i
+
1
2
γ
i
+
2
>
0
,
b
i
,
1
c
i
,
1
b
i
,
2
c
i
,
2
=
η
i
δ
i
γ
i
+
1
2
>
0
,
b
i
,
1
c
i
,
1
b
i
,
3
c
i
,
3
=
η
i
γ
i
+
1
ϕ
i
+
1
ϕ
i
+
2
+
γ
i
+
2
δ
i
h
i
+
1
α
i
+
1
γ
i
+
1
2
γ
i
+
2
>
0
,
b
i
,
2
c
i
,
2
b
i
,
3
c
i
,
3
=
η
i
ϕ
i
+
2
h
i
+
1
α
i
+
1
γ
i
+
1
γ
i
+
2
>
0
,where η
i
=
β
i
β
i
+
1
h
i, δ
i
=
h
i
-
1
+
h
i
+
h
i
+
1
+
2
λ
i
α
i
+
1
+
2
λ
i
+
1
α
i
+
4
λ
i
λ
i
+
1
h
i. From these, we can easily deduce that H
i is nonsingular and all its remaining minors are nonnegative. Therefore, H
i is a nonsingular stochastic and totally positive matrix.Definition 2.
Given a knot vectorU
=
(
u
0
,
u
1
,
…
,
u
n
+
4
) and an array (
λ
1
,
λ
2
,
…
,
λ
n
+
3
)
≜
λ, where all λ
i
∈
(
-
1
/
2
,
1
], i
=
1,2
,
…
,
n
+
3, by using the T-Bézier basis and the coefficients given in (5), functions (3) are defined to be the associated cubic trigonometric B-spline basis functions (T-B-spline basis for short).It is easy to know the shape ofB
i
(
u
) is related to λ
i
+
1
,
λ
i
+
2
,
λ
i
+
3. We refer to the array λ
i
=
(
λ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
) as the shape parameter. For equidistant knots, we refer to the B
i
(
u
) as a uniform T-B-spline basis and refer to the knot vector U as a uniform knot vector. For nonequidistant knots, B
i
(
u
) and U are called a nonuniform T-B-spline basis and a nonuniform knot vector, respectively.Figure1 shows some graphs of uniform T-B-spline basis functions with different shape parameters. As can be seen from the figure, with the increase of the parameter λ
i
+
1, the maximum of B
i
(
u
) decreases and the highest point of the curve moves toward the right. With the increase of λ
i
+
2, the maximum of B
i
(
u
) increases correspondingly. With the increase of λ
i
+
2, the maximum of B
i
(
u
) decreases and the highest point of the curve moves toward the left. When λ
i
+
1
=
λ
i
+
3, the graph of B
i
(
u
) is symmetry.Figure 1
T
-
B-spline basis functions with different shape parameters.


### 2.2. The Property of the Basis Functions

Theorem 3.
TheT-B-spline basis has the following properties:(a)
Partition of unity.∑
i
=
0
n
B
i
(
u
)
=
1 for u
∈
[
u
3
,
u
n
+
1
].
(b)
Nonnegativity. Ifλ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
], then B
i
(
u
)
>
0 for u
∈
(
u
i
,
u
i
+
4
).
(c)
Linear independence. Forλ
i
-
1
,
λ
i
,
λ
i
+
1
∈
(
-
1
/
2
,
1
], i
=
3,4
,
…
,
n
+
1, the set {
B
0
(
u
)
,
B
1
(
u
)
,
…
,
B
n
(
u
)
} is linearly independent on [
u
3
,
u
n
+
1
].
(d)
Normalized total positivity. Foru
∈
[
u
i
,
u
i
+
1
], λ
i
-
1
,
λ
i
,
λ
i
+
1
,
λ
i
+
2
∈
(
-
1
/
2
,
1
], i
=
3,4
,
…
,
n, the system (
B
i
-
3
(
u
)
,
B
i
-
2
(
u
)
,
B
i
-
1
(
u
)
,
B
i
(
u
)
) is a normalized totally positive basis of the space T
λ
i
,
λ
i
+
1.
(e)
Continuity. With a nonuniform knot vector, theT-B-spline basis B
i
(
u
) has C
2 continuity for λ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
] at each of the knots. With a uniform knot vector, B
i
(
u
) has C
3 continuity for λ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
] and C
5 continuity for λ
i
+
1
=
λ
i
+
2
=
λ
i
+
3
=
1 at each of the knots.Proof.
(a) Notice thata
i
+
b
i
,
0
+
c
i
,
0
=
b
i
,
1
+
c
i
,
1
=
b
i
,
2
+
c
i
,
2
=
b
i
,
3
+
c
i
,
3
+
d
i
=
1. For u
∈
[
u
i
,
u
i
+
1
), i
=
3,4
,
…
,
n, we have B
j
(
u
)
=
0 for j
≠
i
-
3
,
i
-
2
,
i
-
1
,
i; then (10)
∑
j
=
0
n
B
j
u
=
∑
j
=
i
-
3
i
B
j
u
=
∑
j
=
0
3
B
i
-
j
,
j
t
i
=
1
.
(b) Whenλ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
], we can see from (5) that a
i
+
3
,
b
i
+
2
,
j
,
c
i
+
1
,
j
,
d
i
>
0 (j
=
0,1
,
2,3). Thus, from the positivity of the T-Bézier basis, we can conclude that B
i
(
u
)
>
0 for u
∈
(
u
i
,
u
i
+
4
).
(c) For anyξ
j
∈
R (j
=
0,1
,
…
,
n), u
∈
[
u
3
,
u
n
+
1
], we define (11)
S
u
=
∑
j
=
0
n
ξ
j
B
j
u
=
0
.For λ
i
-
1
,
λ
i
,
λ
i
+
1
∈
(
-
1
/
2
,
1
],  i
=
3,4
,
…
,
n
+
1, direct computation gives that(12)
S
u
i
=
a
i
ξ
i
-
3
+
b
i
,
0
ξ
i
-
2
+
c
i
,
0
ξ
i
-
1
=
0
,
S
′
u
i
=
π
2
+
λ
i
2
h
i
a
i
ξ
i
-
2
-
ξ
i
-
3
+
h
i
h
i
-
1
d
i
-
1
ξ
i
-
1
-
ξ
i
-
2
=
0
,
S
′
′
u
i
=
π
2
1
+
2
λ
i
2
h
i
2
a
i
ξ
i
-
3
-
ξ
i
-
2
+
h
i
2
h
i
-
1
2
d
i
-
1
ξ
i
-
1
-
ξ
i
-
2
=
0
.Thus, we can obtain the following linear systems of equations with respect to ξ
i
-
3
,
ξ
i
-
2
,
ξ
i
-
1:(13)
a
i
ξ
i
-
3
+
b
i
,
0
ξ
i
-
2
+
c
i
,
0
ξ
i
-
1
=
0
,
a
i
ξ
i
-
2
-
ξ
i
-
3
+
h
i
h
i
-
1
d
i
-
1
ξ
i
-
1
-
ξ
i
-
2
=
0
,
a
i
ξ
i
-
3
-
ξ
i
-
2
+
h
i
2
h
i
-
1
2
d
i
-
1
ξ
i
-
1
-
ξ
i
-
2
=
0
.Since a
i
+
b
i
,
0
+
c
i
,
0
=
1, we have the determinant of the coefficient matrix D
i given by the above linear systems of equations as follows: (14)
D
i
=
a
i
1
c
i
,
0
-
a
i
0
h
i
h
i
-
1
d
i
-
1
a
i
0
h
i
2
h
i
-
1
2
d
i
-
1
=
h
i
h
i
-
1
+
h
i
h
i
-
1
2
a
i
d
i
-
1
>
0
.Therefore, we can conclude that ξ
i
-
3
=
ξ
i
-
2
=
ξ
i
-
1
=
0 for i
=
3,4
,
…
,
n
+
1.
(d) Notice that the system(
T
0
(
t
i
,
λ
i
)
,
T
1
(
t
i
,
λ
i
)
,
T
2
(
t
i
,
λ
i
+
1
)
,
T
3
(
t
i
,
λ
i
+
1
)
) is the optimal normalized totally positive basis of the space T
λ
i
,
λ
i
+
1. Thus, by relation (7) and Theorem 1, we can see that the system (
B
i
-
3
(
u
)
,
B
i
-
2
(
u
)
,
B
i
-
1
(
u
)
,
B
i
(
u
)
) is a totally positive basis.
(e) For anyλ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
] direct computation we have (15)
B
i
k
u
i
+
=
B
i
k
u
i
+
4
-
=
0
,
k
=
0,1
,
2,3
,
5
.
B
i
u
i
+
1
±
=
d
i
,
B
i
u
i
+
2
±
=
1
-
a
i
+
2
-
d
i
+
1
,
B
i
u
i
+
3
±
=
a
i
+
3
,
B
i
′
u
i
+
1
±
=
π
d
i
2
+
λ
i
+
1
2
h
i
,
B
i
′
u
i
+
2
±
=
π
2
+
λ
i
+
2
2
a
i
+
2
h
i
+
2
-
d
i
+
1
h
i
+
1
,
B
i
′
u
i
+
3
±
=
-
π
a
i
+
3
2
+
λ
i
+
3
2
h
i
+
3
,
B
i
′
′
u
i
+
1
±
=
π
2
d
i
1
+
2
λ
i
+
1
2
h
i
2
,
B
i
′
′
u
i
+
2
±
=
-
π
2
1
+
2
λ
i
+
2
2
d
i
+
1
h
i
+
1
2
+
a
i
+
2
h
i
+
2
2
,
B
i
′
′
u
i
+
3
±
=
π
2
a
i
+
3
1
+
2
λ
i
+
3
2
h
i
+
3
2
.When h
i
=
h
i
+
1
=
h
i
+
2
=
h
i
+
3
≜
h, for any λ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
], we have (16)
B
i
′
′
′
u
i
+
1
±
=
π
3
d
i
5
λ
i
+
1
-
2
8
h
3
,
B
i
′
′
′
u
i
+
2
±
=
π
3
5
λ
i
+
2
-
2
a
i
+
2
-
d
i
+
1
8
h
3
,
B
i
′
′
′
u
i
+
3
±
=
π
3
a
i
+
3
2
-
5
λ
i
+
3
8
h
3
.Further, when h
i
=
h
i
+
1
=
h
i
+
2
=
h
i
+
3
≜
h and λ
i
+
1
=
λ
i
+
2
=
λ
i
+
3
=
1, we have (17)
B
i
4
u
i
+
=
B
i
4
u
i
+
4
-
=
0
,
B
i
4
u
i
+
1
±
=
-
3
π
4
d
i
2
h
4
,
B
i
4
u
i
+
2
±
=
3
π
4
a
i
+
2
+
d
i
+
1
2
h
4
,
B
i
4
u
i
+
3
±
=
-
3
π
4
a
i
+
3
2
h
4
,
B
i
5
u
i
+
1
±
=
-
57
π
5
d
i
32
h
5
,
B
i
5
u
i
+
2
±
=
57
π
5
d
i
+
1
-
a
i
+
2
32
h
5
,
B
i
5
u
i
+
3
±
=
57
π
5
a
i
+
3
32
h
5
.This means that the conclusion is correct.So far in the discussion of theT-B-spline basis, we have assumed that each knot is single. On the other hand, the T-B-spline basis also makes sense when knots are considered with multiplicity k
≤
4. For multiple knots, we shrink the corresponding intervals to zero and drop the corresponding pieces of the basis function. For example, if u
i
=
u
i
+
1 is a double knot, then we define (18)
B
i
u
=
B
i
,
1
t
i
+
1
,
u
∈
u
i
+
1
,
u
i
+
2
,
B
i
,
2
t
i
+
2
,
u
∈
u
i
+
2
,
u
i
+
3
,
B
i
,
3
t
i
+
3
,
u
∈
u
i
+
3
,
u
i
+
4
,
0
,
u
∉
u
i
,
u
i
+
4
.As a direct application of functions (3) and (e) in Theorem 3, we have the following corollary which shows the geometric meaning of multiple knots.Corollary 4.
Suppose that aT-B-spline basis B
i
(
u
) has a knot with multiplicity k (k
=
2,3
,
4) at a parameter value u; then at this point the continuity of the T-B-spline basis is reduced from C
2 to C
3
-
k (C
-
1 means discontinuous) for any λ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
]. Moreover, the support interval of the basis is reduced from 4 segments to 5
-
k segments.


## 2.1. The Construction of the Basis Functions

In [7], cubic trigonometric splines are presented for a nonuniform knot vector. These splines are used to define trigonometric spline curves. As special cases, the author also introduces a class of cubic trigonometric polynomial basis functions used to construct trigonometric Bézier curves. The original expression of the basis functions is as follows:(1)
B
i
-
3
u
=
1
-
sin
⁡
t
2
1
-
λ
sin
⁡
t
,
B
i
-
2
u
=
sin
⁡
t
1
-
sin
⁡
t
2
2
+
λ
1
-
sin
⁡
t
,
B
i
-
1
u
=
cos
⁡
t
1
-
cos
⁡
t
2
2
+
λ
1
-
cos
⁡
t
,
B
i
u
=
1
-
cos
⁡
t
2
1
-
λ
cos
⁡
t
.This set of basis functions contains only one shape parameter. In [17], it was further extended to possess two shape parameters.By changing theλ in the last two functions in (1) to μ, a class of cubic trigonometric Bézier basis functions with two shape parameters is defined in [17] as follows. Let λ
,
μ
∈
[
-
2,1
], for t
∈
[
0
,
π
/
2
]; the following four functions are defined to be the cubic trigonometric Bézier basis functions (T-Bézier basis for short), with two shape parameters λ and μ:(2)
T
0
t
;
λ
=
1
-
sin
⁡
t
2
1
-
λ
sin
⁡
t
,
T
1
t
;
λ
=
sin
⁡
t
1
-
sin
⁡
t
2
+
λ
-
λ
sin
⁡
t
,
T
2
t
;
μ
=
cos
⁡
t
1
-
cos
⁡
t
2
+
μ
-
μ
cos
⁡
t
,
T
3
t
;
μ
=
1
-
cos
⁡
t
2
1
-
μ
cos
⁡
t
.In [16], the T-Bézier basis was proved to be the optimal normalized totally positive basis of the space T
λ
,
μ
:
=
S
p
a
n
{
1
,
sin
2
⁡
t
,
(
1
-
sin
⁡
t
)
2
(
1
-
λ
sin
⁡
t
)
,
(
1
-
cos
⁡
t
)
2
(
1
-
μ
cos
⁡
t
)
} for λ
,
μ
∈
(
-
2,1
]. Hence the corresponding cubic trigonometric Bézier curves are suited for conformal curve design. However, the Bézier curve is a single curve segment. When using Bézier curve to describe complex shapes, the problem of joining curve segments smoothly needs to be solved. The Bézier form is the special case of B-spline. B-spline curves consist of many polynomial pieces, offering much more versatility than Bézier curves. Considering the B-spline is more suitable for expressing complex curve and surface, this paper will discuss more generally the B-spline form.Next we will construct a kind of cubic trigonometric spline basis function based on theT-Bézier basis.Given knotsu
0
<
u
1
<
⋯
<
u
n
+
4, we refer to U
=
(
u
0
,
u
1
,
…
,
u
n
+
4
) as a knot vector. Let h
j
=
u
j
+
1
-
u
j, and t
j
(
u
)
=
(
π
/
2
)
(
u
-
u
j
/
h
j
), j
=
0,1
,
…
,
n
+
3; we want to construct the associated spline basis functions as follows:(3)
B
i
u
=
B
i
,
0
t
i
,
u
∈
u
i
,
u
i
+
1
,
B
i
,
1
t
i
+
1
,
u
∈
u
i
+
1
,
u
i
+
2
,
B
i
,
2
t
i
+
2
,
u
∈
u
i
+
2
,
u
i
+
3
,
B
i
,
3
t
i
+
3
,
u
∈
u
i
+
3
,
u
i
+
4
,
0
,
u
∉
u
i
,
u
i
+
4for i
=
0,1
,
…
,
n. Here (4)
B
i
,
0
t
=
d
i
T
3
t
;
λ
i
+
1
,
B
i
,
1
t
=
c
i
+
1,0
T
0
t
;
λ
i
+
1
+
c
i
+
1,1
T
1
t
;
λ
i
+
1
+
c
i
+
1,2
T
2
t
;
λ
i
+
2
+
c
i
+
1,3
T
3
t
;
λ
i
+
2
,
B
i
,
2
t
=
b
i
+
2,0
T
0
t
;
λ
i
+
2
+
b
i
+
2,1
T
1
t
;
λ
i
+
2
+
b
i
+
2,2
T
2
t
;
λ
i
+
3
+
b
i
+
2,3
T
3
t
;
λ
i
+
3
,
B
i
,
3
t
=
a
i
+
3
T
0
t
;
λ
i
+
3
,in which T
j
(
t
;
λ
) for j
=
0,1
,
2,3 are the T-Bézier basis given in (2) and a
i
+
3
,
b
i
+
2
,
j
,
c
i
+
1
,
j
,
d
i for j
=
0,1
,
2,3 are undetermined coefficients.To determine the coefficient values, we impose two conditions onB
i
(
u
), which have C
2 continuity at each knot and form a partition of unity on the interval [
u
3
,
u
n
+
1
]. Then, we can compute the coefficients as follows:(5)
a
i
=
β
i
-
1
h
i
2
α
i
γ
i
,
b
i
,
0
=
β
i
γ
i
+
1
φ
i
-
1
h
i
+
γ
i
ϕ
i
+
1
h
i
-
1
α
i
γ
i
γ
i
+
1
,
b
i
,
1
=
β
i
ϕ
i
+
1
γ
i
+
1
,
b
i
,
2
=
β
i
h
i
+
1
γ
i
+
1
,
b
i
,
3
=
a
i
+
1
,
d
i
=
β
i
+
2
h
i
2
α
i
+
1
γ
i
+
2
,
c
i
,
3
=
β
i
+
1
γ
i
+
2
φ
i
h
i
+
1
+
γ
i
+
1
ϕ
i
+
2
h
i
α
i
+
1
γ
i
+
1
γ
i
+
2
,
c
i
,
2
=
β
i
+
1
φ
i
γ
i
+
1
,
c
i
,
1
=
β
i
+
1
h
i
-
1
γ
i
+
1
,
c
i
,
0
=
d
i
-
1
,where α
i
=
h
i
-
1
+
h
i, β
i
=
1
+
2
λ
i, φ
i
=
h
i
-
1
+
β
i
h
i, and ϕ
i
=
β
i
h
i
-
1
+
h
i, γ
i
=
β
i
h
i
-
2
+
β
i
-
1
β
i
h
i
-
1
+
β
i
-
1
h
i.Foru
∈
[
u
i
,
u
i
+
1
), i
=
3,4
,
…
,
n, we have B
j
(
u
)
=
0 for j
≠
i
-
3
,
i
-
2
,
i
-
1
,
i, and (6)
B
i
-
3
u
=
B
i
-
3,3
t
i
,
B
i
-
2
u
=
B
i
-
2,2
t
i
,
B
i
-
1
u
=
B
i
-
1,1
t
i
,
B
i
u
=
B
i
,
0
t
i
.We can easily check that(7)
B
i
-
3
u
,
B
i
-
2
u
,
B
i
-
1
u
,
B
i
u
=
T
0
t
i
,
λ
i
,
T
1
t
i
,
λ
i
,
T
2
t
i
,
λ
i
+
1
,
T
3
t
i
,
λ
i
+
1
H
i
,where (8)
H
i
=
a
i
b
i
,
0
c
i
,
0
0
0
b
i
,
1
c
i
,
1
0
0
b
i
,
2
c
i
,
2
0
0
b
i
,
3
c
i
,
3
d
i
.Theorem 1.
Whenλ
i
-
1
,
λ
i
,
λ
i
+
1
,
λ
i
+
2
∈
(
-
1
/
2
,
1
], H
i is a nonsingular stochastic and totally positive matrix.Proof.
Whenλ
i
-
1
,
λ
i
,
λ
i
+
1
,
λ
i
+
2
∈
(
-
1
/
2
,
1
], it is obvious that a
i
,
b
i
,
j
,
c
i
,
j
,
d
i are all positive for j
=
0,1
,
2,3. In addition, from (5) we can verify that a
i
+
b
i
,
0
+
c
i
,
0
=
b
i
,
1
+
c
i
,
1
=
b
i
,
2
+
c
i
,
2
=
b
i
,
3
+
c
i
,
3
+
d
i
=
1. This means that H
i is stochastic. With direct computation we have(9)
b
i
,
0
c
i
,
0
b
i
,
1
c
i
,
1
=
η
i
φ
i
-
1
h
i
-
1
α
i
γ
i
γ
i
+
1
>
0
,
b
i
,
0
c
i
,
0
b
i
,
2
c
i
,
2
=
η
i
γ
i
+
1
φ
i
-
1
φ
i
+
γ
i
δ
i
h
i
-
1
α
i
γ
i
γ
i
+
1
2
>
0
,
b
i
,
0
c
i
,
0
b
i
,
3
c
i
,
3
=
η
i
γ
i
+
1
γ
i
+
2
φ
i
-
1
φ
i
h
i
+
1
+
γ
i
+
1
2
φ
i
-
1
ϕ
i
+
2
h
i
+
γ
i
γ
i
+
1
ϕ
i
+
1
ϕ
i
+
2
h
i
-
1
+
γ
i
γ
i
+
2
δ
i
h
i
-
1
h
i
+
1
α
i
α
i
+
1
γ
i
γ
i
+
1
2
γ
i
+
2
>
0
,
b
i
,
1
c
i
,
1
b
i
,
2
c
i
,
2
=
η
i
δ
i
γ
i
+
1
2
>
0
,
b
i
,
1
c
i
,
1
b
i
,
3
c
i
,
3
=
η
i
γ
i
+
1
ϕ
i
+
1
ϕ
i
+
2
+
γ
i
+
2
δ
i
h
i
+
1
α
i
+
1
γ
i
+
1
2
γ
i
+
2
>
0
,
b
i
,
2
c
i
,
2
b
i
,
3
c
i
,
3
=
η
i
ϕ
i
+
2
h
i
+
1
α
i
+
1
γ
i
+
1
γ
i
+
2
>
0
,where η
i
=
β
i
β
i
+
1
h
i, δ
i
=
h
i
-
1
+
h
i
+
h
i
+
1
+
2
λ
i
α
i
+
1
+
2
λ
i
+
1
α
i
+
4
λ
i
λ
i
+
1
h
i. From these, we can easily deduce that H
i is nonsingular and all its remaining minors are nonnegative. Therefore, H
i is a nonsingular stochastic and totally positive matrix.Definition 2.
Given a knot vectorU
=
(
u
0
,
u
1
,
…
,
u
n
+
4
) and an array (
λ
1
,
λ
2
,
…
,
λ
n
+
3
)
≜
λ, where all λ
i
∈
(
-
1
/
2
,
1
], i
=
1,2
,
…
,
n
+
3, by using the T-Bézier basis and the coefficients given in (5), functions (3) are defined to be the associated cubic trigonometric B-spline basis functions (T-B-spline basis for short).It is easy to know the shape ofB
i
(
u
) is related to λ
i
+
1
,
λ
i
+
2
,
λ
i
+
3. We refer to the array λ
i
=
(
λ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
) as the shape parameter. For equidistant knots, we refer to the B
i
(
u
) as a uniform T-B-spline basis and refer to the knot vector U as a uniform knot vector. For nonequidistant knots, B
i
(
u
) and U are called a nonuniform T-B-spline basis and a nonuniform knot vector, respectively.Figure1 shows some graphs of uniform T-B-spline basis functions with different shape parameters. As can be seen from the figure, with the increase of the parameter λ
i
+
1, the maximum of B
i
(
u
) decreases and the highest point of the curve moves toward the right. With the increase of λ
i
+
2, the maximum of B
i
(
u
) increases correspondingly. With the increase of λ
i
+
2, the maximum of B
i
(
u
) decreases and the highest point of the curve moves toward the left. When λ
i
+
1
=
λ
i
+
3, the graph of B
i
(
u
) is symmetry.Figure 1
T
-
B-spline basis functions with different shape parameters.


## 2.2. The Property of the Basis Functions

Theorem 3.
TheT-B-spline basis has the following properties:(a)
Partition of unity.∑
i
=
0
n
B
i
(
u
)
=
1 for u
∈
[
u
3
,
u
n
+
1
].
(b)
Nonnegativity. Ifλ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
], then B
i
(
u
)
>
0 for u
∈
(
u
i
,
u
i
+
4
).
(c)
Linear independence. Forλ
i
-
1
,
λ
i
,
λ
i
+
1
∈
(
-
1
/
2
,
1
], i
=
3,4
,
…
,
n
+
1, the set {
B
0
(
u
)
,
B
1
(
u
)
,
…
,
B
n
(
u
)
} is linearly independent on [
u
3
,
u
n
+
1
].
(d)
Normalized total positivity. Foru
∈
[
u
i
,
u
i
+
1
], λ
i
-
1
,
λ
i
,
λ
i
+
1
,
λ
i
+
2
∈
(
-
1
/
2
,
1
], i
=
3,4
,
…
,
n, the system (
B
i
-
3
(
u
)
,
B
i
-
2
(
u
)
,
B
i
-
1
(
u
)
,
B
i
(
u
)
) is a normalized totally positive basis of the space T
λ
i
,
λ
i
+
1.
(e)
Continuity. With a nonuniform knot vector, theT-B-spline basis B
i
(
u
) has C
2 continuity for λ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
] at each of the knots. With a uniform knot vector, B
i
(
u
) has C
3 continuity for λ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
] and C
5 continuity for λ
i
+
1
=
λ
i
+
2
=
λ
i
+
3
=
1 at each of the knots.Proof.
(a) Notice thata
i
+
b
i
,
0
+
c
i
,
0
=
b
i
,
1
+
c
i
,
1
=
b
i
,
2
+
c
i
,
2
=
b
i
,
3
+
c
i
,
3
+
d
i
=
1. For u
∈
[
u
i
,
u
i
+
1
), i
=
3,4
,
…
,
n, we have B
j
(
u
)
=
0 for j
≠
i
-
3
,
i
-
2
,
i
-
1
,
i; then (10)
∑
j
=
0
n
B
j
u
=
∑
j
=
i
-
3
i
B
j
u
=
∑
j
=
0
3
B
i
-
j
,
j
t
i
=
1
.
(b) Whenλ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
], we can see from (5) that a
i
+
3
,
b
i
+
2
,
j
,
c
i
+
1
,
j
,
d
i
>
0 (j
=
0,1
,
2,3). Thus, from the positivity of the T-Bézier basis, we can conclude that B
i
(
u
)
>
0 for u
∈
(
u
i
,
u
i
+
4
).
(c) For anyξ
j
∈
R (j
=
0,1
,
…
,
n), u
∈
[
u
3
,
u
n
+
1
], we define (11)
S
u
=
∑
j
=
0
n
ξ
j
B
j
u
=
0
.For λ
i
-
1
,
λ
i
,
λ
i
+
1
∈
(
-
1
/
2
,
1
],  i
=
3,4
,
…
,
n
+
1, direct computation gives that(12)
S
u
i
=
a
i
ξ
i
-
3
+
b
i
,
0
ξ
i
-
2
+
c
i
,
0
ξ
i
-
1
=
0
,
S
′
u
i
=
π
2
+
λ
i
2
h
i
a
i
ξ
i
-
2
-
ξ
i
-
3
+
h
i
h
i
-
1
d
i
-
1
ξ
i
-
1
-
ξ
i
-
2
=
0
,
S
′
′
u
i
=
π
2
1
+
2
λ
i
2
h
i
2
a
i
ξ
i
-
3
-
ξ
i
-
2
+
h
i
2
h
i
-
1
2
d
i
-
1
ξ
i
-
1
-
ξ
i
-
2
=
0
.Thus, we can obtain the following linear systems of equations with respect to ξ
i
-
3
,
ξ
i
-
2
,
ξ
i
-
1:(13)
a
i
ξ
i
-
3
+
b
i
,
0
ξ
i
-
2
+
c
i
,
0
ξ
i
-
1
=
0
,
a
i
ξ
i
-
2
-
ξ
i
-
3
+
h
i
h
i
-
1
d
i
-
1
ξ
i
-
1
-
ξ
i
-
2
=
0
,
a
i
ξ
i
-
3
-
ξ
i
-
2
+
h
i
2
h
i
-
1
2
d
i
-
1
ξ
i
-
1
-
ξ
i
-
2
=
0
.Since a
i
+
b
i
,
0
+
c
i
,
0
=
1, we have the determinant of the coefficient matrix D
i given by the above linear systems of equations as follows: (14)
D
i
=
a
i
1
c
i
,
0
-
a
i
0
h
i
h
i
-
1
d
i
-
1
a
i
0
h
i
2
h
i
-
1
2
d
i
-
1
=
h
i
h
i
-
1
+
h
i
h
i
-
1
2
a
i
d
i
-
1
>
0
.Therefore, we can conclude that ξ
i
-
3
=
ξ
i
-
2
=
ξ
i
-
1
=
0 for i
=
3,4
,
…
,
n
+
1.
(d) Notice that the system(
T
0
(
t
i
,
λ
i
)
,
T
1
(
t
i
,
λ
i
)
,
T
2
(
t
i
,
λ
i
+
1
)
,
T
3
(
t
i
,
λ
i
+
1
)
) is the optimal normalized totally positive basis of the space T
λ
i
,
λ
i
+
1. Thus, by relation (7) and Theorem 1, we can see that the system (
B
i
-
3
(
u
)
,
B
i
-
2
(
u
)
,
B
i
-
1
(
u
)
,
B
i
(
u
)
) is a totally positive basis.
(e) For anyλ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
] direct computation we have (15)
B
i
k
u
i
+
=
B
i
k
u
i
+
4
-
=
0
,
k
=
0,1
,
2,3
,
5
.
B
i
u
i
+
1
±
=
d
i
,
B
i
u
i
+
2
±
=
1
-
a
i
+
2
-
d
i
+
1
,
B
i
u
i
+
3
±
=
a
i
+
3
,
B
i
′
u
i
+
1
±
=
π
d
i
2
+
λ
i
+
1
2
h
i
,
B
i
′
u
i
+
2
±
=
π
2
+
λ
i
+
2
2
a
i
+
2
h
i
+
2
-
d
i
+
1
h
i
+
1
,
B
i
′
u
i
+
3
±
=
-
π
a
i
+
3
2
+
λ
i
+
3
2
h
i
+
3
,
B
i
′
′
u
i
+
1
±
=
π
2
d
i
1
+
2
λ
i
+
1
2
h
i
2
,
B
i
′
′
u
i
+
2
±
=
-
π
2
1
+
2
λ
i
+
2
2
d
i
+
1
h
i
+
1
2
+
a
i
+
2
h
i
+
2
2
,
B
i
′
′
u
i
+
3
±
=
π
2
a
i
+
3
1
+
2
λ
i
+
3
2
h
i
+
3
2
.When h
i
=
h
i
+
1
=
h
i
+
2
=
h
i
+
3
≜
h, for any λ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
], we have (16)
B
i
′
′
′
u
i
+
1
±
=
π
3
d
i
5
λ
i
+
1
-
2
8
h
3
,
B
i
′
′
′
u
i
+
2
±
=
π
3
5
λ
i
+
2
-
2
a
i
+
2
-
d
i
+
1
8
h
3
,
B
i
′
′
′
u
i
+
3
±
=
π
3
a
i
+
3
2
-
5
λ
i
+
3
8
h
3
.Further, when h
i
=
h
i
+
1
=
h
i
+
2
=
h
i
+
3
≜
h and λ
i
+
1
=
λ
i
+
2
=
λ
i
+
3
=
1, we have (17)
B
i
4
u
i
+
=
B
i
4
u
i
+
4
-
=
0
,
B
i
4
u
i
+
1
±
=
-
3
π
4
d
i
2
h
4
,
B
i
4
u
i
+
2
±
=
3
π
4
a
i
+
2
+
d
i
+
1
2
h
4
,
B
i
4
u
i
+
3
±
=
-
3
π
4
a
i
+
3
2
h
4
,
B
i
5
u
i
+
1
±
=
-
57
π
5
d
i
32
h
5
,
B
i
5
u
i
+
2
±
=
57
π
5
d
i
+
1
-
a
i
+
2
32
h
5
,
B
i
5
u
i
+
3
±
=
57
π
5
a
i
+
3
32
h
5
.This means that the conclusion is correct.So far in the discussion of theT-B-spline basis, we have assumed that each knot is single. On the other hand, the T-B-spline basis also makes sense when knots are considered with multiplicity k
≤
4. For multiple knots, we shrink the corresponding intervals to zero and drop the corresponding pieces of the basis function. For example, if u
i
=
u
i
+
1 is a double knot, then we define (18)
B
i
u
=
B
i
,
1
t
i
+
1
,
u
∈
u
i
+
1
,
u
i
+
2
,
B
i
,
2
t
i
+
2
,
u
∈
u
i
+
2
,
u
i
+
3
,
B
i
,
3
t
i
+
3
,
u
∈
u
i
+
3
,
u
i
+
4
,
0
,
u
∉
u
i
,
u
i
+
4
.As a direct application of functions (3) and (e) in Theorem 3, we have the following corollary which shows the geometric meaning of multiple knots.Corollary 4.
Suppose that aT-B-spline basis B
i
(
u
) has a knot with multiplicity k (k
=
2,3
,
4) at a parameter value u; then at this point the continuity of the T-B-spline basis is reduced from C
2 to C
3
-
k (C
-
1 means discontinuous) for any λ
i
+
1
,
λ
i
+
2
,
λ
i
+
3
∈
(
-
1
/
2
,
1
]. Moreover, the support interval of the basis is reduced from 4 segments to 5
-
k segments.


## 3. Spline Curves

### 3.1. The Construction of the Curves

Definition 5.
Given control pointsP
i
∈
R
d (d
=
2,3
;
i
=
0,1
,
…
,
n) and a knot vector U
=
(
u
0
,
u
1
,
…
,
u
n
+
4
) and an array λ
=
(
λ
1
,
λ
2
,
…
,
λ
n
+
3
), where n
≥
3 and all λ
i
∈
(
-
1
/
2
,
1
] (i
=
1,2
,
…
,
n
+
3), the curve (19)
P
u
=
∑
j
=
0
n
B
j
u
P
j
,
u
∈
u
3
,
u
n
+
1is called a T-B-spline curve.Obviously, foru
∈
[
u
i
,
u
i
+
1
],  i
=
3,4
,
…
,
n, the curve P
(
u
) can be represented by curve segment (20)
P
u
=
∑
j
=
i
-
3
i
B
j
u
P
j
.Analogous to the cubicB-spline curves, since curve P
(
u
) is generated on the interval [
u
3
,
u
n
+
1
], the choice of the first and last three knots is free, and these knots can be adjusted to give the desired boundary behavior of the curve; see the following descriptions.For open curves, we choose the knot vector(21)
U
=
u
0
=
u
1
=
u
2
=
u
3
,
u
4
,
…
,
u
n
,
u
n
+
1
=
u
n
+
2
=
u
n
+
3
=
u
n
+
4
.This ensures that the points P
0 and P
n are the end points of the curves. An example is given in Figure 2 (left).Figure 2
T-B-spline curves with all λ
i
=
0.To construct closed curves, we can extend the given pointsP
0
,
P
1
,
…
,
P
n by setting P
n
+
1
=
P
0, P
n
+
2
=
P
1, and P
n
+
3
=
P
2 and letting h
j
=
h
n
+
j
+
1 (j
=
1,2
,
3,4). Thus the parametric formula for a closed curve is (22)
P
u
=
∑
j
=
0
n
+
3
B
j
u
P
j
,
u
∈
u
3
,
u
n
+
4
,where B
j
(
u
) (j
=
n
+
1
,
n
+
2
,
n
+
3) are given by expanding (3). An example is given in Figure 2 (right).Based on the properties of theT-B-spline basis, we can know that the curve P
(
u
) has the following properties:(a)
CurveP
(
u
) lies in the convex hull of the points P
i
-
3
,
P
i
-
2
,
P
i
-
1
,
P
i for u
∈
[
u
i
,
u
i
+
1
].
(b)
CurveP
(
u
) has variation diminishing property.
(c)
If a knotu
i has multiplicity k (k
=
1,2
,
3,4), then the curve P
(
u
) has C
3
-
k continuity at this point. With a uniform knot vector, the curve P
(
u
) has C
3 continuity for any λ
i
∈
(
-
1
/
2
,
1
] and C
5 continuity if all λ
i
=
1.


### 3.2. The Representation of an Ellipse

Given a uniform knot vector, and given control pointsP
0
=
(
x
1
+
x
2
/
2
,
y
1
), P
1
=
(
x
1
,
y
1
+
y
2
/
2
), P
2
=
(
x
1
+
x
2
/
2
,
y
2
), and P
3
=
(
x
2
,
y
1
+
y
2
/
2
), where x
1
≠
x
2, y
1
≠
y
2. The corresponding T-B-spline curve with all λ
i
=
0 is a segment of an elliptic arc, whose equation is(23)
P
u
=
x
u
y
u
=
x
1
+
x
2
2
+
x
1
-
x
2
3
cos
⁡
t
y
1
+
y
2
2
+
y
2
-
y
1
3
sin
⁡
t
,
t
∈
0
,
π
2
.According to the method of constructing closed curves given in Section3.1, adding three control points P
4
=
P
0, P
5
=
P
1, and P
6
=
P
2, we can obtain the entire ellipse. When |
x
1
-
x
2
|
=
|
y
2
-
y
1
|, we obtain an entire circle. An example is given in Figure 3.Figure 3
An ellipse represented byT-B-spline curve.


### 3.3. The Representation of a Parabola

Given a uniform knot vector and given control pointsP
0
=
(
x
1
+
x
2
/
2
,
y
1
), P
1
=
(
x
1
,
y
2
), P
2
=
P
0, and P
3
=
(
x
2
,
y
2
), where x
1
≠
x
2, y
1
≠
y
2, the corresponding T-B-spline curve with all λ
i
=
0 is a segment of a parabola, whose equation is(24)
P
u
=
x
u
y
u
=
x
1
+
x
2
2
+
x
1
-
x
2
3
cos
⁡
t
2
y
1
+
y
2
3
+
y
2
-
y
1
3
cos
2
⁡
t
,
t
∈
0
,
π
2
.Adding a control pointP
4
=
P
0, we obtain a symmetric parabolic segment composed of two curve segments. An example is given in Figure 4.Figure 4
A parabola represented byT-B-spline curve.


## 3.1. The Construction of the Curves

Definition 5.
Given control pointsP
i
∈
R
d (d
=
2,3
;
i
=
0,1
,
…
,
n) and a knot vector U
=
(
u
0
,
u
1
,
…
,
u
n
+
4
) and an array λ
=
(
λ
1
,
λ
2
,
…
,
λ
n
+
3
), where n
≥
3 and all λ
i
∈
(
-
1
/
2
,
1
] (i
=
1,2
,
…
,
n
+
3), the curve (19)
P
u
=
∑
j
=
0
n
B
j
u
P
j
,
u
∈
u
3
,
u
n
+
1is called a T-B-spline curve.Obviously, foru
∈
[
u
i
,
u
i
+
1
],  i
=
3,4
,
…
,
n, the curve P
(
u
) can be represented by curve segment (20)
P
u
=
∑
j
=
i
-
3
i
B
j
u
P
j
.Analogous to the cubicB-spline curves, since curve P
(
u
) is generated on the interval [
u
3
,
u
n
+
1
], the choice of the first and last three knots is free, and these knots can be adjusted to give the desired boundary behavior of the curve; see the following descriptions.For open curves, we choose the knot vector(21)
U
=
u
0
=
u
1
=
u
2
=
u
3
,
u
4
,
…
,
u
n
,
u
n
+
1
=
u
n
+
2
=
u
n
+
3
=
u
n
+
4
.This ensures that the points P
0 and P
n are the end points of the curves. An example is given in Figure 2 (left).Figure 2
T-B-spline curves with all λ
i
=
0.To construct closed curves, we can extend the given pointsP
0
,
P
1
,
…
,
P
n by setting P
n
+
1
=
P
0, P
n
+
2
=
P
1, and P
n
+
3
=
P
2 and letting h
j
=
h
n
+
j
+
1 (j
=
1,2
,
3,4). Thus the parametric formula for a closed curve is (22)
P
u
=
∑
j
=
0
n
+
3
B
j
u
P
j
,
u
∈
u
3
,
u
n
+
4
,where B
j
(
u
) (j
=
n
+
1
,
n
+
2
,
n
+
3) are given by expanding (3). An example is given in Figure 2 (right).Based on the properties of theT-B-spline basis, we can know that the curve P
(
u
) has the following properties:(a)
CurveP
(
u
) lies in the convex hull of the points P
i
-
3
,
P
i
-
2
,
P
i
-
1
,
P
i for u
∈
[
u
i
,
u
i
+
1
].
(b)
CurveP
(
u
) has variation diminishing property.
(c)
If a knotu
i has multiplicity k (k
=
1,2
,
3,4), then the curve P
(
u
) has C
3
-
k continuity at this point. With a uniform knot vector, the curve P
(
u
) has C
3 continuity for any λ
i
∈
(
-
1
/
2
,
1
] and C
5 continuity if all λ
i
=
1.


## 3.2. The Representation of an Ellipse

Given a uniform knot vector, and given control pointsP
0
=
(
x
1
+
x
2
/
2
,
y
1
), P
1
=
(
x
1
,
y
1
+
y
2
/
2
), P
2
=
(
x
1
+
x
2
/
2
,
y
2
), and P
3
=
(
x
2
,
y
1
+
y
2
/
2
), where x
1
≠
x
2, y
1
≠
y
2. The corresponding T-B-spline curve with all λ
i
=
0 is a segment of an elliptic arc, whose equation is(23)
P
u
=
x
u
y
u
=
x
1
+
x
2
2
+
x
1
-
x
2
3
cos
⁡
t
y
1
+
y
2
2
+
y
2
-
y
1
3
sin
⁡
t
,
t
∈
0
,
π
2
.According to the method of constructing closed curves given in Section3.1, adding three control points P
4
=
P
0, P
5
=
P
1, and P
6
=
P
2, we can obtain the entire ellipse. When |
x
1
-
x
2
|
=
|
y
2
-
y
1
|, we obtain an entire circle. An example is given in Figure 3.Figure 3
An ellipse represented byT-B-spline curve.


## 3.3. The Representation of a Parabola

Given a uniform knot vector and given control pointsP
0
=
(
x
1
+
x
2
/
2
,
y
1
), P
1
=
(
x
1
,
y
2
), P
2
=
P
0, and P
3
=
(
x
2
,
y
2
), where x
1
≠
x
2, y
1
≠
y
2, the corresponding T-B-spline curve with all λ
i
=
0 is a segment of a parabola, whose equation is(24)
P
u
=
x
u
y
u
=
x
1
+
x
2
2
+
x
1
-
x
2
3
cos
⁡
t
2
y
1
+
y
2
3
+
y
2
-
y
1
3
cos
2
⁡
t
,
t
∈
0
,
π
2
.Adding a control pointP
4
=
P
0, we obtain a symmetric parabolic segment composed of two curve segments. An example is given in Figure 4.Figure 4
A parabola represented byT-B-spline curve.


## 4. Spline Surfaces

### 4.1. The Construction of the Surfaces

Exactly as in the construction ofB-spline tensor product surfaces from B-spline curves, we can construct T-B-spline surface from T-B-spline curve. For T-B-spline surface, the shape parameters in u-direction and v-direction can be different.Definition 6.
Given control pointsP
i
j
∈
R
3 (i
=
0,1
,
…
,
m
;
j
=
0,1
,
…
,
n), two knot vectors U
=
(
u
0
,
u
1
,
…
,
u
m
+
4
), V
=
(
v
0
,
v
1
,
…
,
v
n
+
4
), and two arrays λ
u
=
(
λ
1
u
,
λ
2
u
,
…
,
λ
m
+
3
u
), λ
v
=
(
λ
1
v
,
λ
2
v
,
…
,
λ
n
+
3
v
), where m
,
n
≥
3 and all λ
i
u
,
λ
j
v
∈
(
-
1
/
2
,
1
] (i
=
1,2
,
…
,
m
+
3
;
j
=
1,2
,
…
,
n
+
3), then we can define a T-B-spline surface as follows:(25)
P
u
,
v
=
∑
k
=
0
m
∑
l
=
0
n
B
k
u
;
λ
u
B
l
v
;
λ
v
P
k
l
,
u
∈
u
3
,
u
m
+
1
,
v
∈
v
3
,
v
n
+
1
.Foru
∈
[
u
i
,
u
i
+
1
] and v
∈
[
u
j
,
u
j
+
1
], i
=
3,4
,
…
,
m, j
=
3,4
,
…
,
n, the surface P
(
u
,
v
) can be represented by surface patch (26)
P
u
,
v
=
∑
k
=
i
-
3
i
∑
l
=
j
-
3
j
B
k
u
;
λ
u
B
l
v
;
λ
v
P
k
l
.Most properties of theT-B-spline surfaces follow in a straightforward way from those of the T-B-spline curves. For example, if U and V are single knot vectors, the T-B-spline surfaces have C
2 continuity in both directions. If U and V are uniform knot vectors, the T-B-spline surfaces have C
3 continuity in general and C
5 continuity when all shape parameters are equal to 1.


### 4.2. The Representation of a Surface of Revolution

A surface of revolution is given by(27)
P
u
,
v
=
r
v
cos
⁡
u
,
r
v
sin
⁡
u
,
z
v
T
.For fixed v, an isoparametric line v
=
c
o
n
s
t traces out a circle of radius r
(
v
), called a meridian. Since a circle may be exactly represented by T-B-spline curve, we may find an exact representation of a surface of revolution provided we can represent r
(
v
), z
(
v
) in T-B-spline curve form.The most convenient way to define a surface of revolution is to prescribe the (planar) generating curve, or generatrix, given byg
(
v
)
=
(
r
(
v
)
,
0
,
z
(
v
)
)
T (take the curve in x
o
z plane as an example) and by the axis of revolution, in the same plane as g.Suppose the axis of revolution isz-axis. Assume that g is given by a planar T-B-spline curve with control points G
i
=
(
x
i
,
z
i
) (0
≤
i
≤
m), knot vector U, and shape parameter array λ
u. Denote X
=
[
x
0
,
x
1
,
…
,
x
m
], Z
=
[
z
0
,
z
1
,
…
,
z
n
]. Here X and Z are row vectors formed by the abscissa and ordinate of the control points G
i (0
≤
i
≤
m).By the method given in Section3.2, the T-B-spline curve with control points C
0
=
(
0,1.5
), C
1
=
(
1.5,0
), C
2
=
(
0
,
-
1.5
), C
3
=
(
-
1.5,0
), C
4
=
C
0, C
5
=
C
1, and C
6
=
C
2, uniform knot vector V, and shape parameter array λ
v of all elements are zero which is a unit circle whose center is (
0,0
). Denote C
x
=
[
-
1.5,0
,
1.5,0
,
-
1.5,0
,
1
], C
y
=
[
0,1.5,0
,
-
1.5,0
,
1.5,0
], and I
=
[
1,1
,
1,1
,
1,1
,
1
]. Here C
x and C
y are row vectors formed by the abscissa and ordinate of the control points C
i (0
≤
i
≤
6). I is a row vector with all elements being equal to 1, and it has the same dimension with C
x and C
y.LetP
x, P
y, and P
z be (
m
+
1
)
×
7 matrixes defined by P
x
i
,
:
=
X
(
i
)
C
x,  P
y
(
i
,
:
)
=
X
(
i
)
C
y, and P
z
(
i
,
:
)
=
Z
(
i
)
I, where 1
≤
i
≤
m
+
1, P
∗
(
i
,
:
) denotes the ith row of the matrix P
∗ and ∗
(
i
) denotes the ith element of the vector ∗. Suppose P
x, P
y, and P
z are the coordinate component matrices of the control points P
i
j (0
≤
i
≤
m
,
0
≤
j
≤
6). Then the T-B-spline surface with control points P
i
j, knot vectors U and V, and shape parameter arrays λ
u and λ
v is a surface of revolution. Its generatrix is g
(
v
), and its axis of revolution is z-axis.In this way, we represent exactly “classical” surfaces such as cylinders, spheres, or tori. For example, takingX
=
[
1,1
,
1,1
] and Z
=
[
0,1
,
2,3
], we obtain a cylinder. Taking X
=
[
-
2
,
-
1,1
,
2
] and Z
=
X, we obtain a circular cone. Taking X
=
[
-
1,0
,
1,0
,
-
1
] and Z
=
[
0,1
,
0
,
-
1,0
], we obtain a sphere. Taking X
=
2
[
-
1,0
,
1,0
,
-
1
] and Z
=
[
0,1
,
0
,
-
1,0
], we obtain an ellipsoid of revolution. Taking X
=
[
0
,
-
1,0
,
1
] and Z
=
[
0,1
,
0,1
], we obtain a paraboloid of revolution. Taking X
=
C
x
-
4
I and Z
=
C
y, we obtain a tori. In each case we take uniform knot vector U and shape parameter array λ
u with all elements which are equal to 0. Figure 5 shows the results.Figure 5
Surfaces of revolution represented byT-B-spline surfaces.


## 4.1. The Construction of the Surfaces

Exactly as in the construction ofB-spline tensor product surfaces from B-spline curves, we can construct T-B-spline surface from T-B-spline curve. For T-B-spline surface, the shape parameters in u-direction and v-direction can be different.Definition 6.
Given control pointsP
i
j
∈
R
3 (i
=
0,1
,
…
,
m
;
j
=
0,1
,
…
,
n), two knot vectors U
=
(
u
0
,
u
1
,
…
,
u
m
+
4
), V
=
(
v
0
,
v
1
,
…
,
v
n
+
4
), and two arrays λ
u
=
(
λ
1
u
,
λ
2
u
,
…
,
λ
m
+
3
u
), λ
v
=
(
λ
1
v
,
λ
2
v
,
…
,
λ
n
+
3
v
), where m
,
n
≥
3 and all λ
i
u
,
λ
j
v
∈
(
-
1
/
2
,
1
] (i
=
1,2
,
…
,
m
+
3
;
j
=
1,2
,
…
,
n
+
3), then we can define a T-B-spline surface as follows:(25)
P
u
,
v
=
∑
k
=
0
m
∑
l
=
0
n
B
k
u
;
λ
u
B
l
v
;
λ
v
P
k
l
,
u
∈
u
3
,
u
m
+
1
,
v
∈
v
3
,
v
n
+
1
.Foru
∈
[
u
i
,
u
i
+
1
] and v
∈
[
u
j
,
u
j
+
1
], i
=
3,4
,
…
,
m, j
=
3,4
,
…
,
n, the surface P
(
u
,
v
) can be represented by surface patch (26)
P
u
,
v
=
∑
k
=
i
-
3
i
∑
l
=
j
-
3
j
B
k
u
;
λ
u
B
l
v
;
λ
v
P
k
l
.Most properties of theT-B-spline surfaces follow in a straightforward way from those of the T-B-spline curves. For example, if U and V are single knot vectors, the T-B-spline surfaces have C
2 continuity in both directions. If U and V are uniform knot vectors, the T-B-spline surfaces have C
3 continuity in general and C
5 continuity when all shape parameters are equal to 1.


## 4.2. The Representation of a Surface of Revolution

A surface of revolution is given by(27)
P
u
,
v
=
r
v
cos
⁡
u
,
r
v
sin
⁡
u
,
z
v
T
.For fixed v, an isoparametric line v
=
c
o
n
s
t traces out a circle of radius r
(
v
), called a meridian. Since a circle may be exactly represented by T-B-spline curve, we may find an exact representation of a surface of revolution provided we can represent r
(
v
), z
(
v
) in T-B-spline curve form.The most convenient way to define a surface of revolution is to prescribe the (planar) generating curve, or generatrix, given byg
(
v
)
=
(
r
(
v
)
,
0
,
z
(
v
)
)
T (take the curve in x
o
z plane as an example) and by the axis of revolution, in the same plane as g.Suppose the axis of revolution isz-axis. Assume that g is given by a planar T-B-spline curve with control points G
i
=
(
x
i
,
z
i
) (0
≤
i
≤
m), knot vector U, and shape parameter array λ
u. Denote X
=
[
x
0
,
x
1
,
…
,
x
m
], Z
=
[
z
0
,
z
1
,
…
,
z
n
]. Here X and Z are row vectors formed by the abscissa and ordinate of the control points G
i (0
≤
i
≤
m).By the method given in Section3.2, the T-B-spline curve with control points C
0
=
(
0,1.5
), C
1
=
(
1.5,0
), C
2
=
(
0
,
-
1.5
), C
3
=
(
-
1.5,0
), C
4
=
C
0, C
5
=
C
1, and C
6
=
C
2, uniform knot vector V, and shape parameter array λ
v of all elements are zero which is a unit circle whose center is (
0,0
). Denote C
x
=
[
-
1.5,0
,
1.5,0
,
-
1.5,0
,
1
], C
y
=
[
0,1.5,0
,
-
1.5,0
,
1.5,0
], and I
=
[
1,1
,
1,1
,
1,1
,
1
]. Here C
x and C
y are row vectors formed by the abscissa and ordinate of the control points C
i (0
≤
i
≤
6). I is a row vector with all elements being equal to 1, and it has the same dimension with C
x and C
y.LetP
x, P
y, and P
z be (
m
+
1
)
×
7 matrixes defined by P
x
i
,
:
=
X
(
i
)
C
x,  P
y
(
i
,
:
)
=
X
(
i
)
C
y, and P
z
(
i
,
:
)
=
Z
(
i
)
I, where 1
≤
i
≤
m
+
1, P
∗
(
i
,
:
) denotes the ith row of the matrix P
∗ and ∗
(
i
) denotes the ith element of the vector ∗. Suppose P
x, P
y, and P
z are the coordinate component matrices of the control points P
i
j (0
≤
i
≤
m
,
0
≤
j
≤
6). Then the T-B-spline surface with control points P
i
j, knot vectors U and V, and shape parameter arrays λ
u and λ
v is a surface of revolution. Its generatrix is g
(
v
), and its axis of revolution is z-axis.In this way, we represent exactly “classical” surfaces such as cylinders, spheres, or tori. For example, takingX
=
[
1,1
,
1,1
] and Z
=
[
0,1
,
2,3
], we obtain a cylinder. Taking X
=
[
-
2
,
-
1,1
,
2
] and Z
=
X, we obtain a circular cone. Taking X
=
[
-
1,0
,
1,0
,
-
1
] and Z
=
[
0,1
,
0
,
-
1,0
], we obtain a sphere. Taking X
=
2
[
-
1,0
,
1,0
,
-
1
] and Z
=
[
0,1
,
0
,
-
1,0
], we obtain an ellipsoid of revolution. Taking X
=
[
0
,
-
1,0
,
1
] and Z
=
[
0,1
,
0,1
], we obtain a paraboloid of revolution. Taking X
=
C
x
-
4
I and Z
=
C
y, we obtain a tori. In each case we take uniform knot vector U and shape parameter array λ
u with all elements which are equal to 0. Figure 5 shows the results.Figure 5
Surfaces of revolution represented byT-B-spline surfaces.


## 5. Conclusion

With total positivity, theT-B-spline basis is suitable for conformal design. By using the T-B-spline curves, we can represent ellipses and parabolas exactly. The T-B-spline curves can be C
3 or C
5 continuous by taking equidistant knots. By using the T-B-spline surfaces, we can represent rotating surfaces exactly. The T-B-spline surfaces can also reach C
3 or C
5 continuity. One of our future works is to apply the T-B-spline basis to generate shape preserving interpolation curves.



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*Source: 7067408-2016-02-09.xml*