QUADRATIC NUAT B‐SPLINE CURVES WITH MULTIPLE SHAPE PARAMETERS

MRIDULA DUBE¹, REENU SHARMA²*
¹Department of Mathematics and Computer Science, R. D. University, Jabalpur
²Department of Mathematics, Mata Gujri Mahila Mahavidyalaya, Jabalpur
* Corresponding Author : reenusharma6@rediffmail.com

Received : 15-04-2011     Accepted : 06-05-2011     Published : 01-06-2011
Volume : 3     Issue : 1       Pages : 18 - 24
Int J Mach Intell 3.1 (2011):18-24
DOI : http://dx.doi.org/10.9735/0975-2927.3.1.18-24

Conflict of Interest : None declared

In this paper a new kind of splines, called quadratic non‐uniform algebraic trigonometric B‐splines (quadratic NUAT B‐splines) with multiple shape parameters are constructed over the space spanned by { }. As each piece of the curves is generated by three consecutive control points, they possess many properties of the quadratic B‐spline curves and quadratic trigonometric B‐spline curves. These curves have continuity with non‐uniform knot vector. These curves are closer to the control polygon than the quadratic B‐spline curves and quadratic trigonometric B‐spline curves when choosing special shape parameters. The shape parameters serve to local control in the curves. The changes of one shape parameter will only affect two curve segments. Taking different values of the shape parameters, one can globally or locally adjust the shapes of the curves. The generation of tensor product surfaces by these new splines is straightforward.

Quadratic NUAT B‐spline; shape parameter; continuity; spline surface.

The trigonometric B‐splines were presented in [12] . The recurrence relation for the trigonometric B‐splines of arbitrary order was established in [8] . The construction of exponential tension B‐splines of arbitrary order was given in [7] . It was further shown in [13] that the trigonometric B‐splines of odd order form a partition of a constant in the case of equidistant knots.
In recent years, several bases in new spaces other than the polynomial space have been proposed for geometric modeling in CAGD. For instance, in [10] a basis is constructed for Cm=span {1, cost, ..., cosmt}. In [11] a basis for the space of trigonometric polynomials {1, sint, cost, ..., sinmt, cosmt} is constructed. Some bases are constructed in [9] for the spaces {1, t, cost, sint, cos2t, sin2t}, {1, t, t², cost, sint} and {1, t, cost, sint, tcost, tsint}. In [1] C‐Bézier curves are constructed in the space spanned by {1,t, t², ..., t^n-2, sint, cost}. Non‐uniform algebraic trigonometric B‐splines (NUAT B‐splines), are generated in [14] over the space spanned by {1, t, ..., t^k-3, cost, sint} in which k is an arbitrary integer larger than or equal to 3. But all these curves do not have any shape parameter. In [15,16] C‐B‐splines are presented in the space spanned by {1, t, sint, cost} with parameter α.
In order to improve the shape of the curve and adjust the extent to which a curve approaches its control polygon, some trigonometric curves are presented by using global shape parameters in [2,5,6] . In [3,4] quadratic trigonometric polynomial curves with local shape parameters are constructed. A new kind of B‐splines, quadratic non‐uniform algebraic trigonometric (NUAT) B‐splines with multiple shape parameters, is presented in this paper. The present paper is organized as follows. In section 2, the basis functions with multiple shape parameters of the Quadratic NUAT B‐Spline Curves are established and the properties of the basis functions are shown. In section 3, Quadratic NUAT B‐Spline Curves are given. Continuity of the curves, Open and closed trigonometric, and shape control of the curves are discussed. In section 4, the corresponding Quadratic NUAT Bézier Curves with multiple shape parameters are defined and their properties are given. The Quadratic NUAT B‐Spline Curve approaches to the control polygon by increasing the shape parameters. It is shown in section 5 that the quadratic non‐uniform algebraic trigonometric (NUAT) B‐splines are closer to the control polygon than the quadratic algebraic B‐spline curves, cubic algebraic B‐spline curves and the quadratic trigonometric polynomial curves of [5] when choosing special shape parameters. The corresponding quadratic non‐uniform algebraic trigonometric (NUAT) B‐spline surfaces with multiple shape parameters are defined in section 6. Our results are supported by various numerical examples in each section.

2.1 The construction of the basis functions
Definition 1. Given Knots u₁ < u₂ < ...... < u_n+3 and refer to U = (u₀, u₁, u₂, ...... u_n+3,) as a knot vector. For shape parameters λ_i [ -2, α ], where , let

,







Then the associated quadratic NUAT basis functions with multiple shape parameters are defined to be the following functions:
b_i (u) =

(1)
For i=0,1,2,......,n.

Theorem 1 The basis functions (1) have the following properties:
(a) Non negativity: b_i ≥ 0 for u_i < u < u_i+3
(b) Partition of unity: b_i (u) = 1, u [ u₂, u_n+1 ]
(c) Smallest support property: b_i (u) = 0 for u₀ < u < u_i, u_i+3 ≤ u ≤ u_n+3
(d) Monotonicity: For u [ u_i, u_i+1 ), i=2,3,......n;
b_i-2 (u) = α_i c(t_i),
b_i-1 (u) = 1 - α_i c(t_i) - β_i d(t_i),
b_i (u) = β_i d(t_i)
For a given value of knot u [ u_i, u_i+1 ), b_i-2 (u) and b_i(u) are monotonically decreasing as shape parameters λ_i-1 and λ_i increase respectively and b_i-1(u) is monotonically increasing with the increase in shape parameters λ_i-1 and λ_i respectively.
In view of the properties (a)‐(d) we say that the basis functions has a support on the interval [ u_i, u_i+3 ] i=0,1,2,......,n. For equidistant knots, we refer to the as a uniform basis function. [Fig-1] shows the curves of uniform basis functions for all λ_i = - 1.8 (blue lines), λ_i = 0 (green lines), and λ_i = 1.8 (red lines) for the knots u₀ = 0, u₁ = 1, u₂ = 2, u₃ = 3, u₄ = 4, u₅ = 5, u₆ = 6. The basis functions defined over non‐equidistant knots are called non‐uniform basis functions. [Fig-2] shows the curves of non‐uniform basis functions for all λ_i = - 1.8(blue lines), λ_i = 0(green lines), and λ_i = 1.8 (red lines) for the knots u₀ = 0, u₁ = 0.7, u₂ = 1.5, u₃ = 2.2, u₄ = 5, u₅ = 6.3, u₆ = 7. The basis functions defined over equidistant knots are called uniform basis functions.

Theorem 2 The quadratic NUAT basis functions b_i (u) has C¹ continuity at each of the knots.
Proof‐ Consider the continuity at the knots u_i+1 and u_i+2 (we can deal with the knots u_i and u_i+3 in the same way). Straightforward computation gives that

b_i = β_i b_i = 1 - α_i+1

b_i = 1- β_i+1 b_i = α_i+2

,



,



Since α_j+1 = 1 - β_j, ,
(0 ≤ j ≤ n+1), we get and , k=0,1
The theorem follows.

So far in the discussion of the basis functions, we have assumed that each point is simple. On the other hand, the basis functions also make sense when knots are considered with multiplicity k ≤ 3. For the quadratic NUAT basis functions, with multiple knots, it is worth notice that we shrink the corresponding intervals to zero and drop the corresponding pieces. For example if u_i+1 = u_i+2 is a double knot, then we define
b_i (u) =



Theorem 3 Suppose that a basis function has a knot of multiplicity k (k=2 or 3) at the parameter value u. Then at this point the continuity of the basis function is reduced from C¹ to C^2-k (C^-1 means discontinuity). Moreover the support interval of the basis function is reduced from 3 segments to 4‐k segments.

Proof It is direct application of (1) and the expressions given in the proof of Theorem [Fig-3] shows the curves of the quadratic NUAT basis functions for all λ_i = - 1.8 (blue lines), λ_i = 0 (green lines), and λ_i = 1.8 (red lines) with a double knot (u₀ = 0, u₁ = 1, u₂ = 2, u₃ = 3, u₄ = 3, u₅ = 4, u₆ = 5). [Fig-4] shows the curves of the quadratic NUAT basis functions for all λ_i = -1.8 (blue lines), λ_i = 0 (green lines), and λ_i = 1.8 (red lines) with a triple knot (u₀ = 0, u₁ = 1, u₂ = 2, u₃ = 3, u₄ = 3, u₅ = 3, u₆ = 4).

Definition 2. Given points P_i (i=0, 1, 2, ..., n) in R² or R³ and a knot vector U = (u₀, u₁, ......, u_n+3). Then
T_u = b_j (u) P_j,
n ≥ 2, u [ u₂, u_n+1 ], (3)
is called a quadratic NUAT curve with multiple shape parameters.
Obviously, for u (u_i, u_i+1) (2 ≤ i ≤ n) the curve T(u) can be represented by curve segment
T(u)= b_i-2(u) P_i-2 + b_i-1(u) P_i-1 + b_i(u) P_i (4)
These curves have many properties of the quadratic B‐spline algebraic curves and trigonometric curves of [ 2, 3, 4, 5, 6 ], such as geometric invariance, convex hull property, symmetry, variation diminishing and locality.
Moreover,
T = α_i (P_i-2 - P_i-1) + P_i-1

T = β_i-1 (P_i-1 - P_i-2) + P_i-2 = α_i (P_i-2 - P_i-1) + P_i-1

Analogous to the quadratic B‐spline curves, the choice of the knot vector automatically determines the continuity of the quadratic NUAT curves at each of the knots as shown by the following theorem.
Theorem 4 If a knot u_i has multiplicity K(k=1, 2 or 3) then the quadratic NUAT curves have C^2-k continuity.
Proof It is a direct result of Theorem 2 and Theorem 3.

Since the curve T(u) is generated on the interval [ u₂, u_n+1 ], the choice of the first and last two knots is free and these knots can be adjusted to give the desired boundary behavior of the curve. See the following descriptions.
For an open algebraic trigonometric curve, we choose the knot vector U = (u₀ = u₁ = u₂, u₃, ......, u_n, u_n+1= u_n+2 = u_n+3). This assure that the points P₀ and P_n are points on the curve. Of course, the interior knots can be multiple knots. [Fig-5] shows open NUAT curves for all λ_i=-1.8, 0, 1.8 (red lines) respectively, open trigonometric curves of [5] for λ_i = -1, 0, 1 (blue lines), and the quadratic B‐spline curves (green lines) corresponding to the same control polygon for a non uniform knot vector u₀ = u₁= u₂ = 3, u₃ = 5, u₄ = 5.5, u₅ = 8, u₆ = 9, u₇ = 10, u₈ = u₉= u₁₀ = 11. As the shape parameters λ_i increases, the algebraic trigonometric curves (red lines) approach to the edge of control polygon.
In order to construct closed quadratic algebraic trigonometric curves, we can extend the given points P₀, P₁, P₂, ......, P_n, by setting P_n+1 = P₀,
P_n+2 = P₁ and let u₂ = u_n+3, u₁ = u_n+2(such that T(u₂) = T(u_n+3), T'(u₂) = T'(u_n+3)), u_n+5 ≥ u_n+4. Thus the parametric formula for a closed quadratic NUAT curve is
T(u) = b_j (u) P_j u [ u₂, u_n+3 ] where b_n+1(u) and b_n+2(u) are given by expanding (1).
[Fig-6] shows closed algebraic trigonometric curves for all λ_i = -1.8, 0, 1.8 (red lines) respectively, closed trigonometric curves of [5] for λ_i = - 1, 0, 1 (blue lines), and the quadratic B‐spline curves (green lines) corresponding to the same control polygon for a uniform knot vector.

For u [ u_i, u_i+1 ], t_i = we rewrite (4) as follows
T(u) = b_i-2(u) P_i-2 + b_i-1(u) P_i-1 + b_i(u) P_i
= P_i+j-2rj(t_i)+
(P_i-2 - P_i-1) λ_i-1 sin t_i (sin t_i - 1) +
(P_i - P_i-1) λ_i cos t_i (cos t_i - 1) (5)
Where






Obviously, shape parameters λ_i-1 and λ_i only affect curves on the control edge (P_i-2 - P_i-1) and (P_i - P_i-1) respectively for all i = 2, 3, ..., n + 1. From (5) we can also predict the following behavior of the curves. The curve has local control due to small support of b_i(u). From (5) we can also predict the following behavior of the curves. Change of one control point will alter at most three segments of the curve. So local adjustment can be made without disturbing the rest of the curve. The shape parameters λ_i also serve to effect local control in the curves. As λ_i-1 increases, the curve moves in the direction of the edge (P_i-2 - P_i-1) and as λ_i-1 decreases the curve moves in the opposite direction to the edge (P_i-2 - P_i-1) for all i = 2, 3, ..., n + 1. As the shape parameter λ_i-1 = λ_i the curve moves in the direction of or the opposite direction to the control point P_i-1, when λ_i-1 (= λ_i) increases or decreases for all i = 2, 3, ..., n + 1. [Table-1] shows some computed examples with different values of shape parameters λ_i. These curves are generated by setting λ=(1.8, 1.8, 1.8, 1.8, 1.8, 1.8, 1.8, 1.8, 1.8, 1.8, 1.8) in (a), λ=(‐1.5, ‐0.2, ‐1.8, 1.8, ‐1.5, ‐1, ‐1.5, 1.8, ‐1.8, ‐0.2, ‐1.5) in (b), λ=(0, 0, 0, 0, ‐1.8, 0, ‐1.8, 0, 0, 0, 0) in (c), λ=(1.8, 1.8, ‐1.4, ‐1.8, 1, 1.5, 1, ‐1.8, ‐1.4, 1.8, 1.8) in (d), λ=(‐1, ‐1.4, 1.8, 1.8, 0, ‐1.5, 0, 1.8, 1.8, ‐1.4, ‐1) in (e), λ=(1.8, 1.8, 0.5, ‐1.5, ‐1.5, ‐1.5, ‐1.5, ‐1.5, 0.5, 1.8, 1.8) in (f), λ=(1, 1.2, 1, ‐1.3, ‐1.5, 1.8, ‐1.5, ‐1.3, 1, 1.2, 1) in (g), λ=(‐1, ‐1.8, ‐1.65, ‐1.5, ‐1.5 1.8, ‐1.5, ‐1.5, ‐1.65, ‐1.8, ‐1) in (h), λ=(1.8, 1.8,‐ 1.8, 1.8, 1.8, ‐1.8, 1.8, 1.8, ‐1.8, 1.8, 1.8) in (i), λ=(‐1.8, 1.8, ‐1.8, ‐1.8, 1.8, 1.8, 1.8, ‐1.8, ‐1.8, 1.8, ‐1.8) in (j).

Let u_i < u_i+1, u_i and u_i+1 be double points (u_i is a triple point when i=2 and u_i+1 is a triple point when i=n), thus we have u_i-1 = u_i+1 = 0, so we obtain the Quadratic NUAT Bézier basis functions with multiple shape parameters λ_i




, where

u [ u_i, u_i+1 ],
These basis functions have the following properties:

(1) b_j (u) = 1, for u [ u_i, u_i+1),
(2) For u [ u_i, u_i+1), if λ_i [ -2, α ],
where ,
then b_j (u) > 0 (j = i - 2, i - 1, i)
For the corresponding quadratic NUAT curve (4) in this case, we have
T (u_i) = P_i-2, T (u_i+1) = P_i,





We call the corresponding quadratic NUAT curve (4) as Quadratic NUAT Bézier curve with multiple shape parameters.

Control polygon provides an important tool in geometric modeling. It is an advantage if the curve being modeled tends to preserve the shape of its control polygon.Now we show the relations of the quadratic NUAT curves with the B‐spline curves of degree d (d=2, 3) and the quadratic trigonometric curves [5] corresponding to their control polygon.
Given data points P_i R² or R³ (i=0, 1, 2, ...... , n) and knots u₀ < u₁ < u₂ < ..... < u_n+3. For u [ u_i, u_i+1 ], the associated B‐spline curve of degree d (d=2 and 3) can be given by
B_i(s) = α_i0(s) P_i-2+α_i1(s)P_i-1+α_i2(s)P_i (6)

Where s =
α_i0(s) = α_i(1-s)^d
α_i1(s) = 1 - α_i(1-s)^d - β_i s^d
α_i2(s) = β_i s^d
With a non‐uniform knot vector U, it is easy to show that T_i (u_i) = B_i (u_i) for i = 2, 3, ...... , n+1. From (4) we
T(u)-P_i-1 = b_i-2(u) (P_i-2 - P_i-1) + b_i(u) (P_i - P_i-1)
obtain then taking the norm we obtain
|| T(u)-P_i-1 || ≤ (b_i-2(u) + b_i(u)) max (|| P_i-2 - P_i-1 ||, || P_i - P_i-1 ||) (7)
Since b_i-2(u) and b_i(u) decreases as λ_i-1 and λ_i increases, from (7) we know that quadratic NUAT curve segments approach to their control polygon with the increase of λ_i-1 and λ_i.
b_i-2(u) + b_i(u) get its minimum value at (i.e. and



at λ_i-1 = λ_i = λ i = 0,1, ..., n, we have

[ α_i P_i-2 - (α_i + β_i) P_i-1 + β_i P_i ]

For the associated Quadratic B‐spline curve (d=2) segment B_i(s) we have

[ α_i P_i-2 - (α_i + β_i) P_i-1 + β_i P_i ]

Let , then λ = - 0.20710. From here we know that Quadratic NUAT curves are closer to the control polygons than the Quadratic B‐spline curves when λ > - 0.20710. Similarly we can show that Quadratic NUAT curves are closer to the control polygons than the ‐ (i) cubic B‐spline curves (d=3) when λ > - 1.4142 and (ii) Quadratic trigonometric curves of [5] when λ > - 1.7927

Given control points P_ij (i=0,1,2,....,m; j=0,1,2,....,n) and the knot vectors U = (u₀, u₁, u₂, ...., u_m+3) and V = (v₀, v₁, v₂, ...., v_m+3), using the tensor product method, we can construct quadratic NUAT surface
T (u,v) = b_i (λ₁, i, u) b_j (λ₂, j, v) P_ij ; m, n ≥ 2; u [ u₂, u_m+1 ], v [ v₂, v_n+1 ]
where b_i (λ₁, i, u) and b_j (λ₂, j, v) are the quadratic NUAT basis functions with multiple shape parameters λ_{1, i} and λ_{2, j} respectively. Obviously these surfaces have properties similar to the corresponding quadratic NUAT curves. This surface is C¹ continuous. In addition, since the surfaces have multiple shape parameters, the shape of the surfaces can be adjusted from two direction (in each direction using multiple shape parameters), so they can more conveniently be used in the outline design.

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Table 1- Some computed examples with different values of shape parameters λi