Multi-degree reduction of NURBS curves based on
their explicit matrix representation and polynomial
approximation theory
Cheng Min
Wang Guo-Jin
(E-mail: chengmin@css.zju.edu.cn , wgj@math.zju.edu.cn)
(Institute of Images and Graphics, State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou 310027, China)
Abstract
NURBS curve is one of the most commonly used tools in CAD systems and geometric
modeling for its various specialties, which means that its shape is locally adjustable as well as its
continuity order, and it can represent a conic curve precisely. But how to do degree reduction of NURBS
curves in a fast and efficient way still remains a puzzling problem. By applying the theory of the best
uniform approximation of Chebyshev polynomials and the explicit matrix representation of NURBS
curves, this paper gives the necessary and sufficient condition for degree reducible NURBS curves in an
explicit form. And a new way of doing degree reduction of NURBS curves is also presented, including the
multi-degree reduction of a NURBS curve on each knot span and the multi-degree reduction of a whole
NURBS curve. This method is easy to carry out, and only involves simple calculations. It provides a new
way of doing degree reduction of NURBS curves, which can be widely used in computer graphics and
industrial design.
Keywords: NURBS curves, matrix representation, multi-degree reduction, Chebyshev polynomials
1.
Introduction
As different modeling system coming out ceaselessly, as more and more cross-national corporations
being established, and internet being more and more popularized, it is common to do designs and
machining in different regions. Thus data exchanging, data share and data integration among
geometrical descriptive information of various CAD/CAM systems are getting increasingly frequent.
Different CAD/CAM systems may have quite different requirement and limitations of the degree of the
basis function of curves and surfaces. Furthermore in reality, most CAD/CAM systems usually limit to
deal with only low degree curves and surfaces considering the validity. Thus for the sake of data
exchanging and data integration among curves and surfaces of different degrees, we should do degree
reduction to curves and surfaces of high degree. Besides, in calculations of the functions of curves and
surfaces, degree reduction is also frequently used. Therefore, research on algorithms of degree
reduction has being widely done currently.
On the other hand, NURBS curve is one of the most commonly used mathematical models[1], in the
design/modeling systems of many CAD software design companies for its various specialties, which
means that its shape is locally adjustable as well as its continuity order, and it can represent a conic
curve precisely. So it is obvious that researches on degree reduction of NURBS curves and surfaces are
of great importance in the running and development of modeling systems.
In spite of its great importance, doing degree reduction to NURBS curves involves knot refinement.
Its complicated mathematical expressions often make us flinch. Different from the numerous
researches done in Bézier curves and surfaces, researches in degree elevation and degree reduction of
NURBS curves and surfaces are quite few. We make a simple view of them in the following. By
1
considering each segment of B-spline curves, Piegl and Tiller[2] let each segment of B-spline curves be
represented by a Bézier curve via knot insertion. Thus, algorithms for degree reduction of Bézier
curves can be applied. After each Bézier curve segment is degree reduced, the B-spline curves with the
desired knots are obtained by the knot removal. Wolters, Wu and Farin presented algorithms for degree
reduction of B-spline curves applying the blossoming principle and the least squares method[3]. Yong,
Hu, Sun, Tan and Qin, Huang, gave the necessary and sufficient condition for degree-reducible
B-spline curves. Consequently algorithms for degree reduction of B-spline curves were proposed using
the constrained optimization methods[4-6]. These works are very useful, but it is regrettable that there
are two limitations in using the corresponding methods. Firstly, the degree-reduced curve cannot be
represented in an explicit form. As a result, the approximation error is difficult to be estimated.
Secondly, multi-degree reduction cannot be done in one step. And the step-by-step reduction would
surely produce bigger error, and requires much time.
To avoid the above-mentioned limitations, we have done researches in multi-degree reduction of
NURBS curves in explicit way, and have achieved some results. Applying the explicit matrix
representation of NURBS curves[7,8] and the theory of Chebyshev polynomial approximation[9-12], the
explicit necessary and sufficient condition for degree-reducible NURBS curves is given. Then a new
way of multi-degree reduction is presented in detail, which includes the multi-degree reduction of a
NURBS curve on each knot span and the multi-degree reduction of a whole NURBS curve. The
algorithm is easy to be realized and achieves good result. It obtains the best uniform approximation and
the nearly best uniform approximation respectively.
2. Explicit matrix representation of NURBS curves
Given a knot vector
T = {t i }i¥=-¥ ， t j £ t j +1 , let N i ,k (t ) be the NURBS basis functions of
order k on the given knot vector
T . Let ri = ( wi xi , wi yi , wi z i , wi ), wi > 0, i = 1,2, L n be
n control points under the homogeneous coordinates. Then the NURBS curve of order k
(degree k - 1 ) on the given knot vector T can be represented as
the
n
r (t ) = ( X (t ) ,Y (t ) , Z (t ) , w(t )) = å N i ,k (t )ri ,
i =1
Suppose
t k £ t £ t n +1 , n ³ k .
(1)
t r +1 > t r , now we consider one segment on [t r , t r +1 ) of the NURBS curve (1), which
is under the homogeneous coordinates. If the parameter
t is transformed by the normalized transform
u = (t - t r ) (t r +1 - t r ) , t r £ t < t r +1 ,
the segment on
(2)
[t r , t r +1 ) of the NURBS curve can be conversed into a power form polynomial in the
following form[7,8]
r * (u ) =
r
åN
i = r - k +1
i ,k
(
)
(t )ri = 1, u , L , u k -1 A r ,k ,T (rr - k +1 , rr - k + 2 , L , rr ) ,0 £ u < 1 ,
T
(3)
where the coefficient matrix is
A r ,k ,T = (a j ,i ) k ´k ,
j = 0,1, L , k - 1; i = r - k + 1, r - k + 2, L , r .
2
(4)
Obviously, how to calculate the coefficient matrix is the key problem of power form representation of a
NURBS curve. In Ref.7 and Ref.8, two ways of calculating the explicit expression of the coefficient
matrix is presented. Here we introduce them as the following two lemmas.
2.1
Explicit matrix representation based on generalized difference quotient
Lemma 1.
Let the knot vector be written in the form as
l
(T )
(Ti )
d ( Ti ) i
6444
4l17
4444
8
644444
4
7444444
8
Ti = {t i , t i +1 ,L, t i + k } = {t 1 (Ti ),t 1 (Ti ),L,t 1 (Ti ), L,t d (Ti ) (Ti ),t d (Ti ) (Ti ),L,t d (Ti ) (Ti )} , (5)
where l q (Ti ) > 0, q = 1,2, L, d (Ti ); l1 (Ti ) + l 2 (Ti ) + L + l d (Ti ) (Ti ) = k + 1,
t 1 (Ti ) < t 2 (Ti ) < L < t d (Ti ) (Ti );
Suppose the interval
[t r , t r +1 ) is not empty, then for i = r - k + 1, r - k + 2,L , r , we have
k -1
N i ,k (t ) = N i*,k (u ) = å a ji u j ,u = (t - t r ) (t r +1 - t r ) Î [0,1), t Î [t r , t r +1 ) ,
(6)
j =0
a ji = (t i +k - t i )(t r - t r +1 )
j
d (Ti ) k -1- max( j , k -ll )
å
l =1
t l >t r
å
m =0
a lm (Ti )
(k - 1)!
(t l - t r ) k -1- m - j ,
j!(k - 1 - m - j )!
(7)
where
Here
d lu -1
ö
æ
a ij (Ti ) = b b (i , j ) çç Õ (t v - t u ) lu lv ÕÕ v!÷÷ ,
u =1 v = 0
ø
è 1£u <v £ d
(8)
i = 1,
ì j,
ï
i -1
b(i, j ) = í
i = 1,2, L, d ; j = 0,1, L , li - 1 .
j
+
l
,
i
>
1
,
å
c
ï
c =1
î
(9)
b i (i = 0,1,L, k ) is the algebraic cofactor corresponding to the ith element in the last column
of the matrix
H (T ;1, t ,L , k k ) ,
é1
ê
ê0
êM
ê
ê0
H (T ;1, t ,L, k k ) = ê
êM
ê1
ê
êM
ê0
ëê
2.2
t1
1
t 12 L
2t 1 L
M
0
M
0
L
L
M
M
td
t d2
L
L
M
0
M
0
L
L
ù
t 1k
ú
k -1
kt 1
ú
M
ú
k!
k - l1 +1 ú
t1
ú
(k - l1 + 1) !
ú.
M
ú
ú
t dk
ú
M
ú
k!
t dk -ld +1 ú
(k - l d + 1) !
ûú
Explicit matrix representation based on the Marsden identity
3
(10)
Lemma 2.
[t r , t r +1 ) is not empty, then for i = r - k + 1, r - k + 2,L , r ,
Suppose the interval
we have
k -1
N i ,k (t ) = N i*,k (u ) = å a ji u j , u =
j =0
a ji =
(t r +1 - t r ) j
det G r ,k ,T
k -1
æmö
m= j
è ø
å çç j ÷÷t
m- j
r
t - tr
Î [0,1), t Î [t r , t r +1 ) ,
t r +1 - t r
g m* i , j = 0,1, L , k - 1 .
(11)
(12)
G r ,k ,T is a k ´ k matrix,
Here
G r ,k ,T = ( g m ,i ) k ´k , m = 0,1, L , k - 1; i = r - k + 1, r - k + 2, L , r ，
(-1) m m ! ( k -1- m )
w i ,k
(0) ,
(k - 1)!
w i ,k ( x) = ( x - t i +1 )( x - t i + 2 ) L ( x - t i + k -1 ),k > 1;
g mi =
The element
(13)
(14)
w i ,1 ( x) = 1 ．
(15)
g m*i is the algebraic cofactor corresponding to g mi of G r ,k ,T .
Applying the two lemmas, a NURBS curve can be transformed into a piecewise Ferguson curve in
two ways. This is which our following multi-degree reduction algorithm based on. Taking the length of
the paper into account, hereinafter we only use the conversion matrix
A r ,k ,T in lemma 1 in our
discussion of multi-degree reduction. As for the latter conversion matrix, one can easily get the similar
result.
3.
Necessary and sufficient condition for degree reducible NURBS curves
Based on the lemmas in Section 2, the necessary and sufficient condition for degree reducible
NURBS curves on
[t r , t r +1 ) can be directly obtained. In Ref.4, the necessary and sufficient condition
for degree reducible B-spline curves has been given. But that condition is expressed using the
generalized B divided difference. And a system of equations, which is generated from the condition,
must be solved to get the corresponding degree reduced curve. While in this paper, a degree reducible
condition will be expressed in simple explicit form, and the degree reduced curve can be directly
obtained at the same time without solving a system of equations.
Theorem 1.
The necessary and sufficient condition on the degeneracy of a NURBS curve of order
k on [t r , t r +1 ) is
r
åa
j = r - k +1
where
r = 0,
(16)
k -1, j j
a ij is shown in equation (7). And the degree reduced curve can be expressed as
~
r (t ) = (N r -k + 2,k -1 (t ), N r - k +3,k -1 (t ),L, N r , k -1 (t ) )´
T
A
-1
r , k -1,T
r
r
æ r
ö
ç å a 0, j r j , å a1, j r j ,L , å a k - 2, j r j ÷ .
ç j = r-k +1
÷
j = r-k +1
j = r-k +1
è
ø
4
(17)
4.
Approximation property of Chebyshev polynomials and the conversion between Chebyshev
[9]
bases and power form bases
n is defined as
Chebyshev polynomial of degree
Tn ( x) = cos(n arccos x),
-1 £ x £ 1.
(18)
4.1 The property of the best uniform and the nearly best uniform approximation of Chebyshev
polynomials
Lemma 3.
On the interval
n -1
[- 1,1] ,
the polynomial
g ( x) = å a i Ti ( x) of degree n - 1
i=0
n
achieves the best uniform approximation of the polynomial
f ( x) = å a i Ti ( x) .
i=0
Lemma 4.
f (x) be a polynomial of degree n on [- 1,1] . If represented by Chebyshev
Let
n
bases,
it
can
be
written
f ( x) = å a i Ti ( x)
as
.
Then
the
polynomial
i=0
m
g ( x) = å a i Ti ( x) (m < n - 1) achieves the nearly best uniform approximation of f (x) .
i =0
4.2 Conversion between Chebyshev bases and power form bases
A set of Chebyshev bases is a set of polynomials of degree
n . Based on the conversion between
[12]
Chebyshev bases and Bernstein bases
, the following equation can be obtained.
n
æ æ 2n ö
Tn (2u - 1) = å (-1) n + i çç çç ÷÷
i =0
è è 2i ø
æ nöö n
çç ÷÷ ÷÷ Bi (u ),
èi øø
0 £ u £ 1.
(19)
By expanding the Bernstein basis, equation (19) can be written as
n
n
k
æ 2n ö
n + k æ 2n öæ n - i ö k
÷÷u .
Tn (2u - 1) = å (-1) n +i çç ÷÷u i (1 - u ) n -i = åå (- 1) çç ÷÷çç
i =0
k = 0 i =0
è 2i ø
è 2i øè n - k ø
(20)
Consequently we can get the following theorem.
Theorem 2.
The mutual linear representation between Chebyshev bases
{ }
(0 £ u £ 1) and power form bases u i
Tn = M n C n´n ,
where
n
i =0
{Ti (2u - 1)}in=0
(0 £ u £ 1) can be written in the matrix form as
M n = Tn D n´n , D n´n = C -n´1n ,
Tn = (T0 (2u - 1), T1 (2u - 1),L , Tn (2u - 1) ) ,
5
M n = (1, u , L u n ) ,
(21)
(22)
C n´n
5.
ì i
i + j æ 2 j öæ j - k ö
÷÷ i £ j
ïå (- 1) çç ÷÷çç
= (cij )( n +1)´( n +1) , cij = í k =0
i, j = 0,1, L, n . (23)
è 2k øè j - i ø
ï0
i> j .
î
New algorithm for multi-degree reduction of NURBS curves
The discussion of degree reduction of NURBS curve under the homogeneous coordinates is equal
to that of non-uniform B-spline curve under the affine coordinates. So hereinafter, we will only discuss
degree reduction of non-uniform B-spline curve under the affine coordinates.
5.1 One degree reduction of a segment of a NURBS curve
Suppose the interval
we have
[t r , t r +1 ] is not empty. Then for the NURBS curve segment on [t r , t r +1 ] ,
r
åN
r (t ) =
i = r - k +1
i ,k
(
)
(t )ri = r * (u ) = 1, u , L u k -1 A r ,k ,T (rr - k +1 , rr - k + 2 , L , rr ) .
T
Substitute the equation (21) for the power form bases in the above equation, we can get
(
r * (u ) = (T0 ( 2u - 1), T1 ( 2u - 1), L , Tk -1 ( 2u - 1) ) rr*- k +1 , rr*- k + 2 , L , rr*
Here
(r
k -1
) = å T (2u - 1)r
T
*
i + r - k +1
i
i =0
)
= D ( k -1)´( k -1) A r ,k ,T (rr -k +1 , rr -k + 2 ,L , rr ) .
From lemma 3, we know that the curve of degree k - 2 ,
*
r - k +1
, rr*- k + 2 ,L , rr*
T
T
(24)
k -2
r~ * (u ) = å Ti (2u - 1) ri*+ r - k +1 ,
(25)
is the best uniform degree reduced approximation curve of the curve
r * (u ) = å Ti (2u - 1)ri*+ r - k +1 ,
i=0
k -1
i =0
of which degree is k - 1 .
Then equation (21) and lemma 1 will be used to transform (25) into NURBS form. We have the
following equations:
k -2
(
)
(
r~ * (u ) = å Ti (2u - 1)ri*+ r - k +1 = 1, u , L , u k - 2 C ( k - 2)´( k - 2) rr*- k +1 , rr*- k + 2 , L , rr*-1
i =0
(
= (N r - k + 2, k -1 (t ), N r - k + 3, k -1 (t ), L , N r ,k -1 (t ) ) rr*-*k +1 , rr*-*k + 2 , L , rr*-*1
Here
(r
**
r - k +1
, rr*-*k + 2 ,L , rr*-*1
)
(
T
)
T
=
)
T
r
åN
i =r - k + 2
i , k -1
(t )ri*-*1 . (26)
)
T
= A -r ,1k -1,T C ( k - 2)´( k - 2) rr*-k +1 , rr*-k + 2 , L , rr*-1 .
(27)
Thus the control points of the degree-reduced curve of r (t ) are derived. And the degree-reduced
curve can be written in the form as
r~ (t ) =
r
åN
i = r -k + 2
i , k -1
(t )ri*-*1 .
The above work can be summarized as the following Theorem 3.
Theorem 3.
r (t ) =
As for a NURBS curve segment on
r
åN
i ,k
[t r , t r +1 ] , which is expressed in the form as
(t )ri , the best uniform degree-reduced curve r~ (t ) can be obtained by the equation
i = r - k +1
6
.
(26). Its control points can be calculated by the equation (27), where
the former
(
)
rr*-k +1 , rr*-k + 2 , L , rr*-1 represent
T
k - 1 elements in the column of rr*-k +1 , rr*-k + 2 , L, rr* . And the matrices D , C , and
A is given in equations (21), (23), and (4) respectively. The error bound is
T
ˆ
ˆ
rr* = D
1´( k -1) A r , k ,T (rr - k +1 , rr - k + 2 , L , rr ) , where D1´( k -1) is the last row of the matrix
D ( k -1)´( k -1) .
5.2 Multi- degree reduction of a segment of a NURBS curve
Similar to what we have done in Subsection 5.1, multi-degree reduction of a segment of a NURBS
curve can be done in just one step. Assume that the original curve is of order
k , and we are supposed
k - m (2 £ m £ k - 2) . Our way of doing this is described in the
to reduce it to the order of
following. Firstly transform the given NURBS curve segment into power form. Then represent it by
Chebyshev bases. These two process can be done in one step by the conversion matrices as equation
k - m -1
(24) shows. Then take the partial sum of the Chebyshev series as
å T (2u - 1)r
i
i =0
*
i + r - k +1
. Afterwards,
according to the following formula,
(r
**
r - k +1
)
T
, rr*-*k + 2 , L , rr*-*m
(
)
T
= A -r ,1k -m ,T C ( k -m -1)´( k - m-1) rr*-k +1 , rr*- k + 2 ,L , rr*-m ,
(28)
transform the partial sum back into NURBS representation. Thus the multi-degree reduced curve is
obtained. Based on lemma 4, this multi-degree reduced curve is the nearly best uniform approximation
curve of the original curve. The error bound is
r
år
i = r - m +1
where
*
i
=
k
å
i = k - m +1
T
ˆ (i ) A
D
r , k ,T (rr - k +1 , rr - k + 2 , L , rr ) ,
1´( k -1)
(29)
ˆ (i )
D
1´( k -1) is the i-th row of the matrix D ( k -1)´( k -1) . Obviously as for a NURBS curve segment,
using the way described in this Subsection in one step is equivalent to using the way in Subsection 5.1
to do multi-degree reduction step by step.
5.3 Multi-degree reduction of a whole NURBS curve
For a whole NURBS curve of order k , the adjacent curve segments share k - a control points
(the symbol a represents the multiplicities of the knot where the two curve segment connects).
Therefore to do multi-degree reduction of a whole NURBS curve, one can firstly do multi-degree
reduction to each curve segment respectively. Then calculate the weighted average of the obtained
control points, of which the subscripts are the same. Thus the final multi-degree reduced curve is
obtained. Assume that the original NURBS curve is of order
k , and has n(n ³ k + 1) control
points. Our way of doing multi-degree reduction can be reduced to the following algorithm.
Algorithm 1. (Multi-degree reduction of a whole NURBS curve)
Step1.
Step2.
n - k + 1 segments, each of which
represents the curve segment on [t k , t k +1 ], [t k +1 , t k + 2 ], L , [t n , t n +1 ] , respectively.
Assume that the degree-reduced curve is of order m .
Separate the given whole NURBS curve into
7
Step3.
If k - m = 1 , do as Subsection 5.1 (the corresponding formulae are (24) and (27)).
Else, do as Subsection 5.2 (the corresponding formulae are (24) and (28)).
Then the degree reduced control points of each segment are obtained.
Calculate the weighted average of the obtained control points of which the subscripts are the
same. The detailed process is as follows. (Here we only describe the case of multi-degree
reduction, i.e., m < k - 1 )
3.1 Set the initial value of the array
[r1 , r2 ,L, rn-k +m ]
and
[n1 , n2 ,L, nn-k + m ]
as
0
0 , respectively. The former array represents the value of the control points of the
and
degree-reduced curve obtained finally. And the latter one represents the superposition
times of the control points, which subscripts are the same, of every segment of
degree-reduced curves.
3.2 for (h
3.2.1
3.2.2
= 1, h £ n - k + 1, h + + )
Add the value of the control points of the h-th degree-reduced segment,
calculated by Step2, to rh , rh +1 , L rm + h -1 , respectively.
for ( j
= h, h + 1,L, m + h - 1)
(i = 1, i £ n - k + m , i + + )
n-k + m
According to the control points {ri }i =1
3.3 for
n j = n j + 1.
ri = ri / ni .
calculated by step 3.3, the whole multi-degree
reduced NURBS curve can be rendered.
We should point out that when the difference between curvatures of some NURBS curve segments
is comparatively great, the result obtained by Algorithm 1 will be affected. In that case, we recommend
that the weighted average method be used. The weights of those curve segments of which curvatures
vary greatly should be chosen to be larger.
Step4.
6.
Experiments and results
For the sake of visual clarity, our examples here are planar curves. The control points are expressed
in the homogeneous coordinates of 2D data. In the figure, the original curve and control polygon are
drawn in real line, while the degree reduced curve and control polygon are drawn in dashed.
Example 1.
Degree reduction of a segment of NURBS curve of order 5.
The knot vectors are
control points are
{ri }i5=1
T = {t i }i =1 = {1.1, 2.2 , 3.3 , 4.2 , 4.8 , 6 , 6.7 , 8.5 , 9 ,11} . The original
10
=
{50,40,20} , {10,40,10} , {48,70,12} , {175,100,25} , {65,26,13} .
Fig.1 (a) shows how the degree reduction algorithm in Section 5.1 works. Fig.2 (b) shows how the
degree reduction algorithm in Section 5.2 works, where a NURBS curve segment of order 5 is degree
reduced to that of order 3. We may see that the degree reduced curve almost overlaps the original curve.
Thus the algorithm attains high precision.
a.
Example 2.
Reduce one degree
b. Reduce two degrees in one step
Fig.1: Degree reduction of a segment of NURBS curve
Degree reduction of a whole NURBS curve of order 5.
8
The knot vectors are
T = {t i }i =1 = {1.1, 2 , 3.1, 4.1, 4.9 , 6 , 6.9 , 8.2 , 9.5 ,10.8 ,11.8 ,
17
12.9 ,13.6 ,14.5 ,15.3 ,16 ,16.8 } . The original control points are {ri } 12
i =1 = {15 , 37.5 , 15} ,
{13 , 32 ,10}, {26 , 52 , 13}, {45 , 77 , 15} , {35 , 52 , 10}, {60 , 72 ,15}, {52 , 30 ,10},
{60 , 27 ,10}, {84 , 30 ,12}, {83 , 35 , 10} , {72 , 32 , 8}, {100 , 50 , 10} . Fig.2 shows how Algorithm
1 in Subsection 5.3 works. The whole curve is reduced to order 4.
Fig.2: Degree reduction of a whole NURBS curve
7.
Conclusion
By applying the explicit matrix representation of NURBS curves and the approximation theory of
Chebyshev polynomials, this paper gives the necessary and sufficient condition for degree reducible
segment of a NURBS curve in an explicit form. And a new way of multi-degree reduction of both a
NURBS curve segment and a whole NURBS curve is presented. The necessary and sufficient condition
can be used in fast testing of whether the each segment of the whole NURBS curve is degree reducible.
The algorithm for a whole NURBS curve only involves linear operations. Thus it is easy to carry out,
and can do multi-degree reduction in only one step. It achieves good precision. Examples show how
the algorithm works. It can be fully applied in various modeling systems of NURBS curves. It can also
be supposed to be used in modeling systems of NURBS surfaces after some improvements being made.
Acknowledgements: This work is supported jointly by the National Natural Science Foundation of China (Grant No.60173034),
National Natural Science Foundation for Innovative Research Groups (No.60021201) and the Foundation of State Key Basic
Research 973 Item (Grant No.G1998030600).
References
[1].
Piegl L., Tiller W. The NURBS book. Berlin: Springer, 1995.
[2].
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