Moving Planes and Singular Points of Rational Parametric Surfaces

0.11 MB·English

by  Falai Chen

Moving Planes and Singular Points of Rational
Parametric Surfaces
0 Falai Chena,∗, Xuhui Wangb,a
1
0 aDepartment of Mathematics, University of Science and Technology of China
2 Hefei, Anhui, 230026, China
n bSchool of Mathematics, Hefei University of Technology
a Hefei, Anhui, 230009, China
J

4
1

]
A Abstract
N
In this paper we discuss the relationship between the moving planes of a
.
h rational parametric surface and the singular points on it. Firstly, the inter-
t
a section multiplicity of several planar curves is introduced. Then we derive an
m
equivalent definition for the order of a singular point on a rational paramet-
[
ric surface. Based on the new definition of singularity orders, we derive the

2 relationship between the moving planes of a rational surface and the order
v
0 of singular points. Especially, the relationship between the µ-basis and the
1 order of a singular point is also discussed.
8
2 Keywords: Rational parametric surface; Moving plane; µ-basis; Singular
.
9 point
0
9
0
: 1. Introduction
v
i
X Give an algebraic surface f(x,y,z,w) = 0 in homogeneous form, the
r singular points of the surface are the points at which all the partial deriva-
a
tives simultaneously vanish. Geometrically, A singular point on the surface
is a point where the tangent plane is not uniquely defined, and it embodies
geometricshapeandtopologyinformationofthesurface. Detectingandcom-
puting singular pointshaswideapplicationsinSolidModeling andComputer
Aided Geometric Design (CAGD).
∗
Corresponding author
Email addresses: [email protected](Falai Chen), [email protected]
(Xuhui Wang)
Preprint submitted to January 14, 2010
To find the singular points of a parametric surface P(s,t), one can solve
P (s,t) × P (s,t) = 0. However, the find the orders of singularities, one
s t
has to resort to implicit form. Let f(x,y,z,w) be the implicit equation of
P(s,t). A singular point Q = (x ,y ,z ,w ) of P(s,t) has order r if all
0 0 0 0
the partial derivatives of f(x,y,z,w) with order up to r − 1 vanish at Q,
and at least one of the r-th derivative of f at Q is nonzero. Unfortunately,
converting the parametric form of a surface into implicit form is a difficult
task, which is still a hot topic of research [1, 2, 6, 10, 13, 15]. Other meth-
ods such as generalized resultants [3, 17, 18] to find singular points don’t
need prior implicitization. However, they are not designed for detecting and
computing the singular points of higher order (order ≥ 3). In this paper,
we develop methods to treat singularities of rational parametric surfaces di-
rectly. Specifically, we use moving surfaces technique to study singularities
of rational parametric surfaces and the relationship between the order of the
singularities and moving surfaces.
The remainder of this paper is organized as follows. In Section 2, we
recalls some preliminary results about the µ-basis of a rational surface. The
notion of intersection multiplicity of several planar curves is also introduced.
In Section 3, a new definition for the order of a singular point directly from
the parametric equation of a surface is presented, and the equivalence of the
new definition with the classic definition is proved. In Section 4, we discuss
the relationship between the moving planes and µ-basis of a rational surface
and the singular points of the rational surface. We conclude this paper with
future research problems.
2. Preliminaries
Let R[s,t] be the ring of bivariate polynomials in s, t over the set of real
numbers R. A rational parametric surface in homogeneous form is defined
P(s,t) = a(s,t),b(s,t),c(s,t),d(s,t) , (1)
where a,b,c,d ∈ R[s,t] are(cid:0)polynomials with gcd(a,b,(cid:1)c,d) = 1. In order
to apply the theory of algebraic geometry, sometimes we need to work with
homogeneous polynomials,
P(s,t,u) = a(s,t,u),b(s,t,u),c(s,t,u),d(s,t,u) . (2)
A base point of a ratio(cid:0)nal surface P(s,t) is a parameter pa(cid:1)ir (s0,t0) such
that P(s ,t ) = 0. Note that even if the rational surface P(s,t) is real, the
0 0
base points could be complex numbers and possibly at infinity.
2
A moving plane is a family of planes with parametric pairs (s,t) [15]
A(s,t)x+B(s,t)y +C(s,t)z +D(s,t) (3)
where A(s,t),B(s,t),C(s,t),D(s,t) ∈ R[s,t]. A moving plane is said to
follow the rational surface (1) if
A(s,t)a(s,t)+B(s,t)b(s,t)+C(s,t)c(s,t)+D(s,t)d(s,t) ≡ 0. (4)
The moving plane (3) can be written as a vector form
L(s,t) = A(s,t),B(s,t),C(s,t),D(s,t) .
Let L be the set of the (cid:0)moving planes which follow th(cid:1)e rational surface
st
P(s,t), then L is exactly the syzygy module syz(a,b,c,d) and is a free
st
module of rank 3 [6].
Aµ-basis of therational surface (1) consists ofthree moving planes p,q,r
following (1) such that
[p,q,r] = κP(s,t),
whereκisnonzeroconstantand[p,q,r]istheouterproductofp = (p ,p ,p ,p ),
1 2 3 4
q = (q ,q ,q ,q ), and r = (r ,r ,r ,r ) defined by
1 2 3 4 1 2 3 4
p p p p p p p p p p p p
2 3 4 1 3 4 1 2 4 1 2 3
[p,q,r] = q q q ,− q q q , q q q ,− q q q .
2 3 4 1 3 4 1 2 4 1 2 3
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)r r r (cid:12) (cid:12)r r r (cid:12) (cid:12)r r r (cid:12) (cid:12)r r r (cid:12)!
2 3 4 1 3 4 1 2 4 1 2 3
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
A µ-basis forms(cid:12) a basis for(cid:12) the(cid:12)syzygy mo(cid:12)du(cid:12)le L [6].(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) st (cid:12) (cid:12) (cid:12)
Definition 1. Let f(x,y,z,w) = 0 be the implicit equation of the parametric
surface P(s,t). Then X = (x ,y ,z ,w ) is a singular point of order r or
0 0 0 0 0
an r-fold point if all the derivatives of f of order up to r −1 are zero at X
0
and at least one r-th derivative of f does not vanish at X . Specifically, X
0 0
is a double point if and only if
f (X ) = f (X ) = f (X ) = f (X ) = 0,
x 0 y 0 z 0 w 0
and at least one of the second order derivatives is non-zero.
Todiscusstheorderofsingularpointsonarationalparametricsurface, we
need torecall somepreliminary knowledge abouttheintersection multiplicity
of several curves in P2(C), the projective plane over the complex numbers.
3
Definition 2. [12] Let R be a local ring with maximal ideal m and M be a
finitely generated R-module. Assume R contains k = R/m. For l ≫ 0, the
Hilbert polynomial implies that
e
dim (M/ml+1M) = ld +...,
k
d!
where d = dim(R) and e = e(M) is the multiplicity of M.The refined case
is as follows. Let I be an ideal with msM ⊂ IM for some s, then l ≫ 0
implies that
e
dim (M/Il+1M) = ld +...,
k
d!
where e = e(I,M) is the multiplicity of I ineM.
Accordingtotheabovedefinition, wecandefinetheintersectionmultiplic-
e
ity of several planar curves in P2(C). For planar curves C ,C ,...,C which
1 2 v
are defined by homogeneous equations f (s,t,u) = 0,...,f (s,t,u) = 0 re-
1 v
spectively. Let f ,...,f be the local equation of C ,C ,...,C near point
1 v 1 2 v
p, then the intersection multiplicity of these curves at point p is
e e
m(p) = e(I ,R )
p p
forIp = hf1,f2,...,fviandRp = OP2,p, which is theringofrational functions
defined at p.
Assumee mes ⊂ I ⊂e R, then
• If ms ⊂ J ⊂ I ⊂ R, then e(J,R) ≥ e(I,R).
• If IlJ = Il+1, then e(J,R) = e(I,R), J is a reduction ideal of I.
• If I is generated by a regular sequence, then e(I,R) = dim R/I, I is
k
a complete intersection.
Proposition 3. [12]
1. I has a reduction ideal which is generated by a regular sequence.
2. The regular sequence can be chosen to the generic linear combinations
of the generators of I.
4
3. The order of singular points on rational parametric surfaces
Given a rational parametric surface, we first give a definition about the
order of singular points directly from the parametric equation.
Definition 4. For a rational surface (2), a point X = (x ,y ,z ,w ) (wlog,
0 0 0 0 0
assume w 6= 0) is a r-fold singular point if
0
w a(s,t,u)−x d(s,t,u) = w b(s,t,u)−y d(s,t,u)
0 0 0 0
(5)
= w c(s,t,u)−z d(s,t,u) = 0
0 0
has r + λ intersection points (counting multiplicity) in the (s,t,u) plane,
where the multiplicity is defined in (2), and λ is the number of base point of
the surface.
We will show that the above definition is equivalent to the classic defini-
tion of order of singularities (Definition 1).
Theorem 5. Definition 1 and Definition 4 are equivalent.
Proof. X isanr-foldsingular point onasurfaceifandonlyif, forageneric
0
line passing through X , theline intersects the surface at n−r distinct points
0
besides X , here n is the implicit degree of the surface. Without loss of
0
generality, assume the singular point is at the origin X = (x ,y ,z ,w ) =
0 0 0 0 0
(0,0,0,1). Let the generic line be defined by
l = L ∩L ,
1 2
where
L : α x+α y +α z = 0,
1 1 2 3
L : β x+β y +β z = 0.
2 1 2 3
Consider the two planar curves
C : g = α a(s,t,u)+...+α c(s,t,u) = 0,
1 1 3
D : g = β a(s,t,u)+...+β c(s,t,u) = 0.
2 1 3
Let Z bethe commonzeros of a,b,c, S be thesurface, l− bethe linesegments
by removing the origin from the line l, denote ϕ : (s,t,u) 7→ P(s,t,u), then
C ∩D = ϕ−1(S ∩l−)∪Z.
5
By Bezout’s theorem, one has
n2 = #(S ∩l−)+ dimO /hg ,g i . (6)
p 1 2 p
p∈Z
X
From(3),wecangetthatg andg isareductionidealofha(s,t),b(s,t),c(s,t)i ,
1p 2p p
where p is the intersection point of g = 0,g = 0. Thus,
1 2
e(ha,b,ci ,O ) = e(hg ,g i ,O ) = dim O /hg ,g i
p p 1 2 p p k p 1 2 p
Therefore, (6) is equivalent to
n2 = #(S ∩l−)+ e(ha,b,ci ,O ). (7)
p p
p∈Z
X
Let Z be the point set which satisfies a = b = c = 0 and d 6= 0, Z be
1 2
the point set which satisfies a = b = c = d = 0, then
Z = Z ∪Z , and Z ∩Z = ∅.
1 2 1 2
Therefore, (7) is equivalent to
n2 = #(S ∩l−)+r +λ. (8)
Since the implicit degree of the surface is n2 − λ, we immediately get that
(cid:3)
Definition 1 and Definition 4 are equivalent.
4. Relationship between moving planes and singular points
In this section, we study the relationship between the moving planes and
the order of singular points on a rational parametric surface.
Theorem 6. Let P(s,t,u) be a parametric surface with no base points, and
L(s,t,u) be a moving plane following P(s,t,u). If X = (x ,y ,z ,w ) (as-
0 0 0 0 0
sume w 6= 0) is an r-fold singular point on the surface, then
0
w a(s,t,u)−x d(s,t,u) = 0, w b(s,t,u)−y d(s,t,u) = 0,
0 0 0 0
w c(s,t,u)−z d(s,t,u) = 0, L(s,t,u)·X = 0
0 0 0
have r intersection points (counting multiplicity).
6
Proof. The rational parametric surface P(s,t) has the following special
three moving planes
L := −d(s,t),0,0,a(s,t) ,
1
L := 0,−d(s,t),0,b(s,t) ,
2 (cid:0) (cid:1)
L := 0,0,−d(s,t),c(s,t) ,
3 (cid:0) (cid:1)
andtheybelongtoL (cid:0). Givenamovingpla(cid:1)neL(s,t) = A(s,t),B(s,t),C(s,t)
s,t
D(s,t) followtherationalsurfaceP(s,t),assumeA(s,t),B(s,t),C(s,t),D(s,t)
(cid:0)
arerelatively prime (If they arenot relatively prime, we candeal with it simi-
(cid:1)
larly). Asfourdimensional vectors, L ,L ,L areallperpendiculartoP(s,t),
1 2 3
and L is also perpendicular to P(s,t). Thus, there exist h,h ,h ,h ∈ R[s,t],
1 2 3
and gcd(h ,h ,h ) = 1, such that
1 2 3
hL(s,t) = h L (s,t)+h L (s,t)+h L (s,t) (9)
1 1 2 2 3 3
= (−h d,−h d,−h d,h a+h b+h c).
1 2 3 1 2 3
Since gcd(A,B,C,D) = 1 and gcd(h ,h ,h ) = 1,
1 2 3
h = gcd(−h d,−h d,−h d,h a+h b+h c) = gcd(d,h a+h b+h c).
1 2 3 1 2 3 1 2 3
Thus h|d.
For an r-fold singular point X on the surface, (5) have r intersection
0
points (counting multiplicity):
(p ,...,p ) = (s ,t ,u ),··· ,(s ,t ,u ) , (10)
0 v 0 0 0 v v v
and the multiplicity at p is m(cid:0),i = 0,··· ,v, thus r = m (cid:1)+...+m . From
i i 0 v
(9), we have
h(s,t)l(s,t) = h (s,t)l (s,t)+h (s,t)l (s,t)+h (s,t)l (s,t), (11)
1 1 2 2 3 3
where l(s,t) = L·X ,l (s,t) = L ·X ,i = 1,2,3.
0 i i 0
Now we discuss the two possible cases.
• If the intersection point p is not at infinity, assume p = (s ,t ,1), and
i i i i
from (11), we have
h(s−s ,t−t )l(s−s ,t−t ) = h (s−s ,t−t )l (s−s ,t−t )
i i i i 1 i i 1 i i
+h (s−s ,t−t )l (s−s ,t−t )+h (s−s ,t−t )l (s−s ,t−t ).
2 i i 2 i i 3 i i 3 i i
7
Thus,the ideals generated by local equations of h(s,t,u)l(s,t,u), l (s,t,u),
1
l (s,t,u),l (s,t,u)nearp andlocalequationsofl (s,t,u),l (s,t,u),l (s,t,u)
2 3 i 1 2 3
near p are same.
i
• If intersection point p is at infinity, assume p = (1,t ,0). Since L(s,t) ·
i i i
P(s,t) ≡ 0, its homogeneous form also satisfies:
L(s,t,u)·P(s,t,u) ≡ 0,
and the dehomogenized form also has L(1,t,u) · P(1,t,u) ≡ 0. Similar to
the above analysis, their exist h′,h′,h′,h′ ∈ R[t,u], and gcd(h′,h′,h′) = 1,
1 2 3 1 2 3
such that
′ ′ ′ ′
h l(1,t,u) = h l (1,t,u)+h l (1,t,u)+h l (1,t,u),
1 1 2 2 3 3
where l(s,t,u),l (s,t,u) is the homogeneous form of l(s,t),l (s,t),i = 1,2,3,
i i
andh′|d(1,t,u). Therefore, theidealsgeneratedbylocalequationsofh′l(1,t,u),
l (1,t,u),l (1,t,u),l (1,t,u)nearp andlocalequationsofl (1,t,u),l (1,t,u),
1 2 3 i 1 2
l (1,t,u) near p are also same.
3 i
Since h(p ) 6= 0,h′(p ) 6= 0 (otherwise, p is a base point of P(s,t)),
i i i
and based on the definition of the multiplicity, p ,i = 1,...,v are also the
i
intersection points of
w a(s,t,u)−x d(s,t,u) = 0, w b(s,t,u)−y d(s,t,u) = 0,
0 0 0 0
w c(s,t,u)−z d(s,t,u) = 0, L(s,t,u)·X = 0.
0 0 0
and the intersection multiplicity at p are also same. (cid:3)
i
Remark 1. µ-basis p,q,r of P(s,t) are also three special moving planes
of the surface, and from Theorem 6,
L (s,t,u)·X = 0, L (s,t,u)·X = 0, L (s,t,u)·X = 0,
1 0 2 0 3 0
(12)
p(s,t,u)·X = 0, q(s,t,u)·X = 0, r(s,t,u)·X = 0
0 0 0
also have the r intersection points.
Next we discuss the relationship of the µ-basis and the order of singular
points on a rational parametric surface.
For any moving plane l(s,t) ∈ L , there exist polynomials h (s,t),i =
s,t i
1,2,3, such that l(s,t) = h p+h q+h r (Theorem 3.2 in [6]), thus
1 2 3
hL ·X ,L ·X ,L ·X ,p·X ,q·X ,r·X i = hp·X ,q·X ,r·X i.
1 0 2 0 3 0 0 0 0 0 0 0
8
Similar to the proof of Theorem 6, we can get that
p(s,t,u)·X = q(s,t,u)·X = r(s,t,u)·X = 0 (13)
0 0 0
also have the same multiplicity at each intersection point p ,i = 0,...,v as
i
Equation (5).
An immediate consequence of the above analysis is:
Corollary 7. For a rational parametric surface P(s,t) with no base point
and its µ-basis p,q,r, then X = (x ,y ,z ,w ) is r-fold singular point on
0 0 0 0 0
the surface if and only if the intersection points of
p(s,t,u)·X = q(s,t,u)·X = r(s,t,u)·X = 0
0 0 0
is r (counting multiplicity).
Now we consider the moving surface ideal:
I′ = hdx−a,dy−b,dz −c,dw−1i∩R[x,y,z,s,t].
Theorem 3.4, 3.5 of [6] shows that I′ is a prime ideal and f(x,y,z,s,t) ∈ I′
if and only if f(x,y,z,s,t) = 0 is a moving surface following the rational
surface P(s,t). Moreover, if P(s,t) contains no base point then
′
I = hp,q,ri (14)
where p = p · (x,y,z,1), q = q · (x,y,z,1), r = r · (x,y,z,1). Therefore,
For a rational surface P(s,t) with no base point, and any moving surface
f = 0 following it, f ∈ hp,q,ri. Based on the above analysis, we can improve
Theorem 6 as the following theorem
Theorem 8. For a parametric surface P(s,t,u) with no base point and any
moving surface f(x,y,z,w,s,t,u) = 0 following it. If X = (x ,y ,z ,w )
0 0 0 0 0
(assume w 6= 0) is an r-fold singular point on the surface, then
0
w a(s,t,u)−x d(s,t,u) = w b(s,t,u)−y d(s,t,u)
0 0 0 0
= w c(s,t,u)−z d(s,t,u) = f(x ,y ,z ,w ,s,t,u) = 0
0 0 0 0 0 0
haveandonlyhaver intersectionpoints (countingmultiplicity), wheref(x,y,z,
w,s,t,u) is the homogeneous form of f(x,y,z, w,s,t).
9
5. Conclusion
To make the µ-basis moreapplicable in computing the singular points, we
discuss the relations between moving planes (Specially, µ-basis) and singular
point of the rational surface.
In the future, we will discuss how to detect and compute singular points
on an rational parametric surface based on moving planes (or µ-basis) in an
efficient way.
6. Acknowledgements
This work was partially supported by the NSF of China grant 10671192.
[1] B. Buchberger, Applications of Groebner bases in nonlinear computa-
tional geometry, In Kapur, D. and Mundy, J., (eds), Geometric Reason-
ing, Elsevier Science Publisher, MIT Press, 413–446, 1989.
[2] L. Bus´e, D. A. Cox, and C. D’Andrea, Implicitization of surfaces in P3
in the presense of base points, Journal of Algebra and Its Applications,
Vol.2, 189–214, 2003.
[3] L. Bus´e, M. Elkadi, and A. Galligo, Intersection and self-intersection of
surfaces by means of Bezoutian matrices, Computer Aided Geometric
Design, Vol.25, 53–68, 2008.
[4] F. Chen, J. Zheng, and T. W. Sederberg, The µ-basis of a rational ruled
surface, Computer Aided Geometric Design, Vol.18, 61–72, 2001.
[5] F.Chen, andW.Wang, Revisitingtheµ-basisofarationalruledsurface,
J. Symbolic Computation, Vol.36, No.5, 699–716, 2003.
[6] F. Chen, D. A. Cox, and Y. Liu, The µ-basis and implicitization of a
rational parametric surface, Journal of Symbolic Computation, Vol.39,
689–706, 2005.
[7] F. Chen, L. Shen, and J. Deng, Implicitization and parametrization of
quadratic and cubic surfaces by µ-bases, Computing, Vol.79, No.2-4,
131–142, 2007.
[8] F. Chen, W. Wang, and Y. Liu, Computing singular points of plane
rationalcurves, Journal ofSymbolicComputation, Vol.43, 92–117,2008.
10