An Invariant of Algebraic Curves from the Pascal Theorem

0.24 MB·English

by  Zhongxuan Luo

An Invariant of Algebraic Curves from the Pascal Theorem∗
Zhongxuan Luo†
School of Software, School of Mathematical Sciences, Dalian University of Technology,
Dalian, 116024, China
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0 Sept 5, 2007
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Abstract
6
In 1640’s, Blaise Pascal discovered a remarkable property of a hexagon inscribed in a conic

] - Pascal Theorem, which gave birth of the projective geometry. In this paper, a new geometric
G
invariant of algebraic curves is discovered by a different comprehension to Pascals mystic hexa-
A gram or to the Pascal theorem. Using this invariant, the Pascal theorem can be generalized to
. thecaseofcubic(eventoalgebraic curvesofhigherdegree),thatis,For any given 9 intersections
h
between a cubic Γ3 and any three lines a,b,c with no common zero, none of them is a component
t
a of Γ3, then the six points consisting of the three points determined by the Pascal mapping applied
m to any six points (no three points of which are collinear) among those 9 intersections as well as
the remaining three points of those 9 intersections must lie on a conic. This generalization differs
[
quitea bit and is much simpler than Chasles’s theorem and Cayley-Bacharach theorems.

1
v Keywords: Algebraiccurve;Pascaltheorem;Characteristicratio;Characteristicmapping;Char-
4 acteristic number;spline.
4
3 1 Introduction
1
.
1 Algebraiccurveisaclassicalandanimportantsubjectinalgebraicgeometry. Analgebraicplanecurve
0 is the solution set of a polynomial equation P(x,y)=0, where x and y are real or complex variables,
2 andthe degreeofthecurveisthedegreeofthepolynomialP(x,y). LetP2 betheprojectiveplaneand
1 P be the space of all homogeneous polynomials in homogeneous coordinates (x,y,z) of total degree
: n
v ≤ n. An algebraic curve Γ in the projective plane is defined by the solution set of a homogeneous
n
i
X polynomial equation P(x,y,z)=0 of degree n.
In 1640, Blaise Pascaldiscovered a remarkable property of a hexagon inscribed in a circle, shortly
r
a thereafter Pascal realized that a similar property holds for a hexagon inscribed in an ellipse even a
conic. As the birth of the projective geometry, Pascaltheorem assert: If six points on a conic section
is given and a hexagon is made out of them in an arbitrary order, then the points of intersection of
opposite sides of this hexagon will all lie on a single line. The generalizations of Pascal’s theorem
have a glorious history. It has been a subject of active and exciting research. As generalizations of
the Pascal theorem, Chasles’s theorem and Cayley-Bacharachtheorems in various versions received a
great attention both in algebraic geometry and in multivariate interpolation. A detailed introduction
to Cayley-Bacharachtheorems as well as conjectures can be found in [16,19,30].
The Pascaltheorem can be comprehended in the following severalaspect: first, it is easy to verify
that Pascal’s theorem can be proved by Chasles’s theorem [16] and therefore, probably, Chasles’s
theorem has been regarded as a generalization of Pascal’s theorem in the literature. However, the
Chasles’s theorem and Cayley-Bacharachtheorems have not formally inherited the appearance of the
Pascal theorem, that is the three points joined a line are obtained from intersections of three pair of
∗The project is supported by NNSFC(Nos. 10771028, 60533060), Program of New Century Excellent Fellowship of
NECC,andispartiallyfundedbyaDoDfund(DAAD19-03-1-0375).
†Correspondingauthor: [e-mail]
1
linesinwhicheachlinewasdeterminedbytwopointslyingonaconic;Secondly,thePascaltheoremcan
be used to (geometrically)judge whether or not any six points simultaneously lie on a conic. Another
interestingobservationto the Pascaltheoremisthatit playsa keyroleinrevealingthe instability ofa
linear space S1(∆ ) (the set of all piecewise polynomials of degree 2 with global smoothness 1 over
2 MS
Morgan-Scott triangulation, see figure 5.1 and refer to Appendix 5.1). That is, the Pascal theorem
gives an equivalent relationship between the algebraic and geometric conditions to the instability of
S1(∆ )(refer to Appendix 5.1).
2 MS
Actually,readerswillseethatthePascaltheoremcontainsageometricinvariantofalgebraiccurves,
which is exactly reason that we rake up the Pascal theorem in this paper. In order to get this new
invariantofalgebraiccurves,onemustobservethe Pascaltheoremfromadifferentviewpointinwhich
“arbitrarysixpointsaregivenbyintersectionsofaconicandanythreelineswithoutnocommonzero”
instead of “six points on a conic section is given” (a historical viewpoint) in the Pascaltheorem. This
slightdifferent comprehensionto the Pascaltheorem makesus easily generalize the Pascaltheorem to
algebraic curves of higher degrees and discover an invariant of algebraic curves. Similar to the source
of this paper in which all involved points are the set of intersections between lines and a curve, [20]
has given some interesting results to the special case of the following classical problem: Let X be the
intersectionsetoftwoplanealgebraiccurvesDandE thatdonotshareacommoncomponent. Ifdand
e denote the degrees of D and E, respectively, then X consists of at most d·e points (a week form of
Bezout’s theorem[35]). When the cardinalityof X is exactly d·e, X is calleda complete intersection.
How does one describe polynomials of degree at most k that vanish on a complete intersection X or
onits subsets? The case in which both plane curvesD and E are simply unions of lines and the union
D∪E is the (d×e)-cage in question.
Our main results in this paper are enlightened by studying the instability of spline space and are
provedbysplinemethodandthe“principleofduality”intheprojectiveplane. Thispaperisorganized
as follows: In section 2, some basic preliminaries of the projective geometry are given. In section 3,
some new concepts such as characteristic ratio, characteristic mapping and characteristic number of
algebraic curve are introduced by discussing the properties of a line and a conic. Section 4 gives our
main results for the invariant of cubic and presents a generalization of the Pascal type theorem to
cubic. Moreover, some corresponding conclusions to the case of algebraic curves of higher degrees
(n > 3) are also stated in this section without proofs. The basic theory of bivariate spline , a series
of results on the singularity of spline space and the proof of the main result of this paper are given in
Appendix in the end of the paper.
2 Preliminaries of Projective Geometry
It is well knownthatthe “homogeneouscoordinates”andthe “ principle ofduality”1 arethe essential
tools in the projective geometry. A point is the set of all triads equivalent to given triad (x) =
(x ,x ,x ), and a line is the set of all triads equivalent to given triad [X] = [X ,X ,X ]. By a
1 2 3 1 2 3
suitable multiplication (if necessary), any point in the projective plane can be expressed in the form
(x ,x ,1), which can be shortened to (x ,x ), and the two numbers x and x are called the affine
1 2 1 2 1 2
coordinates. Inotherwords,ifx 6=0,the point(x ,x ,x )inthe projectiveplanecanbe regardedas
3 1 2 3
the point (x /x ,x /x ) in the affine plane. The “principle of duality” in the projective plane can be
1 3 2 3
seenclearlyfromthefollowingresult: ”threepoints(u),(v)and(w)inP2 arecollinear”isequivalentto
”threelines[u],[v]and[w]inP2 areconcurrent”. Infact,thenecessaryandsufficientconditionforthe
both statements is: there are numbers λ,µ,ν, not all zero, such that λu +µv +νw =0(i=1,2,3),
i i i
namely,
u u u
1 2 3
(cid:12) v1 v2 v3 (cid:12)=0.
(cid:12) (cid:12)
(cid:12)(cid:12) w1 w2 w3 (cid:12)(cid:12)
(cid:12) (cid:12)
If(u),(v)aredistinctpoints,ν 6=0.He(cid:12)ncethegeneral(cid:12)pointcollinearwith(u)and(v)canbeformeda
linear combinationof (u) and (v). In other word, a point (u)=(u ,u ,u )∈P2 correspondsuniquely
1 2 3
1Ponceletclaimedthisprincipleashisowndiscovery,butitsnaturewasmoreclearlyunderstoodbyanotherFranch-
man,J.D.Gergonne(1771-1859) [11].
2
to a line [u] = [u ,u ,u ] : u x+u y+u z = 0, while a line [u] = [u ,u ,u ] : u x+u y+u z = 0
1 2 3 1 2 3 1 2 3 1 2 3
corresponds uniquely a point (u) = (u ,u ,u ). We say that a point (u) and the corresponding line
1 2 3
[u] are dual to each other - which is the two-dimensional “principle of duality”. Under this duality, it
follows the following definition.
Definition 2.1 (Duality of planar figure). Let ∆ be a planar figure consisting of lines and points in
the projective plane. A planar figure obtained by the corresponding dual lines and points of the points
∗
and lines in ∆ respectively is called the Dual figure of ∆, denotes by ∆ .
For instance, the dual figure of Fig. ?? is shown in Fig. ??, where [·] represents the corresponding
dual line of the point (·) in Fig. ??.
[l3] [l2] (l4)
[l1] (l1)
(A) [B]
(l3)
[C]
(l2)
(D) (C) (B) [l4] [D]
[A]
Figure 1 Figure 2
3 New Definitions
In what follows, we shall use u to represent a point (u) or a line [u] when no ambiguities exist,
u=<a,b>forthe intersectionpointoflinesaandb,anda=(u,v)forthelinewhichjoinsthepoints
u and v.
l c
First, we review the following properties of a line and a
R
conic. Suppose a line l be cut by any three lines a,b and
b
c with no common zero (see Fig. ??). Let u =< c,a >
,v =< a,b > and w =< b,c >, P =< l,a >,Q =< l,b >
Q
andR=<l,c>. Obviously,thereexistnumbersa ,b (i=
i i
1,2,3)such that P =a u+b v,Q=a v+b w,R=a w+
w 1 1 2 2 3
b u, provided in turn u,v,w, then we have
3
v P
a
u Proposition 3.1.
b b b
1 2 3
· · =−1.
a a a
1 2 3
Figure 3
Proof. Without loss of generality, we assume that u = (1,0,0),v = (0,1,0) and w = (0,0,1). Since
P,Q and R are collinear, hence
a b 0
1 1
(cid:12) 0 a2 b2 (cid:12)=0.
(cid:12) (cid:12)
(cid:12) b 0 a (cid:12)
(cid:12) 3 3 (cid:12)
(cid:12) (cid:12)
It follows that b1 · b2 · b3 =−1. (cid:12) (cid:12)
a1 a2 a3
3
With the same notations, it follows that the necessary and sufficient condition for P,Q and R to
be collinear is b1 · b2 · b3 =−1.
a1 a2 a3
Now let us replace the line l in proposition 3.1 by a conic Γ. There are two intersections between
Γ and each a,b,c. Let {p ,p } =< Γ,a >, {p ,p } =< Γ,b > and {p ,p } =< Γ,c >. Consequently,
1 2 3 4 5 6
there are real numbers {a ,b }6 such that
i i i=1
p =a u+b v p =a v+b w p =a w+b u
1 1 1 , 3 3 3 and 5 5 5 . (3.1)
(cid:26) p2 =a2u+b2v (cid:26) p4 =a4v+b4w (cid:26) p6 =a6w+b6u
We have
Theorem 3.2. Let a conic be cut by any three lines with no common zero. Under the notations above,
we have
b b b b b b
1 2 3 4 5 6
· · =1. (3.2)
a a a a a a
1 2 3 4 5 6
Proof. Let u =< c,a >,v =< a,b >, w =< b,c >. Notice that the duality of the figure composed of
the points {p }6 , u,v,w and the lines a,b,c turns out a planar figure with a structure of Morgan-
i i=1
Scott triangulation with inner edges consists of the dual lines of the points {p }6 , u,v,w (see Fig.
i i=1
6). Note that the six points {p }6 lie on a conic, it is shown from Theorem 5.4 (see appendix 5.1)
i i=1
thatthe spline spaceS1(∆ )(the setofallpiecewise polynomialofdegree2with smoothness1over
2 MS
Morgan-Scotttriangulation∆ )issingular,thatisdimS1(∆ )=7. WhichimpliesfromTheorem
MS 2 MS
5.5 (see appendix 5.1) that Theorem 3.3 thus follows.
On the other hand, Theorem 3.2 can be used to tell whether or not any six points simultaneously
lie on a conic. In fact, let p ∈ P2(i = 1,2,··· ,6) be any six distinct points without any three points
i
are collinear, a =(p ,p ),b=(p ,p ), c=(p ,p ), and u=<c,a>,v =<a,b>, w =<b,c>. Using
1 2 3 4 5 6
the same notations as in (3.2), it follows from the proof of Theorem 3.2 that
Proposition 3.3. For any given six points p ,p ,··· ,p without no three points are collinear, (3.2)
1 2 6
is a necessary and sufficient condition for those six points to be lying on a conic.
Actually, Theorem 3.2 is equivalent to the Pascaltheorem.
[Proof of Pascal theorem.] Letp ∈P2(i=1,2,··· ,6)be anysixdistinctpointswithoutanythree
i
points are collinear. Denoted by a = (p ,p ),b = (p ,p ),c = (p ,p ) and u =< c,a >,v =< a,b >,
1 2 3 4 5 6
w =< b,c >. Without loss of generality, we assume u = (1,0,0),v = (0,1,0) and w = (0,0,1). Since
the 6 points p ∈P2(i=1,2,··· ,6) lie on a conic, (3.2) holds. It is clear that
i
q =<(p ,p ),(p ,p )>=(b b ,−a a ,0)=b b u−a a w,
1 1 2 4 5 4 5 4 5 4 5 4 5
q =<(p ,p ),(p p )>=(a a ,0,−b b )=−b b v+a a u,
2 2 3 5 6 2 3 2 3 2 3 2 3
q =<(p ,p ),(p ,p )>=(0,−b b ,a a )=−b b w+a a v,
3 3 4 1 6 1 6 1 6 1 6 1 6
and (3.2) is equivalent to
b b b b b b
1 6 2 3 4 5
(− )·(− )·(− )=−1. (3.3)
a a a a a a
1 6 2 3 4 5
By Proposition 3.1, three points {q ,q ,q } must be collinear. This is the conclusion of the Pascal
1 2 3
theorem.
Notice thatProposition3.1andTheorem3.2,-1and1 areinvariantsofline andconicrespectively.
We therefore introduce the following definitions.
Definition 3.4 (Characteristic ratio). Let u,v∈P2 be two distinct points (or lines), p ,p ,··· ,p be
1 2 k
points (or lines) on the line (u,v) (or passing through < u,v >), then there are numbers a ,b such
i i
that p =a u+b v,i=1,2,··· ,k. The ratio
i i i
b b ··· ,b
1 2 k
[u,v;p ,··· ,p ]:=
1 k
a a ··· ,a
1 2 k
is called the Characteristic ratio of p ,p ,··· ,p with respect to the basic points (or lines) u,v. If
1 2 k
there are multiple points in the intersection points, the corresponding characteristic ratio is defined by
their limit form.
4
Remark 3.5. For four collinear points u,v,p ,p , while the Characteristic ratio of p ,p with respec-
1 2 1 2
tive to u,v is b1b2, the cross ratio in the projective geometry is defined as a1b2.
a2b2 a2b1
Definition 3.6 (Characteristic mapping). Let u and v be two distinct points, and the line (u,v) join
the points p and q. We call q(or p) the characteristic mapping point of p(or q) with respect to the basic
points u and v if
[u,v;p,q]=1,
and denote q =χ (p) (or p=χ (q)).
(u,v) (u,v)
Apparentlyifqisthecharacteristicmappingpoint(orline)ofp,thenpisthecharacteristicmapping
ofq aswell. Thatis,thecharacteristicmappingisreflexive,i.e., χ ◦χ =I (identity mapping).
(u,v) (u,v)
Geometrically, χ (p) and p are symmetric with respect to the mid-point of u and v.
(u,v)
From the definition of the characteristic mapping, Proposition 3.1 and Theorem 3.2, the property
of the characteristic mapping can be shown in the following corollaries.
Corollary 3.7. Any three points P,Q and R in the projective plane P2 are collinear if and only if
their characteristic mapping points χ (P),χ (Q) and χ (R) are collinear.
(u,v) (v,w) (w,u)
Corollary 3.8. Any six distinct points p ∈P2(i=1,2,··· ,6) lie on a conic if and only if the image
i
of their characteristic mapping χ (p ), χ (p ), χ (p ), χ (p ), χ (p ) and χ (p )
(u,v) 1 (u,v) 2 (v,w) 3 (v,w) 4 (w,u) 5 (w,u) 6
lie on a conic as well.
Bezout’stheorem[35]saysthattwo algebraic curves of degree r and s with no common components
have exactly r·s points in the projective complex plane. In particular, a line l and an algebraic curve
C of degree n without the component l meet in exactly n points in the projective complex plane.
Definition 3.9 (Characteristic number). Let Γ be an algebraic curve of degree n, and a,b,c be any
n
threedistinctlines(withoutcommonzero)wherenoneofthemisacomponentofΓ . Supposethatthere
n
exist n intersections between the each line and Γ, and denoted by {p(a),p(b), p(c)}n the intersections
i i i i=1
between Γ and the lines a,b,c, respectively. Let u=<c,a>,v =<a,b>, w =<b,c>. The number
n
K (Γ ):=[u,v;p(a),··· ,p(a)]·[v,w;p(b),··· ,p(b)]·[w,u;p(c),··· ,p(c)],
n n 1 n 1 n 1 n
independent of a,b and c (See Theorem 4.4 below), is called the characteristic number of algebraic
curve Γ of degree n.
n
Itis obviousfromthe Definition3.9thatifΓ is areducible curveofdegreenandhascomponents
n
Γ and Γ ,n = n + n , then K (Γ ) = K (Γ )· K (Γ ). From the discussion below the
n1 n2 1 2 n n n1 n1 n1 n1
Characteristic number is a global invariant of algebraic curves.
By Definition 3.9, the characteristic numbers of line and conic are -1 and +1 respectively.
Definition 3.10 (Pascal mapping). For any 6 points p ,p ,··· ,p without any three points are
1 2 6
collinear in the projective plane, first define Φ by
Φ({p ,p ,··· ,p })={q ,q ,q },
1 2 6 1 2 3
where q =< (p ,p ),(p ,p ) >,q =< (p ,p ),(p ,p ) > and q =< (p ,p ), (p ,p ) >(i.e. {q }3
1 1 2 4 5 2 2 3 5 6 3 3 4 6 1 i i=1
are the three pairs of the continuations of opposite side of the hexagon determined by {p }6 ). Then
i i=1
the Pascal mapping Ψ to {p ,p ,··· ,p } is defined by
1 2 6
Ψ{p ,p ,··· ,p }:=χ◦Φ{p ,p ,··· ,p }:={χ (q ),χ (q ),χ (q )},
1 2 6 1 2 6 (u,v) 1 (w,u) 2 (v,w) 3
where u=<(p ,p ),(p ,p )>,v =<(p ,p ),(p ,p )> and w =<(p ,p ), (p ,p )>.
1 2 5 6 1 2 3 4 3 4 5 6
NoticethatthePascalmappingonp ,p ,··· ,p givingabovedependsontheorderof{p }6 . One
1 2 6 i i=1
canalsodefinethePascalmappingonp ,p ,··· ,p byu=<(p ,p ),(p ,p )>,w =<(p ,p ),(p ,p )>
1 2 6 2 3 4 5 4 5 1 6
and v =<(p ,p ), (p ,p )> instead, which will not affect the result of the Pascaltheorem (Theorem
2 3 1 6
3.11) giving below. But for the case of higher degrees as stated below, we must insist on u,v and w
being defined as in the definition above.
5
Fig. 4 illustrates the Pascal mapping and
the Pascal Theorem that the three points
χ (q ),χ (q ),χ (q ) are derived by ap-
(u,v) 1 (w,u) 2 (v,w) 3
plying the Pascal mapping to p ,p ,··· ,p . From
1 2 6
Corollary3.7,westatethefollowingversionofPas-
cal theorem in order to generalize it to cubic.
Theorem 3.11 (Pascal theorem). For given 6
pointsp ,p ,··· ,p on aconicsection, the3points
1 2 6
of image of the Pascal mapping on these six points,
Ψ({p }6 ), will all lie on a single line.
i i=1
Figure 4: ThePascal mapping and Pascal Theo-
rem
4 Invariant and Pascal Type Theorem
In this section, we show our main results on the characteristic number to algebraic curves. Conse-
quently, a generalizationof the Pascaltheorem to curves of higher degree is given by the “principle of
duality” and the spline method.
For cubic, we have
Theorem 4.1. The characteristic number of cubic is −1, that is, K (Γ )=−1.
3 3
Proof. See Appendix 5.2.
Let a,b and c be any three distinct lines with no common zero in the projective plane, denoted by
u =< c,a >,v =< a,b >,w =< b,c >. Assume that p ,p ,p are three points on a, p ,p ,p are on
1 2 3 4 5 6
b, and p ,p ,p are on c, then there exist real numbers a ,b ,i=1,2,··· ,9 such that
7 8 9 i i
p =a u+b v p =a v+b w p =a w+b u
1 1 1 4 4 4 7 7 7
 p2 =a2u+b2v , p5 =a5v+b5w and p8 =a8w+b8u (4.1)
 p =a u+b v  p =a v+b w  p =a w+b u,
3 3 3 6 6 6 9 9 9
  
Similar to Proposition 3.3, one can easily show, following the proof of Theorem 4.1, that
Proposition 4.2. The nine points p ,p ,··· ,p lie on a cubic which differs from a·b·c = 0 if and
1 2 9
only if
b b b b b b b b b
1 2 3 4 5 6 7 8 9
· · =−1, (4.2)
a a a a a a a a a
1 2 3 4 5 6 7 8 9
holds.
Now, from Proposition4.2, we provide in the following a new generalizationof the Pascaltheorem
to cubic.
Theorem 4.3. For any given 9 intersections between a cubic Γ and any three lines a,b,c with no
3
common zero, none of them is a component of Γ , then the six points consisting of the three points
3
determined by the Pascal mapping applied to any six points (no three points of which are collinear)
among those 9 intersections as well as the remaining three points of those 9 intersections must lie on
a conic.
Proof. Let {p ,p ,p } = Γ a, {p ,p ,p } = Γ b, {p ,p ,p } = Γ c, and u =< c,a >,v =<
1 2 7 3 3 4 8 3 5 6 9 3
a,b >,w =< b,c >. WithTout loss of generality,Twe assume that u =T (1,0,0),v = (0,1,0) and
w = (0,0,1). It is shown in Theorem 4.1 that those 9 points {p }9 ∈ P2 lying on a cubic implies
i i=1
K (Γ )=−1, or equivalently, (4.2) holds. Notice that
3 3
6
q =<(p ,p ),(p ,p )>=(b b ,−a a ,0)=b b u−a a v,
1 1 2 4 5 4 5 4 5 4 5 4 5
q =<(p ,p ),(p p )>=(a a ,0,−b b )=−b b w+a a u,
2 2 3 5 6 2 3 2 3 2 3 2 3
q =<(p ,p ),(p ,p )>=(0,−b b ,a a )=−b b v+a a w,
3 1 6 3 4 1 6 1 6 1 6 1 6
So applying the Pascal mapping on p ,p ,p ,p ,p ,p , we have
1 2 3 4 5 6
{χ (q ),χ (q ),χ (q )}={−a a u+b b v,a a w−b b u,a a v−b b w}.
(u,v) 1 (w,u) 2 (v,w) 3 4 5 4 5 2 3 2 3 1 6 1 6
Since (4.2) is equivalent to
b b b −b b b −b b b
4 5 7 1 6 8 2 3 9
( ) ·( ) ·( ) =1. (4.3)
−a a a a a a a a a
4 5 7 1 6 8 2 3 9
Thus by Theorem 3.2 and Proposition 3.3, the six points {χ (q ), χ (q ), χ (q ),p ,p ,p }
(u,v) 1 (w,u) 2 (v,w) 3 7 8 9
must lie on a conic.
χ(u,v)(q1) ◦p9
p ◦ χ (q )
7 (w,u) 2
u
p
1
◦
◦p
6
◦p p◦2 p χ(v,w)(q3)
5 ◦8
p◦4 p◦3 v
q w
3
q
2
q
1
Figure 5: Generalization of PascalTheorem
Theorem 4.3 implies that if p ,p ,··· ,p are intersection points between a cubic and and any
1 2 9
three distinct lines where non of them is a component of the cubic (see Fig. 5), the three points
χ (q ),χ (q ),χ (q ) along with p ,p ,p will lie on a conic. Obviously, this is an intrinsic
(u,v) 1 (w,u) 2 (v,w) 3 7 8 9
property of cubic!
Here, let us give an example to illustrate the Pascaltype theorem 4.3. Let a cubic Γ be given by
3
−1120x3+560x2y−60xy2+1008y3−450xyz+1200y2z+580xz2−1514yz2
−729z3=0,
and three lines a : x+z = 0, b : −y+z = 0 and c : −x+z = 0 be given. Then the 9 intersections
between Γ and a,b,c are
3
3 1
p =(−4,−1,4), p =(−1,− ,1), p =( ,1,1),
1 2 3
2 4
1 3 3
p =(− ,1,1), p =(1,− ,1), p =(1,− ,1),
4 5 6
4 2 4
1 47
p =(2,−1,−2), p =( ,1,1), p =(1, ,1),
7 8 9
2 42
7
and u=(0,−1,0),v=(−1,1,1),w=(1,1,1,). By direct computation, we have
5
q =<(p ,p ),(p ,p )>=(−1, ,1),
1 1 2 4 5
2
5
q =<(p ,p ),(p ,p )>=(1, ,1),
2 2 3 5 6
2
q =<(p ,p ),(p ,p )>=(−6,1,1)
3 1 6 3 4
and consequently
5 5
χ (q )=(−1, ,1),χ (q )=(1, ,1),χ (q )=(6,1,1).
(u,v) 1 3 (w,u) 2 3 (v,w) 3
Itiseasytoverifythatthesixpointsχ (q ),χ (q ),χ (q )aswellasp ,p ,p lieonaconic:
(u,w) 1 (v,u) 2 (w,v) 3 7 8 9
4x2+39xy−126y2−65xz+312yz−174z2=0.
Ingeneral,foralgebraiccurvesofdegreen(n≥3),wehaveprovedtheinvariantofalgebraiccurves
and the Pascal type theorem to higher degrees. They are listed in the paper without proofs.
Theorem 4.4. For any algebraic curve Γ of degree n, its characteristic number K (Γ ) is always
n n n
equal to (−1)n.
Withthisinvariant,wemayformulateaPascaltypeTheoremforalgebraiccurvesofhigherdegrees:
Theorem4.5(PascaltypeTheorem). Leta,b,cbeanythreedistinctlineswithnocommonzerointhe
projective plane, and {p(a)}n , {p(b)}n , {p(c)}n be given n points lying on a,b and c, respectively.
i i=1 i i=1 i i=1
Then those 3n points {p(a)}n , {p(b)}n , {p(c)}n lie on an algebraic curve of degree n if and only if
i i=1 i i=1 i i=1
the 3(n−1) points consisting of the three points determined by the Pascal mapping applied to any six
points (no three points of which are collinear) among those 3n intersections as well as the remaining
3(n−2) points of those 3n intersections must lie on an algebraic curve of degree n−1 as well.
In view of the simplicity of the invariant, some known results of algebraic curves (see [35], pp.123)
can be easily understood from our invariant (the characteristic number).
Theorem 4.6. If a line cuts a cubic in three distinct points, the residual intersections of the tangents
at these three points are collinear.
Proof. Let l be a line cutting a given cubic Γ at three points p ,p ,p and l ,l ,l be the three
3 1 2 3 1 2 3
tangents at these points respectively. Denote by q ,q ,q the residual intersections between Γ and
1 2 3 3
l ,l ,l , respectively. Let u =< l ,l >,v =< l ,l >,w =< l ,l >. Then there are real numbers
1 2 3 2 3 1 2 3 1
{a ,b ,c ,d }3 such that p = a v+b w,p = a u+b v,p = a w+b u and q = c v+d w,q =
i i i i i=1 1 1 1 2 2 2 3 3 3 1 1 1 2
c u+d v,q =c w+d u. From Theorem 4.1 and Proposition 4.2, we have
2 2 3 3 3
b c b c b c
( 1)2 1 ·( 2)2 2 ·( 3)2 3 =−1.
a d a d a d
1 1 2 2 3 3
Since p ,p ,p are collinear, then b1 · b2 · b3 =−1. Hence, we have c1 · c2 · c3 =−1, and those three
1 2 3 a1 a2 a3 d1 d2 d3
points q ,q ,q are collinear.
1 2 3
Theorem 4.7. A line joining two flexes of a cubic passes through a third flexes.
Proof. Let p ,p ,p be three flexes of a cubic, and l ,l ,l be the three tangents at these points.
1 2 3 1 2 3
Let u =< l ,l >,v =< l ,l >,w =< l ,l >. Then there are real numbers {a ,b }3 such that
2 3 1 2 3 1 i i i=1
p =a v+b w,p =a u+b v,p =a w+b u. FromTheorem4.1,we have(b1)3·(b2)3·(b3)3 =−1,
1 1 1 2 2 2 3 3 3 a1 a2 a3
hence b1 · b2 · b3 =−1. Which implies p ,p ,p are collinear.
a1 a2 a3 1 2 3
Similar to the proofs of Theorem 4.6 and Theorem 4.7, the following theorem can be also easily
proved by using the invariant that we found.
Theorem 4.8. If a conic is tangent to a cubic at three distinct points, the residual intersections of
the tangents at these points are collinear.
8
5 Appendix
5.1 Bivariate Spline Space over Triangulations
It is well known that spline is an important approximation tool in computational geometry, and it is
widely used in CAGD, scientific computations and many fields of engineering. Splines, i.e., piecewise
polynomials, forms linear spaces that have a very simple structure in univariate case. However, it is
quitecomplicatedtodeterminethestructureofaspaceofbivariatesplineoverarbitrarytriangulation.
Bivariate spline is defined as follows [37]:
Definition 5.1. Let Ω be a given planar polygonal region and ∆ be a triangulation or partition of Ω,
denoted by T ,i=1,2,··· ,V, called cells of ∆. For integer k >µ≥0, the linear space
i
Skµ(∆):={s|s|Ti ∈Pk,s∈Cµ(Ω),∀Ti ∈∆}
is called thesplinespace ofdegreek withsmoothness µ, where Pk is thepolynomial spaceof totaldegree
less than or equal to k.
From the Smoothing Cofactor method [37], the fundamental theorem on bivariate splines was
established.
Theorem 5.2. s(x,y)∈Sµ(∆) if and only if the following conditions are satisfied:
k
1. Foreachinterioredgeof∆,which isdefinedbyΓ :l (x,y)=0,thereexistsaso-called smoothing
i i
cofactor q (x,y), such that
i
p (x,y)−p (x,y)=lµ+1(x,y)q (x,y),
i1 i2 i i
where the polynomials p (x,y) and p (x,y) are determined by the restriction of s(x,y) on the
i1 i2
two cells ∆i1 and ∆i2 with Γi as the common edge and qi(x,y)∈Pk−(µ+1).
2. For any interior vertex v of ∆, the following conformality conditions are satisfied
j
[l(j)(x,y)]µ+1q(j)(x,y)≡0, (5.1)
i i
X
where the summation is taken on all interior edges Γ(j) passing through v , and the sign of the
i j
smoothing cofactors q(j) are refixed in such a way that when a point crosses Γ(j) from ∆ to
i i i1
∆ , it goes around v counter-clockwisely.
i2 j
From Theorem 5.2, the dimension of the space Sµ(∆) can be expressed as
k
k+2
dimSµ(∆)= +τ,
k (cid:18) 2 (cid:19)
where τ is the dimension of the linear space defined by the conformality conditions (5.1).
However, for an arbitrary given triangulation, the dimension of these spaces depends not only on
the topology of the triangulation, but also on the geometry of the triangulation. In general cases, no
dimension formula is known. We say that a triangulation is singular to Sµ(∆) if the dimension of
k
the spline space depends on, in additional to the topology of the triangulation,the geometric position
of the vertices of ∆, and Sµ(∆) is singular when its dimension increases according to the geometric
k
property of ∆. Hence, the singularity of multivariate spline spaces is an important object that is
inevitable in the research of the structure of multivariate spline spaces. For example, Morgan and
Scott’striangulation∆ ([27],seeFig. 6)issingulartoS1(∆ ). Thatistosaythatthe dimension
MS 2 MS
ofthespaceS1(∆ )is6ingeneralbutitincreasesto7whenthepositionoftheinnerverticessatisfy
2 MS
certain conditions.
Now,we takeanexample forS1(∆ ) to intuitively understandTheorem5.2. Letl :α x+β y+
2 MS i i i
γ z =0 (i=1,2,··· ,6),u:α x+β y+γ z =0,v :α x+β y+γ z =0andw :α x+β y+γ z =0
i u u u v v v w w w
9
B
While the singularity of multivariate spline over
any triangulation has not been completely settled,
[l6] [l1] manyresultsonthestructureofmultivariatespline
space in the past 30 years can be found in many
u of references [1–5,7–10,12–14,23,28,29,36,37]. For
(c) (a)
Morgan-Scott’s triangulation, Shi [31] and Diener
v
[14] independently obtained the geometric signifi-
w
[l5] [l2] cance of the necessary and sufficient condition of
(b) dimS1(∆ ) = 7, respectively, and an equivalent
[l ] [l ] 2 MS
4 3
geometric necessary and sufficient condition of sin-
A C
gularity of S1(∆ ) from the viewpoint of projec-
2 MS
tive geometry was obtained in [15].
Figure 6: Morgan-Scott triangulation
in Morgan-Scott triangulation shown in Fig. 6. From Theorem 5.2, the global conformality condition
in S1(∆ ) is
2 MS
λ l2+λ l2+λ u2+λ v2 =0,
1 1 2 2 u v
 λ3l32+λ4l42−λvv2+λww2 =0, (5.2)
 λ l2+λ l2−λ w2−λ u2 =0,
5 5 6 6 w u

where all letters of λ′s are undetermined real constants. Then the dimS1(∆ ) = 6+τ, where τ is
2 MS
thedimensionofthe linearspacedefinedby(5.2). However,the structureofS1(∆ )depends onthe
2 MS
geometric positions of the inner vertices a,b and c, which can be obviously shown from the following
conclusions.
Theorem 5.3 ( [31]). The spline space S1(∆ ) is singular (i.e. dimS1(∆ ) = 7) if and only if
2 MS 2 MS
Aa,Bb,Cc are concurrent, otherwise dimS1(∆ )=6(see Fig.6).
2 MS
Theorem 5.4 ( [15]). Let l (x,y,z) = α x+β y +γ z = 0 (i = 1,2,··· ,6), then the spline space
i i i i
S1(∆ ) is singular (i.e. dimS1(∆ ) = 7) if and only if 6 points {(α ,β ,γ )}6 lie on a conic,
2 MS 2 MS i i i i=1
otherwise dimS1(∆ )=6.
2 MS
Using the principle of duality, an interesting fact is that the equivalent relations in Theorem 5.3
and Theorem 5.4 hold because of the Pascaltheorem!
More Precisely, for the Morgan-Scott triangulation, let
l =a u+b v l =a v+b w l =a w+b u
1 1 1 , 3 3 3 and 5 5 5 (5.3)
(cid:26) l2 =a2u+b2v (cid:26) l4 =a4v+b4w (cid:26) l6 =a6w+b6u,
′ ′
where all a s and b s are constants, then by solving the system of equations in (5.2), we have
i i
Theorem 5.5 ([22],[31]). The spline space S1(∆ ) is singular (i.e. dimS1(∆ )=7) if and only
2 MS 2 MS
if
b b b b b b
1 2 3 4 5 6
· · =1. (5.4)
a a a a a a
1 2 3 4 5 6
In general, for µ ≥ 3, Luo & Chen [24] gave an equivalent condition in an algebraic form to the
singularity of Sµ (∆µ )(µ≥3) as follows: for a given triangulation ∆µ (see Fig. 9), suppose
µ+1 MS MS
l =a u+b v, i=1,2,......µ+1,
i i i
 lj =ajv+bjw, j =µ+2,µ+3,......2µ+2, (5.5)
 l =a w+b u, k =2µ+3,2µ+4,......3µ+3,
k k k

then
10