Theories of contact specified by connection matrices

COMPUTER AIDED GEOMETRIC DESIGN ELSEVIER

Computer Aided Geometric Design 13 (1996) 905-929

Theories of contact specified by connection matrices Ayman W. Habib, Ron N. Goldman * Department (~[Computer Science, Rice University, Houston, TX 77251, USA

Received June 1994; revised May 1995

Abstract We begin by characterizing notions of geometric continuity represented by connection matrices. Next we present a set of geometric properties that must be satisfied by all reasonable notions of geometric continuity. These geometric requirements are then reinterpreted as an equivalent collection of algebraic constraints on corresponding sets of connection matrices. We provide a general technique for constructing sets of connection matrices satisfying these criteria and apply this technique to generate many examples of novel notions of geometric continuity. Using these constraints and construction techniques~ we show that there is no notion of geometric continuity between reparametrization continuity of order 3, (G3), and Frenet frame continuity of order 3, (F3); that there are several notions of geometric continuity between G 4 and F4; and that the number of different notions of geometric continuity between G '~ and F ~ grows at least exponentially with r~.. Keywords: Geometric continuity; Connection matrices; Projective invariance; Reparametrization:

Frenet frame; Group; Contact

1. Introduction The ability to design smooth curves and surfaces is fundamental to geometric modeling and C A G D applications. Several different notions of contact between parametric curves are now available with different degrees of flexibility for designers. Parametric continuity ( C T~) requires the continuity of parametric derivatives, but suffers from its dependence on a specific parametrization. Reparametrization continuity (G 7~) only requires the continuity of parametric derivatives after a suitable reparametrization (Barsky and DeRose, 1984, 1988; Bartels et al., 1987; DeRose, 1985; Goldman and Barsky, 1989; Goldman and Micchelli, 1989; Gregory, 1989). Unlike C ~, G ~ depends only on the intrinsic geometry of the curve rather than on any particular parametrization. Other notions of * Corresponding author. Partially supported by NSF grant CCR-9113239. 0167-8396/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0167-8396(96)00015-5

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contact, such as Frenet frame continuity (F n) (Dyn et al., 1987; Dyn and Micchelli, 1988; Goodman, 1990; Hagen, 1985) and continuity of osculating linear spaces (On), are also invariant under reparametrization. All of these continuity measures can be represented by connection matrices. Here, we begin by specifying conditions under which a notion of geometric continuity can be embodied by connection matrices. We go on to present a set of analytic and geometric properties that all reasonable notions of geometric continuity must satisfy. We then interpret these properties into an equivalent collection of algebraic conditions that must be satisfied by any set of connection matrices representing some form of geometric continuity. Throughout this paper the terms "notion", "measure", "form", "type" and "theory" are used as synonyms when they modify the phrase "geometric continuity". Thus, G n and F n are two different notions, measures, forms, types, or theories of geometric continuity. Also in this paper the phrases "geometric continuity" and "contact" are synonymous; see Section 2.2.1 for further clarification. This paper is divided into two parts. The first part characterizes notions of geometric continuity embodied by connection matrices and introduces a set of intuitive properties that should be satisfied by any reasonable form of geometric continuity. The second part provides new examples of geometric continuity derived using the properties of connection matrices developed in part one. Using these properties and examples, we go on to show that there is no notion of geometric continuity between G 3 and F3; that there are several notions of geometric continuity between G 4 and F4; and that the number of different notions of geometric continuity between G n and F n grows at least exponentially with n.

2. A general theory of contact for parametric curves 2.1. Connection matrices

We are going to consider continuity measures representable by connection matrices. Connection matrices were first introduced by Dyn and Micchelli (1988) to study spline curves joined with different types of geometric continuity at the knots. Recently, spline spaces determined by connection matrices have been studied by several authors (Barry et al., 1993; Boehm, 1985; Dyn et al., 1987; Farin, 1985; Seidel, 1993). Typically these authors fix a connection matrix at each knot and then investigate the linear space of splines determined by a fixed sequence of knots and connection matrices. Our focus in this paper will be different; we study the structure of the sets of connection matrices, rather than the space of splines. Notice that if we allow a set of connection matrices at each knot, then this set of splines is not a linear space, but this problem will be of no concern to us here. In this section, we characterize geometrically what it means for a continuity measure to be embodied by a set of connection matrices. Let f : R 1 --+ •a be an n-times differentiable function and let ~- be a parameter value. We write D n ( f ) ( T ) = [f(T), f ' ( T ) , . . . , f(n)('r)]T.

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A piecewise n-times differentiable function f : R l -+ R d is said to have connection matrix M = (Mij), i, j = 0, 1 , . . . , n, at parameter value r if and only if

D~(f+)(~ -) = M . D~(f-)(-c)

(1)

where f - and f + denote the function f to the left and to the right of ~- respectively. Notice that a curve f(t) = {fl(t), f2(t),..., fd(t)} has the connection matrix M at T iff

D~(f+)(T) = M" Dn(f~-)(T),

i= l,...,d.

Since T is fixed throughout this paper, we shall often omit the parameter 7 from Eq. (1) and simply write D n ( f +) = M D n ( f - ) . We also use the same notation for different curves f , g defined over the same parameter space. Thus, f and g have connection matrix M at parameter value T iff

Dn(f)('c) = M . D,(g)(w). We call n the order of contact. A notion of contact such as G n or F n can be thought of as a binary relation on the set of curves defined in a neighborhood of the parameter T. If H n is some form of contact, then we write (f, g) E H ~ ,,f-->. f and 9 have H '~ contact at T. A notion of contact H n should specify relations between the first n derivatives of f and g. Thus there must exist some functions, R, relating these derivatives such that

(f,g) C H n ¢==~ Dn(f)(T) = R{Dn(g)('r)}. We shall use the notation F ( H n) to denote the set of all such functions R. For example, I'(G n) denotes the set of all reparametrization matrices (Barsky and DeRose, 1984, 1988; Bartels et al., 1987; DeRose, 1985; Goldman and Barsky, 1989; Gregory, 1989), and F ( F ~) denotes the set of all Frenet frame connection matrices (Dyn et al., 1987; Dyn and Micchelli, 1988; Goodman, 1990; Hagen, 1985). We want to investigate under what general conditions the functions in F ( H n) can be represented by connection matrices. Inspecting formula (1) which defines what it means for a curve f : R l ~ R d to have connection matrix M at parameter T, we notice that each of the coordinate functions must have the same connection matrix M at ~-. So, types of continuity that are represented by connection matrices are defined coordinatewise. It follows that if T E F ( H n) is to end up being represented by a connection matrix, T must also be defined coordinatewise. Intuitively, this means that T should perform identically on all coordinate functions. Thus, if f(t) = {fl(t), f2(t),..., fa(t)}, we shall insist that for all T e F(H")

D ~ ( f +) = T { D n ( f - ) }

~

D n ( f +) = T { D , ( f ~ - ) } , i :

l,...,d.

(2)

We shall now determine conditions under which forms of geometric continuity defined coordinatewise can be represented by connection matrices. For this, we introduce the notion of linear invariance. Intuitively, a notion of geometric continuity H n is linearly invariant if H'~-continuous curves preserve their Hn-continuity under all linear transformations. Formally, let

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f : R 1 ---+ R d, and let L : R d --.+ ]~d be a linear transformation. Then we say that H n is linearly invariant if

D n ( f +) : T { D , ~ ( f - ) } ~

Dn(L(f+)) = T{Dn(L(f-)) }

(3)

for all T c T ' ( H ~) and all linear transformations L. Any plausible notion of geometric continuity is expected to be linearly invariant because continuity should be a feature of the curves themselves rather than the coordinate system. Applying a nonsingular linear transformation (L) to a curve is equivalent to applying the inverse transformation (L - l ) to the coordinate system, an operation that must certainly preserve reasonable notions of continuity. If L is a singular linear transformation, then L represents some type of linear projection which should also maintain notions of continuity defined coordinatewise. In fact, we shall insist later on (see Sections 2.2.3 and 2.2.4) that continuity should be preserved under all projective transformations. We now show that, under the assumption that continuity is defined coordinatewise, the set of geometric continuity measures that can be embodied by connection matrices are exactly those that are invariant under linear transformations. T h e o r e m 1. Let H n be a notion of geometric continuity, then: H n can be represented

by connection matrices ~ H '~ is defined coordinatewise and is invariant under linear transformations.

Proof. 3 . Since H n is a notion of geometric continuity that can be represented by connection matrices, it follows immediately from (1) that H n is defined coordinatewise. We need to show that H n is invariant under linear transformations. Suppose that f : ~ l __+ I~d and that ( f + , f - ) E H n at some parameter ~-. Then, at T, we have

D ~ ( f +) = M D n ( f - ) ;

M is an H n connectionmatrix.

Now let L:]~ d --+ 1~d be a linear transformation defined by L(x) = xL, where L is a d × d matrix. Then

Dn(f+)L:MDn(f-)L. So, by the linearity of differentiation

Dn(I+L):MDn(I-L). Hence,

D n ( f +) = M D n ( f - )

~ Dn(L(f+)) = MD~(L(f-)).

<:=. Since H n is defined coordinatewise, it is enough to consider functions f, g : 1~1 _+

[{i.

Now to show that T E F ( H n) can be represented by a connection matrix, we must show that T is linear. That is,

T { c D n ( f ) } = cT{ D n ( f ) } T { D n ( f ) + Dn(g)} -- T { Dn(f) } + T { Dn(g) }.

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Assume that

1)n(f +) = T{Dn(f-)}. Since linear invariance implies scale invariance, scaling by a constant c, we have

Dn(cf+)=T{Dn(cf-)}. Hence, by the linearity of differentiation

cT{D,~(f-)} = T{cDn(i-)}.

(4)

Moreover, since H n is defined coordinatewise,

r { D n ( f - ) } , V,~(g +) = T{D,~(g ) ~D,~(f+,g+)=m{Dn(f-,g-)}.

D n ( f +) :

Applying the shearing transformation

[::]

to both sides of the last equation and employing linear mvariance, we get

D,,((f + +g+),g+)=T{D,~((f-+g-),g

)}-

It follows immediately from (2) that

D,~(f + + g+) = T{Dn(f- + g-) }. But by the linearity of differentiation

Dn(f + + 9+) = D,,(I +) + Dn(g +) ==T{D,~(I-) } + T{Dn(g-) }. Hence

T{D~(f- + 9 - ) }

=

T{D,~(f-) } + T{Dn(g-) }.

(5)

From (4) and (5) it follows that T is linear, so T can be represented by a connection matrix. []

2.2. Fundamental properties of geometric continuiO, From here on we shall deal only with notions of geometric continuity represented by connection matrices. Linear invariance implies that notions of geometric continuity represented by connection matrices are independent of both the scale and the orientation of the coordinate axes. However, arbitrary sets of connection matrices will not always correspond to plausible notions of geometric continuity. We expect notions of geometric continuity to satisfy some additional geometric criteria. Next we pick a generic notion of geometric continuity H n and specify a minimal set of geometric properties that we require H n to have. We shall soon see that these requirements impose some structure on the associated set of connection matrices F(Hn).

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The proposed geometric properties are: 1. H'~-continuity should induce an equivalence relation on the set of curves. 2. Reparametrizations should preserve Hn-continuity. 3. H'~-continuity should be invariant under affine transformations. 4. Hn-continuity should be invariant under perspective projection. 5. H'~-continuity should guarantee the continuity of the first n linear osculating spaces. In the following subsections, we discuss each of the above properties, explain why it is natural, and derive its algebraic consequences for sets of connection matrices. 2.2.1. Equivalence relation A continuity measure H n induces a binary relation on the set of curves defined in a neighborhood of some fixed parameter -r. We write (f, 9) c H n if and only if f and 9 meet with Hn-continuity at 7-. We claim that any reasonable notion of contact H n must be an equivalence relation. That is, H n should satisfy the three basic properties of equivalence relations: 1. Reflexivity: (f, f ) E H n. Consider a curve f that is C n at 7-. Whatever notion of continuity H n represents, we expect f to be Hn-continuous at T. Thus (f, f ) E H n. Alternatively, we expect D n ( f ) = Dn(g) ~ ( f , g ) C H n so certainly (f, f ) E H n. 2. Symmetry: (f, g) ~ H a ~ (g, f ) ~ H n. For any reasonable notion of geometric continuity H n, if two curves (f, g) are H ncontinuous at T, then (g, f ) should also be Hn-continuous at T. That is, continuity should depend only on the curves, not their order. 3. Transitivity: (f, g), (g, h) 6 H n => (f, h) E H n. So far we have been using the word continuity loosely to describe the property that parametric derivatives are related somehow at a point. This notion is better described as contact because we can always find curves whose derivatives satisfy some specified relation (match for example) but are not, by our intuition, smooth (see Fig. 1). We prefer to employ the term continuity here, however, because it is widely used and accepted in the CAGD literature for similar notions. Intuitively, contact is a transitive relation. All the standard notions of geometric continuity--G n, F n and O n - - i n d u c e transitive relations. We shall show that together these requirements on H n force the set of connection matrices F ( H n) to form a group. We begin with a useful lemma. L e m m a 1. Let M be an arbitrary (n + 1) × (n + 1) matrix. Then there is a function f ; ~1 ___).~n+l such that." D , ( f ) ( r ) = M. Proof. To construct f ( t ) = {f0(t), fl ( t ) , . . . , fn(t)} such that Dn (f~(t))('r) = {Mo~, M l i , • • •, M n i } T,

A.W. Habib, R.N. Goldman / Computer Aided Geometric Design 13 (1996) 905-929

J

911

g(t)

t

Fig. 1. Two curves that have high order contact but their join is not smooth. we treat the ith column of M as the Taylor coefficients of fi(t) at t = ~-. That is, we pick .fi (t) =

L

(t - ~-)~

Mki" k----S]-"

k=O NOW,

D n ( f ) ( r ) = {{Moo, M l O , . . , , M n o } T , . . . , { M o n , M , n , . . . , M n n } T} = M.

[]

Proposition 1. H n is an equivalence relation ¢# the corresponding set of connection matrices F ( H n) form a group. In particular: (i) H n is reflexive ¢==> I E F ( H n ) ; (ii) H n is symmetric ¢==> ( M E F ( H n) =:> M -1 E F ( H n ) ) ; (iii) H n is transitive ,'. :, (M, g E F ( H n) ~ M N E F ( H n ) ) . Proof. We shall prove only part (ii) since the proofs of (i) and (iii) are very similar to the proof of (ii). =:~: Let M E F ( H n ) . By the previous lemma we can pick f and g so that Dn(f)(~-) = I and Dn (g)(~-) = M. Hence by construction, Dn(g)(T) = M . Dn(f)(~-). Since M E F ( H "~) it follows that (g, f ) E H n, so by symmetry (f, g) E H a. Hence, Dn(f)('r) = g . Dn(g)(~-)

for some N E F ( H n ) .

Substituting the definitions of D n ( f ) and Dn(g), we obtain I=N.M; SO,

M-l:N~r(Hn). ~:: If (g, f ) E H '~, then Dn(g) = M . D , , ( f ) for some M E F ( H n ) . Hence D n ( f ) = M - ~ D n ( g ) , so ( f , g ) E H '~. []

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2.2.2. Reparametrization invariance We want H' 0 as reparametrization functions at "I-. These conditions guarantee that a reparametrization function h has a local inverse and that h -1 is also a reparametrization function. We say that Hn-continuity is invariant under reparametrization if (f, g) e H n ~ (f, g o h) •

g n

where h is a reparametrization function. If, as we generally assume, H '~ is symmetric, we can reparameterize f as well. Notice that if H n is reparametrization invariant and h is a reparametrization function, then

(f, 9 o h) • H ~ ~ (f, (g o h) o h - ' ) • H n (f,g) •

by reparametrization invariance

g n

SO,

( f , g ) E H n .: ". (f, g o h ) E H n. If we reparameterize a curve f ( t ) by a function h, the relation between the parametric derivatives of f at "I- before and after reparametrization can be computed by repeated application of the Chain rule. Moreover, D n ( f o h)(T) depends linearly on D n ( f ) ( T ) . Thus there is a matrix R(h)(7) depending only on the derivatives of h at 7- such that

D n ( f o h)(r) = R ( h ) ( r ) . D ~ ( f ) ( r ) .

(6)

Recall that G n refers to reparametrization continuity. That is, two curves f and g have Gn-continuity at ~- iff f is a reparametrization of g at T. So,

(f, 9) e G ~ ¢==~ D ~ ( f ) ( r ) = D~(9 o h)(r) = R ( h ) ( r ) . Dn(g)(r) for some reparametrization function h at T. It follows that F ( G n) is the set of all matrices on the form R ( h ) ( r ) where h is a reparametrization function. These matrices are often referred to as/3-matrices because of their relation to the theory of ¢3-splines (Goldman and Barsky, 1989; Hohmeyer and Barsky, 1989). Proposition 2. Suppose H n is an equivalence relation (i.e., F ( H n) is a group). Then

H n is invariant under reparametrization ~

ff(G n) C_ ['(Hn).

Proof. =~: Since all matrices in F ( G n) are of the form R(h)(7) for some reparametrization function h, to show that I ' ( G '~) C F ( H n) it suffices to show that

D n ( f ) ( w ) = R ( h ) ( w ) . Dn(g)(T) ~ But by (6)

( f , g ) E H n.

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D , ~ ( f ) ( r ) = R ( h ) ( r ) . D n ( g ) ( r ) ¢==>D,~(f)(r) = D,~(g o h)(r) ~=> (f, 9 o h) ~ H '~ since I • F ( H n)

(.f, 9) E H '~ by reparametrization invariance. ~ . This is straightforward since F ( H ~) is a group.

2.2.3. Invariance under affine transformations We. have shown that if H '~ is a notion of geometric continuity represented by connection matrices then H ¢~ is invariant under linear transformations. However, this is not enough. What we really want is that H '~ should be invariant under all affine transformations. That is, not only do we expect H'~-continuity to be independent of the orientation of the coordinate axes, but we also expect H n-continuity to be independent of the choice of the coordinate origin. Again, we demand this invariance because by moving the origin we are not changing the curves, but only their representation. The following proposition shows how invariance of H~-continuity under affine transformations is related to the struclure of the connection matrices in F(H'~). Proposition 3. The following 3 conditions are equivalent: (1) H '~-continuity is invariant under affine transformations; (2) D , ( e ) = M . D,~(c) for all constant functions c and all 3~I < F(H'~), • (3), Mio = 5io ]br all M ~ F(H'~). Proof. 1 <~ 2. Let A ( x ) = x A + c be an affine transformation and let M ~! F(H'~). Suppose that

Dn(.f) = M . Dn(g)

for some M.

Then by the linearity of differentiation,

D,~(f A) = M . D n ( g A ) . Hence,

D,~(f A + c) = M . D ~ ( g A + e) ¢==> D~(c)

M . D,,(c).

2 ,~ 3. This follows immediately since [e, 0 . . . . , 0 1 7 = M . [ e , 0 , . . . , 0 ]

T ~:~

M~0=~0.

[]

As a consequence of invariance under affine transformations, we get invariance under embeddings. We say that Hn-continuity is invariant under embeddings iff for all matrices M ~ F ( H ~) and every constant c, (f, g) have connection matrix M at -r =~ ((f, e), (g, c)) have connection matrix M at r. Invariance under embeddings is an essential property of contact because it asserts that continuity does not depend on the ambient space of the curves but only on their intrinsic geometry.

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Corollary 1. H n is invariant under arbitrary embeddings tee H • is invariant under affine transformations.

Proof. The proof follows directly from part (2) of Proposition 3.

[]

This result comes as no surprise because we can think of each translation of •d as an embedding of II~d into a translated copy of itself in I~d+l . 2.2.4. Invariance under projective transformations

We expect any reasonable continuity measure to be preserved under perspective projection because ultimately what we actually see is not the curve itself but rather a projection of the curve onto either a graphics screen or the retina. Projective invariance has recently received much attention; both G n and F n have been shown to be projectively invariant (Goldman and Micchelli, 1989; Pottmann, 1989). A general projective transformation A is a map A: { x l , x 2 , . . . , x n } ~ { x ~ ' , x ~ , . . . , x ~ } such that: , aklxl+ak2x2+".+ak~xn+bk xk = dlxl + d2x2 + + dnxn + c

k= 1,...,n. '

Let: Xk = a k l X l + a k 2 X 2 + ' ' ' + a k n X n + b k ~

k---- l , . . . , n ,

xn+ l** = dlXl + d2x2 + " • • + dnxn + c. Then, x~ = x*k*/Xn*+l. Now we can form the transformation A by composing four transformations: A - / - / 4 oI/3 o / / 2 o//1 where H1 : { X l , X 2 , ' " , X n }

~ {Xl,X2,'",Xn,O},

//2 : x l ~ x 2 ~ ' " ~ X n ~ 0

~

Xl ~X2 ~ ' " ~ X n ~Xn+l

. 2. ,.' ' - ., X . n . , X. n +. l ). I--+ . {Xl,X2,...,Xn, l} /"/3 : { Xl. ,X 1-I4 : { X ; , X ~ , . . . ,X~, 1) ~

,

{ x ; , x ~ , . . . ,x~}.

Notice that /I1 is an embedding, 172 is an affine transformation, and /74 is an orthogonal projection. Each of these transformations preserves continuity under our previous criteria (orthogonal projection works because continuity is defined coordinatewise). All that remains to guarantee invariance under projective transformations is invariance under transformations of the form

//.3: {Yl,Y2,...,Yn, Yn+I) v-+ { Y l / Y n + I , Y 2 / Y n + I , . . . , Y n / Y n + I , 1) which represent the perspective projection of the point {Yt, Y2,..., Yn, Yn+l} from the origin onto the plane yn+l = 1. Thus, to insure projective invariance, we need to insure that if f* = (f, w) and g* = (g, u) are two curves such that

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D,~(f*)(T) = M . D,~(g*)('r),

915

M 6 r(Hn),

then their perspective projections ( f / w , g/u) are related by

D n ( f /W)(T) = N " Dn(g/u)(~-)

for some N E F(H'~).

Notice that N # M. In general, the connection matrix changes under perspective projection because, unlike affine transformations such as translations or rotations which alter only the relationship between the curve and the coordinate system, perspective changes the actual shape of the curve. Thus, we expect the connection matrix to change under perspective transformations. We insist only that the resulting connection matrix belongs to the same group F(H'~). In fact, it is known that the only group for which N = M is F ( G n) (Goldman and Micchelli, 1989). Below we give the conditions under which a group of connection matrices represents a notion of continuity that is projectively invariant. We begin by recalling the definition of the Leibniz matrix L(w)('c), the matrix that results from applying Leibniz's rule to the product w • f, i.e.,

D,~(w. f)(-c) = L ( w ) ( r ) . D,~(f)(r).

(7)

By Leibnitz's rule: (S) =

O,

i < j.

Thus the matrix L(w)(T) is lower triangular and is given by o

o

L(w)('r) =

o

o ..

o

w"(~-)

2W'(T)

w(r)

...

0

:

:

:

"..

:

D

f

l

W(T)

kZ/ Proposition 4. An affinely invariant notion of geometric continuit), H '~ is projectively

invariant iff" Dn(w) : M . D~(u), M 6 F ( H '~) ~ ( r ( 1 / w ) . M . L(u)) 6 F ( H ~ ) . Proof. Let f* = (f, w) and g* • (9, u) be two curves meeting at a parameter value 7with Hn-continuity represented by the connection matrix M 6 -P(Hn). Then at r

D n ( f ) = M . D,~(g),

D,~(w) = M . D,~(u).

Therefore by (7)

D,~(f /w) = L ( 1 / w ) . D n ( f ) = L ( 1 / w ) . M . Dn(g) = L ( 1 / w ) . M . L(u)" D,,(g/u)

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so ( f / w ) , (9/u) meet at ~- with connection matrix L ( 1 / w ) . M . L(u). Hence, H n is projectively invariant iff:

Dn(W) : M . Dn(u), M C I ' ( H ~) ~ ( L ( i / w ) . M . L(u)) E F ( H n ) .

[]

2.2.5. Continuity of osculating linear spaces Geometrically, the continuity of the first n osculating linear spaces (On-continuity) is the weakest notion of nth order contact. If the osculating linear spaces fail to agree, then there can be no sensible geometric contact between the curves. At parameter T, On-continuous curves have common position (co), tangent line, and higher order osculating linear spaces. Thus, P(O n) is the group of (n + 1) x (n + 1) nonsingular lower triangular connection matrices with first column [ 1 , 0 , . . . , 0] T. Because On-continuity is so weak, we expect that any plausible notion of contact H n will imply On-continuity. That is, we expect that Hn-continuity ~ On-continuity, or equivalently H n _C O n. In terms of connection matrices,

c on

r(/-/")

r(On).

That is, H n c_ O n <=~ all M E F ( H n) are ( n + 1) x ( n + 1) nonsingular lower triangular connection matrices with Mio = 5i0.

2.2.6. Summary and commentary To summarize, Table 1 shows the five analytical properties we require on a sensible geometric continuity measure H n, together with the corresponding algebraic conditions on the set of connection matrices F ( H n ) . Underlying these properties, we proved in Section 2.1 that a notion of geometric continuity H '~ can be represented by a set of connection matrices F ( H n) iff H n is defined coordinatewise and is invariant under linear transformations. Properties (1) and (5) in Table 1 establish the basic correspondence between our intuition and the meaning of geometric continuity. The first says that geometric continuity is some form of contact which is an equivalence relation. The last asserts that contact must at least insure the continuity of osculating linear spaces and so relates contact back to geometry. Properties (2) and (3) mean that H n is an intrinsic property independent of both curve parametrization (2) and coordinate system (3). The fourth property relates Table 1 Property of Hn-contact

Property of F(H ~)

(1) Equivalence relation

F(H ~) is a group

(2) Invariant under reparametrization

F(a '~) C r ( H n)

(3) Invariant under affine maps

For all M E F(Hn), Mio = 6i¢1

(4) Invariant under projective maps

M • r ( H n ) , D~(w) =

MDn(u)

(Z(1/w). M. L(u)) • F(H ~) (5) H n =~ O n

For all M • F(Hn), Mij = 0 (j > i)

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geometric continuity to vision; it says that a mathematically smooth curve should look smooth. The five properties listed above for a generic notion of continuity H 7~ are not all independent since both property (4) and (5) imply property (3). However, the five properties of F(H '~) are independent and together they insure the equivalent set of properties of H n. Moreover, these properties of F(H '~) are cumulative; the first k properties in the list for F(H n) are equivalent to the hth property in the list for H n. We shall discuss these five properties further in the next section along with some examples illustrating their independence; for additional examples see (Habib and Goldman, 1994). Although these five properties are the defining characteristics of geometric continuity, nevertheless it is sometimes useful during the investigation of geometric continuity to study sets of connection matrices that lack one or more of these properties. For example, in the study of rational curves, we may wish to investigate curves that are not smooth. but which become smooth under the canonical perspective projection H : I R d+t --~ R
( i, lt)

fa(t), fd+t (t)) = \ f-~+7(-t)''''' .f,,+, (t) ]"

Let F(L n) denote the group of all (n + 1) x (n + 1) Leibnitz matrices L(w)(7-), where w(m) # 0. Then both Hohmeyer and Barsky (1989) and Goldman and Micchelli (1989) show that the curves which become G n under this canonical projection are precisely those curves with connection matrices in

F(Gr')F(L '~) = { M I ~ S = G L

where G c r ( G ' ) ,

L E P(LT~)}.

More: generally, if H n is a projectively inwtriant form of geometric continuity, then the curves that become H n under this canonical projection are precisely those curves with connection matrices in F(H~)F(Ln). This result is proved for F ~ in Theorem 11 of (Goldman and Micchelli, 1989); the general result for H " follows in much the same manner, Notice, however, that matrices in F(Hn)F(L '~) do not satisfy property (3) because F ( L :~) C F(Hn)F(L n) and matrices in F(L '~) violate this constraint. Thus the set F(H'~)F(L ~) is not affinely invariant. This makes sense because the canonical projection is actually the perspective projection from the origin onto the plane .m,t+l - 1. Thus the canonical projection depends on the choice of the origin, and so the set _P(H'~)F(L '~) is not affinely invariant. Notice too that if F(H n) satisfies the tive characteristic properties of geometric continuity listed above, then the set F(Hn)F(E ~) does indeed satisfy each of the other four characteristic properties of sets of connection matrices. Property (5) follows because both F ( H '~) and F(L n) contain only lower triangular matrices and property (2) holds because F ( G ''~) C_ F(H 7~) C F(H'~)F(Ln). The fourth property is more subtle, but can proved using the fact that F ( H 7~) satisfies this constraint. Finally property (1) follows from the equality F(H'~)F(L '~) = F(Ln)F(H ~) which, in turn, is a consequence of property (4). Thus the connection matrices in F ( H ' ) F ( L ' ) define a weak form of geometric continuity that is not affinely invariant but does satisfy the four remaining constraints. We close this section by observing that even though our entire analysis has been per-

918

A.W. Habib, R.N. Goldman /Computer Aided Geometric Design 13 (1996) 905-929

formed only for parametric curves, almost everything carries over to parametric surfaces as well, though sometimes conditions on entries of M must be replaced by conditions on sub-blocks of M.

3. N e w theories of contact

3.1. Groups of connection matrices representing new notions of geometric continuity Up to this point, the only groups of connection matrices we have encountered are those representing the three standard notions of geometric continuity: G n, F n and O n. Each of these groups satisfies all the properties listed in Section 2.2. This is to be expected because these notions of contact are based on intrinsic geometric characteristics of curves. The most subtle property to verify is projective invariance. Goldman and Micchelli (1989) give an analytic proof for Frenet frame continuity (Fn); Pottmann (1989) provides a more geometric argument which he then uses to introduce additional projectively invariant notions of geometric continuity. We are now going to present a general technique for generating groups of connection matrices representing new notions of geometric continuity. In Sections 3.2 and 3.3 we shall use this technique to confirm the existence of classes of geometric continuity strictly between the classical notions of G n, F n and O n. To simplify our notation, we shall often omit the zeroth row and column of a connection matrix since by properties (3) and (5) of the previous section, the zeroth row and column of any connection matrix M are fixed (Mi0 = 3i0, M0i --- 30i). Before proceeding, recall that continuity under reparametrization, G n, is represented by the group F ( G n) of fl-matrices. Leaving off the zeroth row and column, we have G E F ( G n) ¢~ G is represented by an n x n matrix /3~

0

/32

G =

N

.

.

.

.

.

.

.

.

.

.

.

.

3/31/32 :

/3n

"

"

"

......

...

0

"..

:

,

/3?

where the entries of G are computed from the Chain Rule. The matrices in F ( G n) have a rich structure. Each G E F(G n) is specified uniquely by the entries Gil = /3i, i = 1 , . . . , n , and each entry Gij, i /> j /> 1 depends only on /31, /32, . . . , /3i-j+1 as can be verified by the Chain Rule. The matrices in F ( F n) are less constrained; only the zeroth column and the diagonal entries are restricted to be of the same form as G, everything else--below the diagonal--is free. To construct sets of connection matrices that have all the properties listed in Section 2.2, we find the hardest requirements to satisfy are the group property and the property corresponding to projective invariance. Shortly we shall describe a technique for building

A.g~ Habib, R.N. Goldman/Computer Aided GeometricDesign 13 (1996) 905-929

919

sets of connection matrices that satisfy both of these properties. We begin by constructing groups of matrices containing F(Gn). Let G E F(G '~) be an n × n matrix and consider matrices M of the form

where the entries of M marked 9 match those in G, blank entries above the diagonal are zero, on the diagonal are free but nonzero, and below the diagonal are arbitrary. We call the seed matrix G a parent matrix for M, and refer to the lower triangular sub-matrix of M that matches that of the parent G as a diagonal-block. In fact, we shall allow arbitrarily many diagonal-blocks, of variable size, possibly overlapping, as long as each block comes from the same parent. Notice that parent matrices are not unique since each /3i in the first column of G affects only entries of distance ) (i - 1) from the diagonal of G. Formally, we introduce: (S-construction). Fix G E F(G~). Let 1 <~ rl < .-. < rp <~ n, and I <~ sl < . " < sp <~ n be two monotonic sequences of positive integers with sz ~< rt lt'or

Definition

l = 1 , . . . , p. Suppose M is a nonsingular lower triangular matrix of the same size as G such that for I = 1 , . . . , p

Suppose further that all the other entries of M below the diagonal (except for the fixed zeroth column) are free and the diagonal entries of M are nonzero. Then we call G a parent matrix of M, and denote the set of all such matrices M as the parent matrix G varies over F(G '~) by Sn [(rl, sl), (r2, s2), •. •, (rp, sv) ]. The parameters of Sn[(r'l, sl), (r2, s 2 ) , . . . , (rp, sr~)] have the following interpretation: • n denotes that matrices M E Sn are of size n x n, or equivalently (n + 1) x (n + 1) when restoring the zeroth row and column. • p denotes that there are p lower triangular sub-matrices of M E S,~ matching those of the parent G E F ( G n) (p can be zero for none). • (ri, si) denote the indices of the vertex of the ith diagonal-block of M ~ S~ that matches a diagonal-block in the parent G E F(G~). For example, omitting the zeroth row and column, we have 0 *

0

0

*

0

0

*

~

o

$4 [(4, 3)] = *

g

*

9

g

*

6/~2/32

~314

A.W. Habib, R.N. Goldman/Computer Aided Geometric Design 13 (1996) 905-929

920

Notice too that by construction: P

Sn[(Tl,81),...,(I'p, Sp)] C ~ S[(rj,Bj)]. j=l

Containment is strict because matrices in ["]~=z S[(rj, sj)] may have diagonal-blocks with different parents, while matrices in Sn[(rl, S l ) , . . . , (rp, Sp)] must have each diagonalblock come from the same parent. Next we show that the sets of matrices built using the S-construction form groups. Proposition 5. The set of matrices S n [ ( r l , s l ) , ( r 2 , s 2 ) , . . . ,(rp, Sp)] forms a group. Moreover: 1. G is a parent of M ~ G - l is a parent of M - l ; 2. G is a parent of M, H is a parent of N ~ G . H is a parent of M . N. Proof. We can ignore the zeroth row and column since these are fixed. Moreover it suffices to prove the proposition for individual diagonal-blocks because overlapping diagonal-blocks come from the same parent. But then the result is straightforward because, by construction, these blocks are nonsingular, lower triangular, and lie along the diagonal. [] Not only do the sets of matrices generated using the S-construction form groups, these sets also satisfy most of the other properties listed in Section 2.2. Properties (3) and (5) are immediate from the construction. Property (2), F ( G n) C S, follows easily because we can instantiate free entries in M E S to the corresponding entries in its parent G c F(Gn), or in other words because G is a parent of G. It remains to determine which sets built using the S-construction correspond to projectively invariant notions of continuity. For notational convenience we define:

sn,k

s,), ..,

sp)] = sn [(k, 1),

with the restriction that for i = 1 , . . . ,p and 0 ~< (r/ - si) ~< k. We shall show shortly that these sets correspond to projectively invariant notions of geometric continuity. Graphically this construction corresponds to sets of matrices of the form

where entries marked 9 match those in G and for clarity we have restored the the zeroth row and column of M. Again we may have arbitrarily many (possibly none and possibly overlapping) diagonal-blocks but each diagonal-block must be no bigger than the one in the upper left corner which includes the zeroth column; everything else below the diagonal, except for the zeroth column, is free and diagonal entries are nonzero.

A.V~ Habib, R.N. Goldman /Computer Aided Geometric Design 13 (1996) 905-929

921

Notice that the three standard notions of geometric continuity are instances of the S,~,k-construction since: • F ( G n) : S,~[(n, 1)] = Sn,n[()], • I-'(F n) = &~[(1, 1), (2, 2 ) , . . . , (n, ,~)] : S,,,, [(2, 2 ) , . . . , (n, ,,,)], •

r'(o",)

s.[O]

=

=

&,,[O].

Other examples will appear later in Section 3.3.2; see also (Habib and Goldman, 1994). Next we show that groups of connection matrices of the form S,~,k[(rj,sl), . . . , (rp, sv) ] correspond to projectively invariant notions of geometric continuity. We begin with a simple observation. Let L, M and N be three lower triangular matrices, and let P = L . M • N. Then

P,.s =

E

L,.#M
(9)

i = s j=s

This follows since for arbitrary matrices L, M, N

i=0 j=0

and L, M, N are lower triangular. In particular, P,,, depends on M,,, and entries to the right and above but no other entries in M. The next proposition uses the result by Goldman and Micchelli (1989) which states that G'
G e V(Gn), Dn(w)(r) = G. D , ( u ) ( r ) ~ G = L ( 1 / w ) . G. L(u)

(10)

where the matrix L(w) is the Leibniz matrix defined by Eqs. (7) and (8),

Proposition 6. Let M • Sn,k[(rl, sl), (r2, s2),... , (rp, sp)] with parent matrix G c F ( G n) and let D,~(w) = M . Dn(u). Then N : L ( 1 / w ) . M . L(u) E S,,,k[(r,, s,), (r2, s 2 ) , . . . , (rp,sp)] Moreover, G is a parent of N.

Proof. Construct w* such that: .O~(w*)

=

G. Dn(u).

Since the first k + 1 rows of M and G are identical and

D,~(w) = M . D,,(u), it follows that :

j :

o,...,k.

Together with (8) and the fact that ( l / w ) (p) depends only on to (e), j = 0 , . . . ,p, this implies that

A. W. Habib, R.N. Goldman /Computer Aided Geometric Design 13 (1996) 905-929

922

Lij(1/w*) = Lij(1/w),

0 ~ (i-j)

(11)

<~ k.

By (10) we know that L(1/w*) • G. L(u) = G. Thus, by (9), we have r

N~s = E

i

~

L~,i(1/w)Mi,jLj,~(u),

i=s j=s r i

G~ = E i~8

E

L~,i(1/w*)Gi,jLj,~(u).

j~$

Let (ro, s0) = (k,0) and consider the entries N~,~, where sq <<. s <<. r <. rq and q = 0 , . . . ,p. From (11) and the fact that rq - sq <~ k, it follows that

Lr,i(1/w) = L~,i(1/w*),

sq ~ i ~ r ~ rq.

Moreover by the construction of M

Mij = G i j ,

8q <<.j <<.i <<.rq.

Hence,

Nrs = Grs,

Sq <<.s <~ r <~rq and q = 0 , . . . , p .

It follows that N E S n , k [ ( r l , S l ) , . . . , (rp, sp)], and that G is a parent of N.

[]

T h e o r e m 2. The set of matrices Sn,k[(rl, s l ) , . . . , (rp, Sp)] represents a projectively in-

variant notion of geometric continuity. Proof. From the construction of S.,k and Propositions 5 and 6, it follows that the set S,~,k satisfies all five properties listed in Section 2.2 and so represents a projectively invariant notion of geometric continuity. [] In fact, we can say more about matrices resulting from the projective invariance constraint. Let M E Sn,k[(rl, s l ) , . . . , (rp, Sp)] and construct M ' so that the entries of M ' are the same as the entries of M except at some of the vertices of diagonal-blocks. We claim that the value of MCk,s k is preserved under the construction corresponding to projection as shown by the proposition below.

Proposition 7. Let M E { j~t ~ mrq sq M~jI --- Mij

S,~,k[(rl, sl), (r~, s 2 ) , . . . , (rp, sp)] and construct M ' so that:

for some values of q in { 1 otherwise.

, p},

If Dn(w) = M ' . Dn(u) and N' = L ( 1 / w ) . M ' . L(u), then

u'q, q -- M'q, q,

q=

Moreover, N~r,~ -= M~, s whenever M~, s is an entry in a diagonal-block of M'. Proofi By Proposition 6, we already know that whenever M~r,s is an entry in a diagonalblock of M ~ then N~,~ = M~,s. It remains to show that

N~q,~ = M ~ , ~

whenM~.,~

-¢M~,~.

A.W. Habib, R.N. Goldman/Computer Aided GeometricDesign 13 (1996) 905-929

923

By (9) rq

N:. sq = E

i

E

M~,jLj,sq(u)L,.q,i(1/w).

(12)

{ : S q .j~Sq

Now let G be a parent matrix of M. Then, as in the proof of Proposition 6, we can build w* such that

L~j(1/w*) =- L~j(1/w),

0 <<.(i - j) .~ k,

(13)

and for all r, s,

Gr,~ = ~

~

Gi,jLj,,(u)L~.,i(1/w*).

(14)

i=S j ~ s

By construction,

.~/[~j = Mij = G~j,

Sq <<.j ~ i <<.rq - 1 a n d s q + l

<~j <<.i = rq.

Let/~4~'q,sq = G~q,s, + Ar~,sq. Then by (12) and (13) rq

N~r~,~, = E

i

E

GijLjsq(u)Lr,,i(1/w*) + Lr,,,.q(1/w)Lsq,s,(u)Ar,,s,.

(15)

i:Sq j=Sq

Now by (8) we have, (1/w) = 1/w

and

Lsq,s (u) = u.

Since Dn(w) = M ' . Dn(u) and M~o -- l, it follows that at the point of evaluation w -- u; hence

Lrq,,. (1/w)L~,sq(u)-- 1. Thus by (14) and (15)

From the proof of Proposition 6 we conclude that entries in diagonal-blocks of matrices in Sn,k[(rl, s l ) , . . . , (rp, Sp)] are invariant under projection. Proposition 7 says something more. Let S~,k[(rl,sl),..., (rp, Sp)] be a set of matrices defined the same way as Sn,k[(rl, s l ) , . . . , (rp, sp)] except that the constraints on the entries (ri, si), for some values of i in {1 . . . . ,p}, are relaxed so that the (ri,si)th entry is either free or depends only on entries in diagonal-blocks. This set need not form a group nor contain F(Gn). But Proposition 7 asserts that if such a set forms a group and contains F(G'~), then it represents a projectively invariant notion of geometric continuity. We shall see later on (Examples 1 and 2) that this result actually comes in handy for proving the projective invariance of notions of geometric continuity embodied by groups of connection matrices.

924

A. W. Habib, R.N. Goldman /Computer Aided Geometric Design 13 (1996) 905-929

3.2. F n C H n C 0 n We start by showing that there are notions of geometric continuity H n that lie strictly between F n and O n by constructing groups of connection matrices F ( H n) such that

F ( F n) C F ( H n) C r ( o n ) . E x a m p l e 1. Consider the geometric continuity measure H n with connection matrices F ( H ~) of the form

M=

1

0

0

fll

0

/32

0

/3n

0

• " "

0

...

0

...

o

. . . . . .

±/3?

It is easy to check that F(H'*) satisfies all our criteria. The only nontrivial property to verify is projective invariance which follows from Proposition 7. The group F ( H 2) is represented by the set of connection matrices

M=

1

0

0

0

/31

0

0

/32

Let

D n ( f +) = M D n ( f - ) ,

M C F(H2).

Then, f+ = f-,

f+' =/31f-',

-2 - - tt

f + " = / 3 2 f - ' 4-/31 a*

,

so the curvature vectors t~ are related by t~+ = ( f + ) ' x ( f + ) " = / 3 1 f - ' × (/32f-' 4-/32f - ' ) [If+'ll 3 11/31f-'H 3

= ::l:t~-.

Notice that F 2 continuity is stronger than H 2 because F 2 requires matching the curvatures exactly so that ~+ = ~ - . Other examples can easily be constructed that satisfy our algebraic requirements by freeing some entries along the diagonal. Using the S notation we have F n C_ C_ 0

3.3.

Gn C H n C F n

Now we turn our attention to continuity measures between G n and F n. We know that /-'(G 2) = E ( F 2 ) , so there can be no groups in between. What groups of connection

A.W. Habib, R.N. Goldman / Computer Aided Geometric Design 13 (1996) 905-929

925

matrices lie between F ( G n) and F ( F n) for n ~> 3? We address this question first for continuity of order 3 and later on for order r~, > 3. 3.3.1. G 3 C H 3 C F 3 Here we prove there is no notion of geometric continuity that satisfies our criteria and lies strictly between G 3 and F 3. From the point of view of geometric invariants like curvature and torsion, this result is important because most curves in C A G D lie in 3-space.

T h e o r e m 3. There are no geometric continuit), measures H 3 strictly between G 3 and F 3. Proof. The proof proceeds by showing that there is no group of 4 x 4 matrices F ( H 3) strictly between F ( G 3) and F ( F 3 ) . We do this by adding one arbitrary non-~3-matrix, A E ] ' ( F 3) - F(G3), to F ( G 3) and showing that we can then obtain any other matrix in F ( F 3) using only multiplication by matrices in F(G3). For notational convenience, we shall again adopt the convention of omitting the zeroth row and column of the connection matrices since these are fixed. Consider an arbitrary matrix A E F ( F 3) -- F(G3); then

A=

[

/it

0

3:

312

/~3

A--

0

9: 33

0

,

~>o,

5¢0.

0

1

•

3/il/i2 -k ~5 /i~

So A can be factored as /il

] o loll 0

0 3/31/32

•

Call the second factor D. Now pick any other matrix C E F ( F 3) - F(G3); then

C=

71

0

72

712

73

37172+cr

=

72

7~

•

1

•

Since the first factor is a/i-matrix and hence already in F(G3), it suffices to show how to construct the second factor from D and the 3-matrices using matrix multiplication and inversion. We proceed in the following manner: 1. If el = 5//i~ and e2 = cr/'),~ have the same signs, go to the next step. Otherwise, notice that -I D -l z

1

1i =

0

1

•

926

A.W. Habib, R.N. Goldman/Computer Aided Geometric Design 13 (1996) 905-929

But D -1 E F ( H 3) since F ( H 3) is a group. Let ej = -5//3~. Then e~ has the same sign as e2 = ~r/~,~. 2. Compute

i i 0 0]l :ll i10 1 0

P =

=

c~2

C~l2

0

o~3

3 oq o~2

ct

0

1

--o~2el

O:le 1

•

1

el

0

'

c~1

0

0

c~2

c~

0

o~3

3ct I o~2

o!~

]

•

Now P c F ( H 3) since each of its factors is in F ( H 3 ) . If we pick al = e2/el > 0 and ce2 = 0, then we have

P=

0

1

,

0 which is all that is required to compute the matrix C.

[]

In fact Theorem 3 proves a stronger result than stated. The proof shows that there are no g r o u p s of 4 x 4 connection matrices strictly between / ' ( G 3) and _F(F3). Thus even if we drop the requirement of projective invariance we still cannot find a notion of geometric continuity strictly between G 3 and F 3. 3.3.2. G 4 C H 4 C F 4

Although there are no groups of connection matrices between F ( G 3) and F ( F 3 ) , the situation changes quickly for higher orders. E x a m p l e 2. An interesting example appears in (Pottmann, 1989) where he considers the tangent surface defined by the lines tangent to a given curve. Pottmann constructs connection matrices for parametric curves by requiring that these tangent surfaces meet smoothly. His connection matrices for continuity of order 4 are given by

M=

[ /31

0

0

0

/32

/32

0

0

/33

/35

/34

/36

0 3(/31/35 -- fl~/32)

/34

It is easy to verify that this set of matrices forms a group and that this group contains F(G4). The remaining properties of geometric continuity other than projective invariance

A. IV.. Habib, R.N. Goldman / Computer Aided Geometric Design 13 (1996) 905-929

927

are immediate from the construction. Projective invariance follows because by Proposition 7 the values directly below the main diagonal remain invariant under projection. Notice that this is an example of a set of connection matrices that satisfies all the properties listed in Section 2.2 but is not generated by the S-construction. E x a m p l e 3. If in Example 2 we pick/35 = 3/31/32, w e get another continuity measure, represented by the set of matrices of the form fll

M=

0

0

0

o

o

fd3

3~1&

~

o

&

f15

6f12f12

f14

This set is just $4,3[(4,3)], so by Theorem 2 it too satisfies all five properties listed in Section 2.2. The S-construction provides us with two additional examples of projectively invariant notions of geometric continuity between G 4 and F4: $4,2[(4,3)] and ,5'4,3[(4, 4)]. Whether these four groups exhaust all projectively invariant notions of geometric continuity between G 4 and F 4 is still an open question.

3.3.3. Higher order continuity We now show that the number of new notions of geometric continuity between G ~ and F '~ grows very rapidly with n. T h e o r e m 4. The number of new notions of geometric continuity between G n and F '~ increases at least exponentially with n (when ~, >~5). Proof. Let fn-1 denote the number of notions of continuity between G '~-1 and F n-I Then we can build at least 2 f n - l new notions of continuity between G n and F n as follows: Let F(G n-l) C F ( H n-l) C F ( F n-l) and define F*(H '~-1) to be the collection of all (n + 1) × (n + 1) lower triangular matrices M* such that:

M~,j=Mi,j, i , j = 0 , . . . , T z

1

forsome M c F(H'~-l),

M,*~,o = o,

m~,n#O, M/*,n=0,

i=0,...,n-1.

Then F * ( H n - l ) represents a projectively invariant notion of geometric continuity of order n. However F*(H n-l) is not contained in F(F'~). To correct this defect, let

S' ( H " - ' ) = F * ( H " - ' ) c~ &,~ E(,~,~)i

s 2 ( m '-') = r * ( H ' - ' ) n

S,,,,[(,,,.-- 1)].

By construction both SI(Hn-1),S2(H "-1) C F(F'~). Moreover Sl(H '~-1) and S2(H n-l) are intersections of pairs of groups representing projectively invariant no-

928

A . ~ Habib, R.N. Goldman/Computer Aided Geometric Design 13 (1996) 905-929

tions of geometric continuity of order n and hence they also represent projectively invariant notions of geometric continuity of order n. Notice too that if K n-I and H n-1 are two distinct notions of geometric continuity between G '~-I and F n-l, then SJ(K n-l) ¢ SJ(H n-l) because the intersection with Sn,j affects only the nth row for j = 1,2; entries along the diagonal in the ( n - 1)st row are already constrained since, by assumption, F(Kn-1), F(H -l) C F(Fn-1). Hence, f , ~> 2fn-1. Thus the number fn of notions of geometric continuity between G n and F n is at least O(2'~). []

4. Conclusions and open questions We have presented a new algebraic approach to the construction of sets of connection matrices representing different notions of geometric continuity. We introduced five properties that must be satisfied by all reasonable geometric continuity measures. Focusing on these properties, we were able to develop schemes to create many new forms of geometric continuity that qualify under our criteria. Many questions, however, remain unanswered and need further investigation. In some cases, we are able to identify invariants associated with continuity measures from the corresponding connection matrices. Finding these invariants in general is an interesting problem that needs further study. Also consider the reverse question: given a collection of geometric invariants, how can we find the most general form of geometric continuity and the associated group of connection matrices that preserves the continuity of these invariants? Pottmann (1989) defines new notions of continuity for curves by considering the continuity of their associated tangent surfaces. How his sets of connection matrices relate to the groups created by the S-construction is still an open question. More generally, what precisely is the geometric meaning of the notions of geometric continuity derived from the S-construction? Another interesting problem is to find the set of all transformations that preserve a given notion of geometric continuity. That is, for a notion of continuity H n characterize the set of transformations T such that (f, 9) ~ H'~ ~ (T o f, T o 9) C H n. If H n ¢ G n are there any differentiable transformations that work other than projections? Finally, properties of splines developed using the new continuity measures introduced here need to be investigated.

Acknowledgements This work was supported in part by National Science Foundation grant CCR-9113239. We would also like to thank Joe Warren, Tom Lyche, and Phil Barry for reading a preliminary version of this paper and providing many helpful suggestions.

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