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Some remarks on geometric continuity of rational surface patches
Computer
Aided
Geometric
Design 9 (1992)
143
143-157
North-Holland
Some remarks on geometric continuity of rational surface patches Bert Jiittler Technische Vniversitiit Dresden, Germany
Peter Wassum Technische Hochschule Darmstadt, Germany Received
January
1991
Revised November
1991
Abstract Jiittler, Aided
B. and P. Wassum, Geometric
The geometric geometric
Some
Design 9 (1992)
continuous
continuous
joint
joint
remarks
between
between
continuity
of rational
surface
patches,
Computer
two rational
surface patches is considered as a special case of the a criterion concerning the in R’. Furthermore,
two hypermanifolds
independence
of the equations
is developed.
By solving the continuity
resulting from the necessary and sufficient
continuity
are determined.
parameter
more than the conditions
These
conditions
conditions
conditions
for one of the patches,
generally
provide
explicit
for geometric conditions
five free form-parameters
continuity
for geometric
such yielding
one
known up to now.
Keywords. Geometric
continuity,
continuity
explicit representation
conditions,
on geometric
143-157.
geometric
continuity of Bizier
of hypercones,
rational
Bezier surface patches, explicit
control points.
Introduction
In Computer Aided Geometric Design curves and surfaces are often described by polynomial resp. by rational parametric representations, e.g. by BCzier curves and surfaces (cf. [Farin ‘901). As an advantage, rational representation schemes allow the exact description of conic sections, which traditionally play an important role in technical applications. In order to describe more complex surfaces, several surface patches have to be assembled for one surface. The joint between two patches should be geometric continuous of order k (k = 1, 2,. . . ), i.e. in case of k = 1 with continuous tangential plane (GC’) or in case of k = 2 furthermore with continuous Dupin indicatrix (CC’). Different conceptions of geometric continuity have been derived [Hoschek & Lasser ‘891, [Degen ‘881, [Pottmann ‘891. Basicly, these conceptions are equivalent to the differential geometric concept of order of contact introduced by Cauchy [Cauchy 18261, [Scheffers ‘231. Conditions for geometric continuity are essential in order to construct geometric continuous joints between surface patches. In 1982, Farin found an elegant construction of a Correspondence bereich
to: Bert Jiittler
Mathematik,
und Peter Wassum,
Arbeitsgruppe
c/o
Prof. Hoschek,
Differentialgeometrie
und Kinematik,
Technische
Germany.
0167-8396/92/$05.00
0 1992 - Elsevier
Science Publishers
Hochschule
Schlossgartenstr.
B.V. All rights reserved
Darmstadt.
7, W-6100
Fach-
Darmstadt,
144
B. Jiittler, P. Wassum / Geometric continuity
CC’-joint between polynomial surface patches [Farin ‘821. Necessary and sufficient conditions for geometric continuity of polynomial surface patches were developed by Liu and Hoschek [Liu & Hoschek ‘891, [Wassum ‘901. Using an example, DeRose showed in [DeRose ‘901 the general independence of the equations derived from these conditions. Degen developed explicit conditions for geometric continuity (i.e., solved them for one of the patches [Degen ‘901. In case of rational surface patches, necessary and sufficient conditions for geometric continuity of order one as well as explicit sufficient conditions providing four free formparameters are known [Liu ‘901, [Vinacua & Brunet ‘891. Discussing this topic, DeRose [DeRose ‘901 also showed the general independence of the equations resulting from the necessary and sufficient conditions. The topic of geometric continuity is thoroughly considered in [Wassum ‘911. Geometric interpretations for construction methods ensuring geometric continuity are developed there. In this paper an unusual way to the conditions for geometric continuity is presented. Above all, the case of a joint between two patches with agreeing polynomial degrees along their common boundary curve will be considered. This case shows best the occuring phenomena and all other cases can be covered using analogous deductions. In [Kruppa ‘571, the theory of curves is considered as a special case of the theory of tangent surfaces. Analogously to this, in this paper the geometric continuity of rational surfaces will be interpreted as a special case of the geometric continuity of selected hypermanifolds in R”. Furthermore, a criterion for the independence of the equations derived from the necessary and sufficient conditions will be developed. The continuity conditions will be solved for one of the patches and thereby explicit conditions for geometric continuity will be found. These conditions generally provide five free form-parameters. Thus, they have one parameter more than the conditions known up to now.
1. Geometric continuity of rational surface patches Let a rational surface patch be given by its Cartesian coordinates neous coordinates x:
x resp. by its homoge-
x,(r, s)
&[O, 1]2-,R3:(r,
s) -+x(r,
S) = X0(: s) ,
I 1 xz(r9
s)
x3(r9
s)
(1-l) (x is also defined on a domain R, [O,112c R c R2, and is restricted to [0, 112.This guarantees the regularity of 5 at the boundary x(a[O, l]*).) The four functions Xi(U, c) (i = 0, 1, 2, 3) are polynomials in r and s with highest degree in r equal to n and in s equal to m. For example, w can be written as a rational BCzier surface patch: x( r, s) = E i-0
k B,Y( r)B,!‘( s)bij
where B,k( r) = (:)I’( 1 - t)k-‘,
t E
[O,
11
j=O
bij E R4\{0}.
(1.2)
B. Jiittler, P. Wassum / Geometric continuity
Usually, the control points bij are written in the form bij = Wij
.
with weights wij. Of course, this description excludes infinite points. The given surface patch x is to be continued along its edge r = 0 by a second rational surface patch. This second patch is described by its Cartesian coordinates y resp. by its homogeneous coordinates y: Y,(U, L’)
1 y : [O, 112+ IX3: (u, c) -+y(u,
c) =
Y,(U? u)
Yo(Ut L’)
1 Y,(U9U) 1
(Again, y is also defined on a domain U, [0, 112c II c R2, and is restricted to [O, ll’.) The four function< yi (i = 0, 1, 2, 3) are polynomials in u and u with highest degree in II equal to n and in u equal to m*. The second patch y should join with r along its edge u = 0. For example, y can be written as rational BCzier surface patch: y(u, u) = g
i=”
2 B,Y’*(u)B,F’(u)b~j,
b~jER4/{0).
(1.5)
j=l)
The joint along the common, boundary curve r = u = 0 should be geometric continuous in the following differential geometric sense of order of contact: Definition. The surface patches x and y have contact of order k along the boundary curves r = u = 0 (i.e., they have a CC&-joint) if-a regular parametric transformation
U+R:(u,
c) +(r(u,
c), s(u, u))
with r(0, c) =O
(1.6)
exists, where the partial derivatives of x and y of 0-, . . . , k-th order with respect to u and c are identical along the common boundary curve: (&j(Q.)'Y(U,
L?)I(o,c)=(D~,)i(DL.)jx(r(u' L'),s(u9 li'))l(O.O a
for i+jgk,
i, jEN
where D, := g i
The condition r(0, c) = 0 guarantees x(0, s). The Co-condition
the coincidence
1
.
of the boundary
curves ~(0, c) and
X(0, u) =y(O, c)
(1.8)
is also demanded in order to simplify the calculation. Therefore, tion (1.6) has to satisfy ~(0, u) = L’ in addition to
(1.7)
r(0, c) = 0.
the parametric
transforma(1.9)
(If the Co-condition (1.8) is not satisfied, in general x(0, ~1)and ~(0, L’) can only differ by a constant factor p. This follows from the degrees presumed for the polynomials Xi and Yj (i, j=O,..., 3). If p f 1 holds, let y’ := py and calculate with y’ instead of y.)
B. Jiittler. P. Wassum / Geometric continiciry
116
Resulting from the differentiation of rational expressions, the equations for the patch y obtained from the above definition prove out to be very complicated. In order to avoid this, conditions for geometric continuity of patches x and y being described by their homogeneous coordinates will be deduced in the next section. The following abbreviation for the vector polynomiais (i.e., the vector-valued polynomials) of the derivatives of the homogeneous coordinates x resp. y with respect to r resp. u (i.e., crosswise to the common boundary curve) is introduced in order to simplify the notation and with respect to the definition of the strip element (see Section 3.3):
‘x=‘x(L’) :=(D,)‘x(r, S)I(().(.) (i=O, ‘y=‘y(r)
:=(D,,)‘y(ll,
l,...
L’)I(“.l,) (i=O,
l,...
), ).
(1.10)
These vector polynomials take the form
‘x( L’) = (mm_!i)! ,t B;(C) A&,.,
and
‘Y(L’)
= Cmz’i,!
r==O
where AObj,k:= bj.k
,g Bkn((‘) , 0
A’&?.,
A”’ ‘bj,k := A’b,, ,,k - A’bjsk,
and
(1.11)
if x and y are written as rational BCzier surface patches. The abbreviation (1.10) allows a shorter description of the derivatives along the common boundary curve. For example: (1.12)
‘i( C) = Q( D,)‘x( r, s) I (“.(.).
2. Development
of conditions
for geometric
continuity
2.1. Geometric continuity of hypercones in R4
The three-dimensional real projective space can be considered as the set of all one-dimensional subspaces of [w4 (cf. [Schaal ‘761). Then the surface patches x and y correspond canonically to two hypercones X and Y in R4: X(r, s, t) :=t-x(r,
s) (t#O)
and
Y(U, L’, W) := w*y(u,
L’) (W # 0).
(2.1)
Here, hypercone always means the system of all lines connecting a fixed point (the point 0 = [0, 0, 0, OIT>with the points of a two-dimensional director manifold (x(r, s) resp. y(u, L,)). (Another kind of hypercones in Iw4 is the system of all 2-planes connecting a fixed line with the points of a directrix.) The images of all points of the hypercones X and Y under the projection
zo
,:lR4\((0}xR3)-,R3:
!I [) :;
z3
ZI
4;
z2
(2.2)
z3
lie on the surface patches x and y. This projection can be interpreted as intersection of X resp. Y with the hyperplane z,, = l-(and deletion of 0-th coordinates). The following theorem describes the correspondence between the regularity of a surface patch and the regularity of the hypercone linked to it: Theorem. The surface patch x is regular at its point x(rO, ~“1, if and only if the hypercone X is regular at all points X(r,, SO, t) with t # 0 and xO(rO, s,,) # 0 holds.
B. Jiittler, P. Wassum / Geometric
continuity
147
Proof. First, the case xJT”, sO) f 0 is considered.
Then it has to be two vectors x., and x., are linear dependent if and only if the three i.e. x.,, x., and x, are linear dependent (‘,z’ means the derivation linear dependence of xSrr x., and x follows immediately from the and x., as
shown that at (T,~, SJ the vectors X.,, X., and X.,, with respect to z.) The linear dependence of x.,
(2.3) hold. In order to proof this inversion, let a nontrivial linear combination of the null vector be given: (YX.,+ px,, + yx = 0 The 0-th coordinate
( LY,p, y E R, not all equal to 0).
(2.4)
of this equation reads:
ax,., + px”.x + yx, = 0.
(2.5)
The coefficients (Y, p, y are considered as unknown. Then, every solution of (2.5) and therefore the linear combination (2.4) can be written as linear combination of the basis (Y,= -x0, p, = 0, y, =xO., and a2 = 0, & = -x0, yz =xws of the space of solutions: 0=
-K,x~x.,
-
~~x~~x.,
+
(~,x~.,
+
~~x,~,,)x
(K,,
K~ E
R).
(2.6)
Therefore, the linear dependence of x9, and r., follows from (2.3). Obviously, the assertion holds for x,,(~s, s,J = 0. 0 The geometric continuity of hypercones is analogously defined with the geometric continuity of surfaces: Definition. The hypercones X and Y have contact of order k along the two-dimensional boundary manifolds r = u = 0 (i.e., they have a GC“-joint) if a regular parametric transformation
I/x (R\(O}) -+Rx(R\(O}):(u,
L',w)-+(u,L',
w), s(u, L',w), t(u, u, w))
with r(0, u, w) = 0
(2.7)
exists, where the partial derivatives of 0-, . . . , k-th order with respect to u, c and w are identical along the common boundary manifold:
( DJi( D,)‘(D,)‘Y(u,0, w) I (0.c.w) =
mA QJ’(QJ’Wr ( u, c, w),
for i+j+l
s(u,
0,
w), r(u,
L’,
w)) l(0.r.w)
i, j, IEN.
(2.8)
The condition r(0, c‘, w) = 0 guarantees the coincidence of the two-dimensional boundary manifolds X(0, U, w) and Y(0, S, t). The parametric transformation (2.6) has to satisfy ~(0, u, w) = u
and
~(0, c, w) = w
in addition to
r(0, u, w) = 0.
(2.9) This follows from the Co-condition (1.8). The connection between the geometric continuity of surfaces and the geometric continuity of the hypercones corresponding to them will be studied in the next section.
148
B. Jiittler. P. Wussum
/ Geometric
continuity
2.2. The correspondence between the geometric continuity of surfaces and the geometric continuity of hypercones
More appropriate than the definition given in Sections 1 and 2.1 is another definition of geometric continuity to study the correspondence between the geometric continuity of surfaces and the geometric continuity of hypercones: The surface patches 3 and y (the hypercones X and Y) have contact of order k along the boundary curves (along the iwo-dimensional boundary manifolds) r = u = 0 (i.e., they have a CC’-joint) if the following assertion holds for every curve a = a(q) on the surface 5 (A =,4(q) on the hypercone X): If ~(0) (A(O)) lies on the boundary curve (on the two-dimensional boundary manifold) r = 0 and a(q) (A(q)) is regular in a neighbourhood of q = 0, then a second curve _b=b(q) (B = B(q)) exists on the surface patch y (on the hypercone Y), which is regular in a neighbourhood of q = 0, and Definition.
lim -&n(q) 141-+c lqlk
(
lim 7
1
lr11’0 191
-_b(q)II
(2.10)
=o
II A(q) -B(q) II = 0
(2.11)
i
holds. The conditions of the above definition obviously hold, if a parametric transformation (1.6) resp. (2.7) exists. On the other hand, a parametric transformation can be constructed by considering selected curves on the surface x resp. on the hypercone X. The exact proof of the equivalence of both definitions for geometric continuity will not be shown here, cf. [Scheffers ‘231. The following theorem describes the correspondence between geometric continuity of surfaces and geometric continuity of the hypercones linked to them: Theorem. If the surface patches x and y resp. the hypercones X and Y are regular along their boundary curces resp. along their bound&y manifolds r = u = 0, then the surface patches x and y bar-e contact of order k (i.e., a GC"-joint ) there, if and only if the hypercones X and Y also hat; contact of order k there. Proof. (-) Let a GCk-joint of the surface patches x and y along the boundary curve r = II = 0 be given. Then a regular parametric transformation (1.6) exists, where the partial
derivatives of x and y of 0-,.. ., k-th order with respect to II and 1’ are identical along the boundary curve (1.7). -Obviously, and
Y’(rl’, L”, w’) := w’. (2.12)
are two other regular parametric representations of the hypercones parametric transformation (1.6) can be extended by r’( Ii’, L”, w’) = r(u’,
L*‘),
s’(u’,
L-‘, w’) =s(u’,
v’)
and
X and Y. The given t’(u’,
u’, w’) = w’
(2.13)
B. Jiittler, P. Wussum
/ Geometric
continuity
149
to a parametric transformation (2.7) where the partial derivatives of X’ and Y’ of 0-, . . . , k-th order with respect to u, c and w are identical along the common two-dimensional boundary manifold r’ = u’ = 0 (2.8). Therefore, X’ and Y’, i.e. also X and Y, have contact of order k there. (+> The second definition of geometric continuity is more appropriate to proof this inversion than the first one. Let a GCk-joint of the hypercones X and Y along the two-dimensional boundary manifold r = u = 0 be given. Then it has to be shown that the following assertion holds for every curve ~(4) on the surface x: If a(O) lies on the boundary curve r = 0 and a is regular in a neighbourhood of 4 = 0, then a second curve b(q) exists on the surface y, which is regular in a neighbourhood of 4 = 0, and (2.10) holds. Obviously, if a(q) is a curve on 5 which is regular in a neighbourhood of q = 0 and a(O) lies on the boundary curve r = 0, then the curve
44) :=
i-($)1
(2.14)
I
lies on the hypercone a curve
X and A(O) lies on the two-dimensional
boundary manifold r = 0. Then
B(q) =
(2.15)
exists on the hypercone Y, which is regular in a neighbourhood of 4 = 0, and (2.11) holds. This follows from the GCk-joint assumed for X and Y. The curve b*(q)
:= -
1 (2.16)
b,(q) _b(q)
is a curve on the surface y and regular in a neighbourhood
-L(ll) 141k -
of 4 = 0. On the other hand
--_b*(q)II
IM(q) -B(q)11 + I”‘?;&’
llls(rIll)
(2.17)
holds. Moreover, the equation be(q) = 1 +o( lqlk),
i.e.
for b,(q) results from (2.11). Therefore,
lim q-+0
b,(q) - 1 I&
(2.18)
=O
(2.10) holds for a(s) and b*(q).
q
The geometric continuity of surfaces is equivalent to the geometric continuity of the hypercones corresponding to them. The conditions for geometric continuity of hypercones resulting from (2.8) will be applied to surfaces in the next section.
150
B. Jiittler, P. Wassum /
2.3. Conditions for geometric
continuity
Geometric continuity
of first and second order
The Co-condition (1.8) required for the surfaces transmits to the hypercones: (2.19)
y I (O.l~,W) =x I (O.r.w)So, now it reads oy =0x,
(2.20)
using the abbreviations introduced Y*, I (o.~.,~)
=
(rdar
+
in (1.10). In addition to (2.191, the three equations
sd.,
+
tq,X.t)
y,,. I (OJ~.W) = ( r.t.X.r + s.,.X., + t-,X.,) y.,
I (O.l~.W) =
( rqwX.r
+
s-,X.,
+
I (o.~.~)~
(2.21)
I (o,~.,,,,) and
(2.22)
t.,X.,)
(2.23)
I (o,~..~~
and the condition of regularity det
I
t*u
r’lc rqp
Sh s,,.
t,,
rqw
s,,
t.w
I
(2.24)
#O
(O.C~.W)
have to be satisfied for geometric continuity of first order. The three equations (2.211, (2.22) and (2.23) result from (2.8). The two last equations (2.22) resp. (2.23) hold with r.r(O, c, w) = 0,
s.JO,
u, w) = 1
and
t.w(O, c, w) = 0
rvw(O, c, w) = 0,
s.,(O,
L’, w) = 0
and
t.,(O,
resp.
(2.25)
u, w) = 1
(2.26)
as they follow from derivation of (2.13) with respect to 1’ resp. W. Therefore, t-JO,
(2.27)
L’, w) # 0
still has to be satisfied as condition of regularity. The equation w.‘y(u)
=rJO,
v, w) *w*‘x(s(O,
u, w))
+s.,,(O,
0, w)*w*“i(s(O,
u, w))
+ t.,,(O, 0, w) *Ox(s(O, u, w))
(2.28)
results from (2.21) using the definition of the hypercones (2.1) and involving the abbreviations introduced in (1.10). Let t.,(O, u, w) := W *i.,(O, u, W).
(2.29)
‘y(u) = rqu(O, c, w) *‘x(u) +s.,(O, 0,w) *'i(u)+ r*.,(O, u,w) *Ox(u)
(2.30)
Therefore,
follows from dividing of (2.28) by w and as (2.9) holds. The left side of this equation does not depend on w. Then, r.,(O, c, w), sJ0, L’,w) and i.,(O, u, w> also do not depend on w as ‘x, Ox and Of are linear independent. (This linear independence results from the regularity of X.1 Therefore, the GC’-condition reads (2.31)
‘y = rYu’x + s.,Oi + ivuox,
where all quantities are functions of L’. Analogously, the GC2-condition ‘y =
r.uu’x + s.,,,Oi + i.,,Ox + ( r.u)2 2x + ( s.u)20i + 2r.“s.,,‘i
+ 2r.,,i.,‘x
+ 2s.,i.,‘f
(2.32)
B. Jiittlrr, P. Wassum
/ Geometric
151
continuity
follows. It can as well be written as *y = F.Jx
+ s1.,,,“x+ i.,,‘)x + (r.“)* *x + (s.,,)*ox + 2r,,s.,,‘x,
(2.33)
* Again, all quantities are functions of c’. where i.,,,, := T.~~,+ 2r.,,i.U and i.,,,, := s.,,,, + 2s.,t.,,. The coefficients T.&,and s.,, are already determined by the contact of first order. In order to simplify notations, the part already determined in (2.33) will be called := (r.J2 *x + (S.,,)%
K=K(c)
(2.34)
+ 2r.‘,S,“‘x.
A recurrent representation of the GCk-conditions is developed in [Wassum ‘9Ia, b]. The systems of linear equations resulting from the conditions for geometric continuity of first and second order will be studied in the next section.
3. Explicit conditions
for geometric
continuity
3. I. Explicit conditions for geometric continuity of first order The alternating product
(e. el e2 e3\ N=N(c):=‘xr\‘xA’x=det
- ‘x - Of \ -ox-,
(3.1)
(e,: i-th unit vector of R4) of the three vector polynomials ‘x, Of and Ox yields a normal vector N. This vector satisfies the three equations (N, ‘x) = (N, “x) = (N, Ox) = 0
(3.2)
and its length is equal to the three-dimensional volume of the parallelepiped spanned by ‘x, Ox and Ox (cf. [Reichardt ‘571). The normal vector N is described by a vector polynomial, where the highest degree in c is 3n - 2, although the highest degrees in ~3of the vector polynomials ‘x, aA and Ox are n, n - 1 and n respectively. This results from the relation of the two factors Ox and Ox = D,.‘x in the alternating product. The equation (N, ‘y) = 0
(3.3)
follows by inner multiplication of condition (2.30) for a CC’-joint between the surface patches x and y by N and it is equivalent to this condition. Let
(3.4) Then the following system of homogeneous linear equations for the vector polynomial ‘y (i.e., for the first row of unknown control points of the patch y) can be derived from (3.3):
c
i+j=k 0 %2j3g
n0,iYo.j+n,,iY,,j+n*,iY*,j+n3,iY3,j=O 2
(~=OV**.*~~-~)*
(35)
152
8. Jiittler,
F’. Wussum
/ Geomrtric
continuity
This system consists of 4n - 1 equations with 4n + 4 unknowns. Therefore it has at least five linear independent solutions. In general, the degree in I* of the normal vector N is 3n - 2 and N is irreducible (i.e., its coordinate functions have no common linear factors). The next theorem yields a criterion of the number of linear independent solutions of (3.5): Theorem. If the degree in c of N really is 3n - 2 (i.e., N cannot be described by a uector polynomial, where the degree in L’ is only (3n - 31, system (3.5) has more than fioe linear independent solutions if and only if N is reducible (i.e., its coordinate functions haoe common linear factors). Proof. (+I
Let six linear independent
solutions
’ P0.i liCli(L’)
=
;;,;
.I
(i=
C”+Ti(C)
1,...,6)
(3.6)
\ P3.i 1
of system (3.5) be given. (The degree in L’of the vector polynomials ri(l*) in this representation is assumed to be n - 1.) Then three linear independent, linear combinations qi = qj( u) = aill + pi11 + y,l, + ail, + EjlS + ljI, (i = 19 29 3;
aj,
Pjt
Yj,
sj,
Ej9
lj
E
R)
of the given solutions exist, where the coefficients homogeneous linear equations
(3.7) aj, pi, -yj, Sj, ej, lj satisfy the system of
as this system consists of three equations with six unknowns. The first, second and third coordinate function of the vector polynomials qj, obtained from (3.7), are polynomials, where the highest degree in u is only n - 1 as (3.8) holds. The vector polynomials qj have to satisfy the last equation of system (3.5). Without loss of generality, let n,,3n_3 z 0 be assumed. Therefore, also the 0-th coordinate functions of the vector polynomials qj are polynomials, where the highest degree in L’ is only n - 1. The alternating product ni=ti(v)
=q* Aq*Aq3
(3.9)
of the vector polynomials qj is described by a vector polynomial i, where the highest degree in u is 3n - 3 and this is always (for all c E R) linear dependent on N. Thus, N(u) = K(u)N(L’) is reducible. (c) Let N(c) = K(u~‘?(u) be reducible. Then equation (3.3) is equivalent to (ni, ‘y) = 0.
(3.10)
A system of homogeneous linear equations follows from above. It consists of at most 4n - 2 equations with 4n + 4 unknowns. Therefore it has at least six linear independent solutions. 0 Analogously it is easy to show that the following assertion holds: If the degree in u of the normal vector N is really 3n - 2, system (3.5) has exactly 5 + g linear independent solutions, where g is the degree in u of the largest common divisor of the coordinate functions of N.
153
B. Jiittler, P. Wassum / Geometric continuity
solutions ‘y,(c) (i = 1,. . . , 5) of equation (3.3) (and therefore
The five linear independent of system (3.5)) are known: ‘y,(c)
‘y,(c)
=‘x(L’),
'y,(u) = uOf(u),
ly,(Ly) =u(n%(L’)
and
'y,(u) ="_r(u)
=“i( L’),
(3.11)
-lPf(L’)).
(Note that the highest degree in L’of ‘y,(v) is only n.) If the highest degree in c of the normal vector N is really 3n - 2 and N is irreducible, each solution of system (3.5) can be written as a linear combination of the five solutions (3.11). Therefore the following condition for a Ccl-joint between the surface patches x and y holds: Theorem. If the highest degree in c’ of the normal cector N is really 3n - 2 and N is irreducible, the considered surface patches x and y hate contact of first order (i.e., a CCL-joint 1 along their common boundary curve, if and onlyif (2.20) holds and ‘y can be written as ‘y(u)
=r.,(
where
u)'x(c)
+s.,(c)Oi(u)
+i,U(u)ox(u),
r+,,(u) = P~,~, s-t,(u) = UI.0 + Ul,lL'+ q,2c2, i.,(u) = 7,,. + T*,,U
(PI.07 al,iT
71,j
E Iw)
(3.12)
and 71,1 +
nu1.2
(3.13)
= 0
and the condition of regularity P,.~ # 0 (cf. (2.27)) hold too.
The equations for the first row of unknown control points derived from the above theorem will be developed in Section 4. 3.2. Conditions for geometric continuity of second order The equation (3.14) (N, ‘Y) = (N, K) follows from condition (2.33) for a GC2-joint between the surface patches x and y by inner multiplication by the normal vector N = N(u) and it is equivalent to this condition-(K = K(c) was introduced in (2.34X) Analogously to the case of the CCL-joint, a system of inhomogeneous linear equations for the vector polynomial 2y (i.e., for the second row of the unknown control points of the patch y) results from (3.14). The corresponding homogeneous system is again provided by system (3.5). In general, the solutions of (3.5) are well known. Now, a special solution of (3.14) has to be found. The vector polynomial K = K(U) does not represent such a solution because its degree in u is equal to n + 2. Let ;h”( u) =
P2.0 + PZ.IL’,
= a2.0 + @2,,U + u2.2u2
fdu) Lj)
i.,,(
= T2,0 + T2,,U
+
q,3u3,
+ T2,2u2
(PZ.i*
u2.jy
‘2.k
E ‘).
(3.15)
Then, the following conditions result by comparing the coefficients of v”+’ and u”+~ on the left- and right-hand side of equation (2.33): P2.1 = n”2.3 (n
-
2nP,,ou,.29
+ ?2.2 =
-n(n
-
1)u2,3
+ ?2,2 =
nu2.2
+ 72,l
=
-2n(n
-
w1.2)2,
-(n -
-
l)(n
w7,,,~,,2.
-
w,z)29
(3.16)
B. Jiirtler,
154
P. Wassum/ Geometric
continuity
The coefficients pz.,, u2.3 and T~_~are determined by these conditions. One of the two coefficients u2,2 and TV,, as well as the coefficients pz.O, gzVO,c2,,, ~~,rr can be chosen arbitrarily. These five free form-parameters correspond to the five solutions (3.11) of the homogeneous system. Therefore, the following theorem holds: Theorem. If the highest degree in c of the normal rector N is really 3n - 2 and N is irreducible, then the surface patches x and y hate contact of second order (i.e., a GC2-joint) along their common boundary curce r = u = 0, if and only if they hare contact of first order there and ‘y can be written as 2y( c) = F#,“( Lq’ x ( L3) + s1.,,,( c)“.+( c) + i.,J
c)“x( L’) + (r.,(c))’
+ (s,,( Ls))‘Of( c) + 2r.,( c)s.“( c)‘i(
2x(c) (3.17)
/I),
where the coefficients satisfy (3.12), (3.13), (3.15) and (3.16).
The conditions (2.201, (3.12) and (3.17) for geometric introducing the strip element in the next section.
continuity will be summarized
by
3.3. The strip element
The conditions (2.20), (3.12) and (3.17) for geometric continuity of 0-th, first and second order can be summarized as follows: 1
OY ly
=
i., + s.,
L, +f.,,D,. Ill2Y where r.” = P,,~,
D, +stDL?
q,2L’2,
i.,
TI.IL’,
r*,o
+
(9
0
r ‘u
0
?.,, + 2r.,s.,D,
s., = (T,,0 + @,.IL’+ =
0
rt II
“‘X
-
(3.18)
‘x 2X
1
(b)
A = P2.0 A Shu = a20, + U2,IC + u2.2u2 r’ltu
i ‘UU = 72.0
+ 729 (9
(a) (b) (c) and the
+ P2.1~3 + u2.,v3 7 -I- 72,2v2.
(6)
(6)
: All parameters can be chosen arbitrarily. : One of these parameters can be chosen arbitrarily. : All parameters are determined.
conditions (3.13) and (3.16) hold.
The hypervectors (Ox, ‘x, 2x,. . . IT and (‘y, ‘y, ‘y,. . . IT can be called the strip elements of the surface patches x and y along the boundary curves r = u = 0, analogously to the curve elements of a curve in a point. Then, equation (3.18) describes a transformation of these strip elements. Conditions for geometric continuity of higher than second order can be developed by continuing the studies of Sections 2 and 3.
155
B. Jiittler, P. Wassum / Geometric continuity
4. Construction
of a Ccl-joint
between two BCzier surface patches
The conditions for the control points of the patch y resulting from (3.12) will be developed in this chapter. If x is written as BCzier surface patch, then the five sohttions (3.11) of (3.5) read
‘yr( ti)
=‘X(
L’) =
5
*n(b,,j
B,!‘( L’)
- b,,j),
j=O ‘Y~(G)=~~(L’)=
i B,!‘(C) j-0
.(("-j)(bo,j+,-bO,j)+i(bO.j-bo,j-,)),
CBi”(~).j(bo,j-bo,j-,),
‘Y~(L’)=u~~(u)=
j=O
‘Yq(U) =OX( L’) = t By( L’) * bo,j, j=O
kBy(c)*jbo,j_,.
‘Y~(L’)=L’(~‘x(u)-u’~(u))=
(4.1)
j-0
The O-th and the first derivative with respect to u of y are (cf. (1.11)) I3;( ~1)* bcj
“y( L!) = i
and
j=O 'Y(~) =
~
Bi"(U)
.~*(b~j-b~j).
(4.2)
j=O
r= u = 0
The identity of the control points of x and y at the boundary bcj = bo,j
(j = 0,. . . , n)
(4.3) directly follows from the Co-condition (2.20). (If (1.8) does not hold, then (4.3) and (4.5) are the equations for the control points of y’. The control points of y follow from division by CL.) From (3.12) resp. from ry=
&JYj
(fc;ER)
(4.4)
i-0
the representation bcj=bo,j
+o(b,,j-b0.j)
(j=O,...,
n; a, P,
+Pbo,j+ Y,
6,
YLbo,j+‘Lbo.j-l n n
+’ =bo.j+ n
1
(4.5)
E E q.
for the first row of unknown control points of y is obtained with some calculations. A geometric interpretation of (4.5) can be given in R 4. The equation (4.5) takes the inhomogeneous form n-j j j W;“,j= WO,j+ a( -Wo,j+ 1 and n n n Wl,j
_b,,j= -!-
-
wO,j)
+
pwO,j
+
YswO,i
+
SvwO,j_*
+
E
wO,jbO,j + "(wl.j_bl,j- wO,j_bO,j)+ PwO,j-bO.j+ YLwO.j_bO,j
n
w;".j
+S'WO,j-lbO,j-1 n (j=O,...,
n-j + E pwO,j+ n
n; a, P, Y, 6, E E W with control points denoted likewise (1.3).
dO,j+
1
1
(4.6)
156
B. Jiittler, P. Wassum /
Geometric
continuity
In (4.51, only CY< 0 make sense. Patch y is not regular along the boundary II = 0 for (Y= 0 and for (Y> 0 patch x will be continued by patch y in the undesired direction.
Conclusion
In this paper, explicit necessary and sufficient conditions for geometric continuity between two adjacent rational surface patches with the same degree along their common boundary curve have been developed. These conditions yield a description of the set of all patches joining geometric continuously with a given patch. Out of this set, a particular patch can be selected with respect to background-information (e.g., to conditions provided by requirements from an approximation or interpolation problem). The conditions obtained in this paper can be generalized to the case of the geometric continuous joint between two patches with different degrees along their common boundary curve. Then the degrees of the coefficients in (3.12) and in (3.17) have to be adapted. Generally, the system of linear equations (3.5) resulting from the necessary and sufficient conditions has exactly 5 + 3h linear independent solutions, where the highest degree in c of ‘y (resp. 2y> is n + h. These solutions read ci ‘x(u) (i = 0,. . . , h), ~‘j ‘x(c) (j = 0,. . . , h), tek ‘i(u>(k=O ,..., h+l)anduhf’ (n’x(c> - u”i(.(o)). Of course, in this case the homogeneous coordinates of the common boundary curve can differ by linear factors. The strip elements of triangular rational surface patches are different from the strip elements of the tensor-product rational surface patches: the degrees of the vector polynomials ‘x resp. jy (for a triangular rational surface patch x resp. y) are n -i resp. n -i. The methods developed in this paper can be adapted to this strip elements. The solutions resulting from a reducible normal vector are studied in [Wassum ‘91aI.
References Cauchy, A. (1826).
Leqons sur les applications
et Bibliothique
du calcul infinitesimal
B la gCom&ie,
De Bure f&es,
Librairie
du Roi
du Roi, Paris.
Degen,
W. (1988),
Degen,
W. (1990),
Some remarks on Bizier Explicit
continuity
curves, Computer
conditions
Aided
for adjacent
Geometric
Bezier
Design 5, 259-268.
surface patches,
Computer
Aided
Geometric
Design 7, 181-190. DeRose,
T. (1990),
Aided Farin,
Necessary
Geometric
G. (1982),
and sufficient
conditions
for tangent
plane continuity
of Bezier
surfaces,
Computer
Design 7, 16.5-179.
A construction
for visual C’-continuity
of polynomial
surface
patches,
Computer
Graphics
and
Image Processing 20, 272-282. Farin, G. (1990), Hoschek, Kruppa,
Geometric D.
Surfaces forComputer
E. (19571, Analytische
Liu, D. (1990), Liu,
Curces and
J. and Lasser, D. (1989), GC’
Grundlagen
und konstruktice
continuity
conditions
Design, Academic
Datencerarbeitung,
Dtfferentialgeometrie,
between
two adjacent
Springer,
rational
Press, New York, 2nd edition. Teubner,
Stuttgart,
277-286.
Wien.
Bizier
surface
patches,
Computer
Aided
and triangular
Bezier
Design 7, 151-163.
and Hoschek,
J. (1989),
GC’-continuity
surface patches, Computer-Aided Pottmann,
Aided Geometric
der geometrischen
H. (1989),
Projectively
conditions
between
adjacent
rectangular
Design 21 (4), 194-200. invariant
classes of geometric
continuity
for CAGD,
Computer
Aided
Geometric
Design 6, 307-321. Reichardt,
H. (1957).
Schaal, H. (1976), Scheffers,
Vorlesungen i&r
Vektorrechnung
G. (1923).
Anwendung
Berlin und Leipzig, Vinacua,
601-611.
der Differential-
Geometric,
Deutscher
Vol. 2, Vieweg,
und lntegralrechnung
Verlag
der Wissenschaften,
Berlin.
Braunschweig.
auf die Geometrie,
Walter
de Gruyter
& Co.,
3rd edition.
A. and Brunet,
Schumaker,
Vektor- und Tensorrechnung.
und Analytische
L., eds.,
P. (19891, A construction Mathematical
Methods
for VC’
continuity
in Computer
Aided
of rational Geometric
B&er Design,
patches, in: Lyche, T. and Academic
Press, Boston,
B. Jiitrler, P. Wassum / Geometric continuify
Wassum, P. (1990), Conditions and constructions for CC’-/CC’- continuity between patches. to be published in: Hagen, H., ed., Curce and Surface Modeling, SIAM, Wassum, P. (1991a), Bedingungen und Konstruktionen zur geometrischen Stetigkeit tive Basistransformationen, Ph.D. thesis, TH Darmstadt. Wassum, P. (1991b), Geometric continuity between adjacent rational BCzier surface
157 adjacent integral BCzier surface Philadelphia, PA. und Anwendung auf approximapatches, to be published.
Aided
Geometric
Design 9 (1992)
143
143-157
North-Holland
Some remarks on geometric continuity of rational surface patches Bert Jiittler Technische Vniversitiit Dresden, Germany
Peter Wassum Technische Hochschule Darmstadt, Germany Received
January
1991
Revised November
1991
Abstract Jiittler, Aided
B. and P. Wassum, Geometric
The geometric geometric
Some
Design 9 (1992)
continuous
continuous
joint
joint
remarks
between
between
continuity
of rational
surface
patches,
Computer
two rational
surface patches is considered as a special case of the a criterion concerning the in R’. Furthermore,
two hypermanifolds
independence
of the equations
is developed.
By solving the continuity
resulting from the necessary and sufficient
continuity
are determined.
parameter
more than the conditions
These
conditions
conditions
conditions
for one of the patches,
generally
provide
explicit
for geometric conditions
five free form-parameters
continuity
for geometric
such yielding
one
known up to now.
Keywords. Geometric
continuity,
continuity
explicit representation
conditions,
on geometric
143-157.
geometric
continuity of Bizier
of hypercones,
rational
Bezier surface patches, explicit
control points.
Introduction
In Computer Aided Geometric Design curves and surfaces are often described by polynomial resp. by rational parametric representations, e.g. by BCzier curves and surfaces (cf. [Farin ‘901). As an advantage, rational representation schemes allow the exact description of conic sections, which traditionally play an important role in technical applications. In order to describe more complex surfaces, several surface patches have to be assembled for one surface. The joint between two patches should be geometric continuous of order k (k = 1, 2,. . . ), i.e. in case of k = 1 with continuous tangential plane (GC’) or in case of k = 2 furthermore with continuous Dupin indicatrix (CC’). Different conceptions of geometric continuity have been derived [Hoschek & Lasser ‘891, [Degen ‘881, [Pottmann ‘891. Basicly, these conceptions are equivalent to the differential geometric concept of order of contact introduced by Cauchy [Cauchy 18261, [Scheffers ‘231. Conditions for geometric continuity are essential in order to construct geometric continuous joints between surface patches. In 1982, Farin found an elegant construction of a Correspondence bereich
to: Bert Jiittler
Mathematik,
und Peter Wassum,
Arbeitsgruppe
c/o
Prof. Hoschek,
Differentialgeometrie
und Kinematik,
Technische
Germany.
0167-8396/92/$05.00
0 1992 - Elsevier
Science Publishers
Hochschule
Schlossgartenstr.
B.V. All rights reserved
Darmstadt.
7, W-6100
Fach-
Darmstadt,
144
B. Jiittler, P. Wassum / Geometric continuity
CC’-joint between polynomial surface patches [Farin ‘821. Necessary and sufficient conditions for geometric continuity of polynomial surface patches were developed by Liu and Hoschek [Liu & Hoschek ‘891, [Wassum ‘901. Using an example, DeRose showed in [DeRose ‘901 the general independence of the equations derived from these conditions. Degen developed explicit conditions for geometric continuity (i.e., solved them for one of the patches [Degen ‘901. In case of rational surface patches, necessary and sufficient conditions for geometric continuity of order one as well as explicit sufficient conditions providing four free formparameters are known [Liu ‘901, [Vinacua & Brunet ‘891. Discussing this topic, DeRose [DeRose ‘901 also showed the general independence of the equations resulting from the necessary and sufficient conditions. The topic of geometric continuity is thoroughly considered in [Wassum ‘911. Geometric interpretations for construction methods ensuring geometric continuity are developed there. In this paper an unusual way to the conditions for geometric continuity is presented. Above all, the case of a joint between two patches with agreeing polynomial degrees along their common boundary curve will be considered. This case shows best the occuring phenomena and all other cases can be covered using analogous deductions. In [Kruppa ‘571, the theory of curves is considered as a special case of the theory of tangent surfaces. Analogously to this, in this paper the geometric continuity of rational surfaces will be interpreted as a special case of the geometric continuity of selected hypermanifolds in R”. Furthermore, a criterion for the independence of the equations derived from the necessary and sufficient conditions will be developed. The continuity conditions will be solved for one of the patches and thereby explicit conditions for geometric continuity will be found. These conditions generally provide five free form-parameters. Thus, they have one parameter more than the conditions known up to now.
1. Geometric continuity of rational surface patches Let a rational surface patch be given by its Cartesian coordinates neous coordinates x:
x resp. by its homoge-
x,(r, s)
&[O, 1]2-,R3:(r,
s) -+x(r,
S) = X0(: s) ,
I 1 xz(r9
s)
x3(r9
s)
(1-l) (x is also defined on a domain R, [O,112c R c R2, and is restricted to [0, 112.This guarantees the regularity of 5 at the boundary x(a[O, l]*).) The four functions Xi(U, c) (i = 0, 1, 2, 3) are polynomials in r and s with highest degree in r equal to n and in s equal to m. For example, w can be written as a rational BCzier surface patch: x( r, s) = E i-0
k B,Y( r)B,!‘( s)bij
where B,k( r) = (:)I’( 1 - t)k-‘,
t E
[O,
11
j=O
bij E R4\{0}.
(1.2)
B. Jiittler, P. Wassum / Geometric continuity
Usually, the control points bij are written in the form bij = Wij
.
with weights wij. Of course, this description excludes infinite points. The given surface patch x is to be continued along its edge r = 0 by a second rational surface patch. This second patch is described by its Cartesian coordinates y resp. by its homogeneous coordinates y: Y,(U, L’)
1 y : [O, 112+ IX3: (u, c) -+y(u,
c) =
Y,(U? u)
Yo(Ut L’)
1 Y,(U9U) 1
(Again, y is also defined on a domain U, [0, 112c II c R2, and is restricted to [O, ll’.) The four function< yi (i = 0, 1, 2, 3) are polynomials in u and u with highest degree in II equal to n and in u equal to m*. The second patch y should join with r along its edge u = 0. For example, y can be written as rational BCzier surface patch: y(u, u) = g
i=”
2 B,Y’*(u)B,F’(u)b~j,
b~jER4/{0).
(1.5)
j=l)
The joint along the common, boundary curve r = u = 0 should be geometric continuous in the following differential geometric sense of order of contact: Definition. The surface patches x and y have contact of order k along the boundary curves r = u = 0 (i.e., they have a CC&-joint) if-a regular parametric transformation
U+R:(u,
c) +(r(u,
c), s(u, u))
with r(0, c) =O
(1.6)
exists, where the partial derivatives of x and y of 0-, . . . , k-th order with respect to u and c are identical along the common boundary curve: (&j(Q.)'Y(U,
L?)I(o,c)=(D~,)i(DL.)jx(r(u' L'),s(u9 li'))l(O.O a
for i+jgk,
i, jEN
where D, := g i
The condition r(0, c) = 0 guarantees x(0, s). The Co-condition
the coincidence
1
.
of the boundary
curves ~(0, c) and
X(0, u) =y(O, c)
(1.8)
is also demanded in order to simplify the calculation. Therefore, tion (1.6) has to satisfy ~(0, u) = L’ in addition to
(1.7)
r(0, c) = 0.
the parametric
transforma(1.9)
(If the Co-condition (1.8) is not satisfied, in general x(0, ~1)and ~(0, L’) can only differ by a constant factor p. This follows from the degrees presumed for the polynomials Xi and Yj (i, j=O,..., 3). If p f 1 holds, let y’ := py and calculate with y’ instead of y.)
B. Jiittler. P. Wassum / Geometric continiciry
116
Resulting from the differentiation of rational expressions, the equations for the patch y obtained from the above definition prove out to be very complicated. In order to avoid this, conditions for geometric continuity of patches x and y being described by their homogeneous coordinates will be deduced in the next section. The following abbreviation for the vector polynomiais (i.e., the vector-valued polynomials) of the derivatives of the homogeneous coordinates x resp. y with respect to r resp. u (i.e., crosswise to the common boundary curve) is introduced in order to simplify the notation and with respect to the definition of the strip element (see Section 3.3):
‘x=‘x(L’) :=(D,)‘x(r, S)I(().(.) (i=O, ‘y=‘y(r)
:=(D,,)‘y(ll,
l,...
L’)I(“.l,) (i=O,
l,...
), ).
(1.10)
These vector polynomials take the form
‘x( L’) = (mm_!i)! ,t B;(C) A&,.,
and
‘Y(L’)
= Cmz’i,!
r==O
where AObj,k:= bj.k
,g Bkn((‘) , 0
A’&?.,
A”’ ‘bj,k := A’b,, ,,k - A’bjsk,
and
(1.11)
if x and y are written as rational BCzier surface patches. The abbreviation (1.10) allows a shorter description of the derivatives along the common boundary curve. For example: (1.12)
‘i( C) = Q( D,)‘x( r, s) I (“.(.).
2. Development
of conditions
for geometric
continuity
2.1. Geometric continuity of hypercones in R4
The three-dimensional real projective space can be considered as the set of all one-dimensional subspaces of [w4 (cf. [Schaal ‘761). Then the surface patches x and y correspond canonically to two hypercones X and Y in R4: X(r, s, t) :=t-x(r,
s) (t#O)
and
Y(U, L’, W) := w*y(u,
L’) (W # 0).
(2.1)
Here, hypercone always means the system of all lines connecting a fixed point (the point 0 = [0, 0, 0, OIT>with the points of a two-dimensional director manifold (x(r, s) resp. y(u, L,)). (Another kind of hypercones in Iw4 is the system of all 2-planes connecting a fixed line with the points of a directrix.) The images of all points of the hypercones X and Y under the projection
zo
,:lR4\((0}xR3)-,R3:
!I [) :;
z3
ZI
4;
z2
(2.2)
z3
lie on the surface patches x and y. This projection can be interpreted as intersection of X resp. Y with the hyperplane z,, = l-(and deletion of 0-th coordinates). The following theorem describes the correspondence between the regularity of a surface patch and the regularity of the hypercone linked to it: Theorem. The surface patch x is regular at its point x(rO, ~“1, if and only if the hypercone X is regular at all points X(r,, SO, t) with t # 0 and xO(rO, s,,) # 0 holds.
B. Jiittler, P. Wassum / Geometric
continuity
147
Proof. First, the case xJT”, sO) f 0 is considered.
Then it has to be two vectors x., and x., are linear dependent if and only if the three i.e. x.,, x., and x, are linear dependent (‘,z’ means the derivation linear dependence of xSrr x., and x follows immediately from the and x., as
shown that at (T,~, SJ the vectors X.,, X., and X.,, with respect to z.) The linear dependence of x.,
(2.3) hold. In order to proof this inversion, let a nontrivial linear combination of the null vector be given: (YX.,+ px,, + yx = 0 The 0-th coordinate
( LY,p, y E R, not all equal to 0).
(2.4)
of this equation reads:
ax,., + px”.x + yx, = 0.
(2.5)
The coefficients (Y, p, y are considered as unknown. Then, every solution of (2.5) and therefore the linear combination (2.4) can be written as linear combination of the basis (Y,= -x0, p, = 0, y, =xO., and a2 = 0, & = -x0, yz =xws of the space of solutions: 0=
-K,x~x.,
-
~~x~~x.,
+
(~,x~.,
+
~~x,~,,)x
(K,,
K~ E
R).
(2.6)
Therefore, the linear dependence of x9, and r., follows from (2.3). Obviously, the assertion holds for x,,(~s, s,J = 0. 0 The geometric continuity of hypercones is analogously defined with the geometric continuity of surfaces: Definition. The hypercones X and Y have contact of order k along the two-dimensional boundary manifolds r = u = 0 (i.e., they have a GC“-joint) if a regular parametric transformation
I/x (R\(O}) -+Rx(R\(O}):(u,
L',w)-+(u,L',
w), s(u, L',w), t(u, u, w))
with r(0, u, w) = 0
(2.7)
exists, where the partial derivatives of 0-, . . . , k-th order with respect to u, c and w are identical along the common boundary manifold:
( DJi( D,)‘(D,)‘Y(u,0, w) I (0.c.w) =
mA QJ’(QJ’Wr ( u, c, w),
for i+j+l
s(u,
0,
w), r(u,
L’,
w)) l(0.r.w)
i, j, IEN.
(2.8)
The condition r(0, c‘, w) = 0 guarantees the coincidence of the two-dimensional boundary manifolds X(0, U, w) and Y(0, S, t). The parametric transformation (2.6) has to satisfy ~(0, u, w) = u
and
~(0, c, w) = w
in addition to
r(0, u, w) = 0.
(2.9) This follows from the Co-condition (1.8). The connection between the geometric continuity of surfaces and the geometric continuity of the hypercones corresponding to them will be studied in the next section.
148
B. Jiittler. P. Wussum
/ Geometric
continuity
2.2. The correspondence between the geometric continuity of surfaces and the geometric continuity of hypercones
More appropriate than the definition given in Sections 1 and 2.1 is another definition of geometric continuity to study the correspondence between the geometric continuity of surfaces and the geometric continuity of hypercones: The surface patches 3 and y (the hypercones X and Y) have contact of order k along the boundary curves (along the iwo-dimensional boundary manifolds) r = u = 0 (i.e., they have a CC’-joint) if the following assertion holds for every curve a = a(q) on the surface 5 (A =,4(q) on the hypercone X): If ~(0) (A(O)) lies on the boundary curve (on the two-dimensional boundary manifold) r = 0 and a(q) (A(q)) is regular in a neighbourhood of q = 0, then a second curve _b=b(q) (B = B(q)) exists on the surface patch y (on the hypercone Y), which is regular in a neighbourhood of q = 0, and Definition.
lim -&n(q) 141-+c lqlk
(
lim 7
1
lr11’0 191
-_b(q)II
(2.10)
=o
II A(q) -B(q) II = 0
(2.11)
i
holds. The conditions of the above definition obviously hold, if a parametric transformation (1.6) resp. (2.7) exists. On the other hand, a parametric transformation can be constructed by considering selected curves on the surface x resp. on the hypercone X. The exact proof of the equivalence of both definitions for geometric continuity will not be shown here, cf. [Scheffers ‘231. The following theorem describes the correspondence between geometric continuity of surfaces and geometric continuity of the hypercones linked to them: Theorem. If the surface patches x and y resp. the hypercones X and Y are regular along their boundary curces resp. along their bound&y manifolds r = u = 0, then the surface patches x and y bar-e contact of order k (i.e., a GC"-joint ) there, if and only if the hypercones X and Y also hat; contact of order k there. Proof. (-) Let a GCk-joint of the surface patches x and y along the boundary curve r = II = 0 be given. Then a regular parametric transformation (1.6) exists, where the partial
derivatives of x and y of 0-,.. ., k-th order with respect to II and 1’ are identical along the boundary curve (1.7). -Obviously, and
Y’(rl’, L”, w’) := w’. (2.12)
are two other regular parametric representations of the hypercones parametric transformation (1.6) can be extended by r’( Ii’, L”, w’) = r(u’,
L*‘),
s’(u’,
L-‘, w’) =s(u’,
v’)
and
X and Y. The given t’(u’,
u’, w’) = w’
(2.13)
B. Jiittler, P. Wussum
/ Geometric
continuity
149
to a parametric transformation (2.7) where the partial derivatives of X’ and Y’ of 0-, . . . , k-th order with respect to u, c and w are identical along the common two-dimensional boundary manifold r’ = u’ = 0 (2.8). Therefore, X’ and Y’, i.e. also X and Y, have contact of order k there. (+> The second definition of geometric continuity is more appropriate to proof this inversion than the first one. Let a GCk-joint of the hypercones X and Y along the two-dimensional boundary manifold r = u = 0 be given. Then it has to be shown that the following assertion holds for every curve ~(4) on the surface x: If a(O) lies on the boundary curve r = 0 and a is regular in a neighbourhood of 4 = 0, then a second curve b(q) exists on the surface y, which is regular in a neighbourhood of 4 = 0, and (2.10) holds. Obviously, if a(q) is a curve on 5 which is regular in a neighbourhood of q = 0 and a(O) lies on the boundary curve r = 0, then the curve
44) :=
i-($)1
(2.14)
I
lies on the hypercone a curve
X and A(O) lies on the two-dimensional
boundary manifold r = 0. Then
B(q) =
(2.15)
exists on the hypercone Y, which is regular in a neighbourhood of 4 = 0, and (2.11) holds. This follows from the GCk-joint assumed for X and Y. The curve b*(q)
:= -
1 (2.16)
b,(q) _b(q)
is a curve on the surface y and regular in a neighbourhood
-L(ll) 141k -
of 4 = 0. On the other hand
--_b*(q)II
IM(q) -B(q)11 + I”‘?;&’
llls(rIll)
(2.17)
holds. Moreover, the equation be(q) = 1 +o( lqlk),
i.e.
for b,(q) results from (2.11). Therefore,
lim q-+0
b,(q) - 1 I&
(2.18)
=O
(2.10) holds for a(s) and b*(q).
q
The geometric continuity of surfaces is equivalent to the geometric continuity of the hypercones corresponding to them. The conditions for geometric continuity of hypercones resulting from (2.8) will be applied to surfaces in the next section.
150
B. Jiittler, P. Wassum /
2.3. Conditions for geometric
continuity
Geometric continuity
of first and second order
The Co-condition (1.8) required for the surfaces transmits to the hypercones: (2.19)
y I (O.l~,W) =x I (O.r.w)So, now it reads oy =0x,
(2.20)
using the abbreviations introduced Y*, I (o.~.,~)
=
(rdar
+
in (1.10). In addition to (2.191, the three equations
sd.,
+
tq,X.t)
y,,. I (OJ~.W) = ( r.t.X.r + s.,.X., + t-,X.,) y.,
I (O.l~.W) =
( rqwX.r
+
s-,X.,
+
I (o.~.~)~
(2.21)
I (o,~.,,,,) and
(2.22)
t.,X.,)
(2.23)
I (o,~..~~
and the condition of regularity det
I
t*u
r’lc rqp
Sh s,,.
t,,
rqw
s,,
t.w
I
(2.24)
#O
(O.C~.W)
have to be satisfied for geometric continuity of first order. The three equations (2.211, (2.22) and (2.23) result from (2.8). The two last equations (2.22) resp. (2.23) hold with r.r(O, c, w) = 0,
s.JO,
u, w) = 1
and
t.w(O, c, w) = 0
rvw(O, c, w) = 0,
s.,(O,
L’, w) = 0
and
t.,(O,
resp.
(2.25)
u, w) = 1
(2.26)
as they follow from derivation of (2.13) with respect to 1’ resp. W. Therefore, t-JO,
(2.27)
L’, w) # 0
still has to be satisfied as condition of regularity. The equation w.‘y(u)
=rJO,
v, w) *w*‘x(s(O,
u, w))
+s.,,(O,
0, w)*w*“i(s(O,
u, w))
+ t.,,(O, 0, w) *Ox(s(O, u, w))
(2.28)
results from (2.21) using the definition of the hypercones (2.1) and involving the abbreviations introduced in (1.10). Let t.,(O, u, w) := W *i.,(O, u, W).
(2.29)
‘y(u) = rqu(O, c, w) *‘x(u) +s.,(O, 0,w) *'i(u)+ r*.,(O, u,w) *Ox(u)
(2.30)
Therefore,
follows from dividing of (2.28) by w and as (2.9) holds. The left side of this equation does not depend on w. Then, r.,(O, c, w), sJ0, L’,w) and i.,(O, u, w> also do not depend on w as ‘x, Ox and Of are linear independent. (This linear independence results from the regularity of X.1 Therefore, the GC’-condition reads (2.31)
‘y = rYu’x + s.,Oi + ivuox,
where all quantities are functions of L’. Analogously, the GC2-condition ‘y =
r.uu’x + s.,,,Oi + i.,,Ox + ( r.u)2 2x + ( s.u)20i + 2r.“s.,,‘i
+ 2r.,,i.,‘x
+ 2s.,i.,‘f
(2.32)
B. Jiittlrr, P. Wassum
/ Geometric
151
continuity
follows. It can as well be written as *y = F.Jx
+ s1.,,,“x+ i.,,‘)x + (r.“)* *x + (s.,,)*ox + 2r,,s.,,‘x,
(2.33)
* Again, all quantities are functions of c’. where i.,,,, := T.~~,+ 2r.,,i.U and i.,,,, := s.,,,, + 2s.,t.,,. The coefficients T.&,and s.,, are already determined by the contact of first order. In order to simplify notations, the part already determined in (2.33) will be called := (r.J2 *x + (S.,,)%
K=K(c)
(2.34)
+ 2r.‘,S,“‘x.
A recurrent representation of the GCk-conditions is developed in [Wassum ‘9Ia, b]. The systems of linear equations resulting from the conditions for geometric continuity of first and second order will be studied in the next section.
3. Explicit conditions
for geometric
continuity
3. I. Explicit conditions for geometric continuity of first order The alternating product
(e. el e2 e3\ N=N(c):=‘xr\‘xA’x=det
- ‘x - Of \ -ox-,
(3.1)
(e,: i-th unit vector of R4) of the three vector polynomials ‘x, Of and Ox yields a normal vector N. This vector satisfies the three equations (N, ‘x) = (N, “x) = (N, Ox) = 0
(3.2)
and its length is equal to the three-dimensional volume of the parallelepiped spanned by ‘x, Ox and Ox (cf. [Reichardt ‘571). The normal vector N is described by a vector polynomial, where the highest degree in c is 3n - 2, although the highest degrees in ~3of the vector polynomials ‘x, aA and Ox are n, n - 1 and n respectively. This results from the relation of the two factors Ox and Ox = D,.‘x in the alternating product. The equation (N, ‘y) = 0
(3.3)
follows by inner multiplication of condition (2.30) for a CC’-joint between the surface patches x and y by N and it is equivalent to this condition. Let
(3.4) Then the following system of homogeneous linear equations for the vector polynomial ‘y (i.e., for the first row of unknown control points of the patch y) can be derived from (3.3):
c
i+j=k 0 %2j3g
n0,iYo.j+n,,iY,,j+n*,iY*,j+n3,iY3,j=O 2
(~=OV**.*~~-~)*
(35)
152
8. Jiittler,
F’. Wussum
/ Geomrtric
continuity
This system consists of 4n - 1 equations with 4n + 4 unknowns. Therefore it has at least five linear independent solutions. In general, the degree in I* of the normal vector N is 3n - 2 and N is irreducible (i.e., its coordinate functions have no common linear factors). The next theorem yields a criterion of the number of linear independent solutions of (3.5): Theorem. If the degree in c of N really is 3n - 2 (i.e., N cannot be described by a uector polynomial, where the degree in L’ is only (3n - 31, system (3.5) has more than fioe linear independent solutions if and only if N is reducible (i.e., its coordinate functions haoe common linear factors). Proof. (+I
Let six linear independent
solutions
’ P0.i liCli(L’)
=
;;,;
.I
(i=
C”+Ti(C)
1,...,6)
(3.6)
\ P3.i 1
of system (3.5) be given. (The degree in L’of the vector polynomials ri(l*) in this representation is assumed to be n - 1.) Then three linear independent, linear combinations qi = qj( u) = aill + pi11 + y,l, + ail, + EjlS + ljI, (i = 19 29 3;
aj,
Pjt
Yj,
sj,
Ej9
lj
E
R)
of the given solutions exist, where the coefficients homogeneous linear equations
(3.7) aj, pi, -yj, Sj, ej, lj satisfy the system of
as this system consists of three equations with six unknowns. The first, second and third coordinate function of the vector polynomials qj, obtained from (3.7), are polynomials, where the highest degree in u is only n - 1 as (3.8) holds. The vector polynomials qj have to satisfy the last equation of system (3.5). Without loss of generality, let n,,3n_3 z 0 be assumed. Therefore, also the 0-th coordinate functions of the vector polynomials qj are polynomials, where the highest degree in L’ is only n - 1. The alternating product ni=ti(v)
=q* Aq*Aq3
(3.9)
of the vector polynomials qj is described by a vector polynomial i, where the highest degree in u is 3n - 3 and this is always (for all c E R) linear dependent on N. Thus, N(u) = K(u)N(L’) is reducible. (c) Let N(c) = K(u~‘?(u) be reducible. Then equation (3.3) is equivalent to (ni, ‘y) = 0.
(3.10)
A system of homogeneous linear equations follows from above. It consists of at most 4n - 2 equations with 4n + 4 unknowns. Therefore it has at least six linear independent solutions. 0 Analogously it is easy to show that the following assertion holds: If the degree in u of the normal vector N is really 3n - 2, system (3.5) has exactly 5 + g linear independent solutions, where g is the degree in u of the largest common divisor of the coordinate functions of N.
153
B. Jiittler, P. Wassum / Geometric continuity
solutions ‘y,(c) (i = 1,. . . , 5) of equation (3.3) (and therefore
The five linear independent of system (3.5)) are known: ‘y,(c)
‘y,(c)
=‘x(L’),
'y,(u) = uOf(u),
ly,(Ly) =u(n%(L’)
and
'y,(u) ="_r(u)
=“i( L’),
(3.11)
-lPf(L’)).
(Note that the highest degree in L’of ‘y,(v) is only n.) If the highest degree in c of the normal vector N is really 3n - 2 and N is irreducible, each solution of system (3.5) can be written as a linear combination of the five solutions (3.11). Therefore the following condition for a Ccl-joint between the surface patches x and y holds: Theorem. If the highest degree in c’ of the normal cector N is really 3n - 2 and N is irreducible, the considered surface patches x and y hate contact of first order (i.e., a CCL-joint 1 along their common boundary curve, if and onlyif (2.20) holds and ‘y can be written as ‘y(u)
=r.,(
where
u)'x(c)
+s.,(c)Oi(u)
+i,U(u)ox(u),
r+,,(u) = P~,~, s-t,(u) = UI.0 + Ul,lL'+ q,2c2, i.,(u) = 7,,. + T*,,U
(PI.07 al,iT
71,j
E Iw)
(3.12)
and 71,1 +
nu1.2
(3.13)
= 0
and the condition of regularity P,.~ # 0 (cf. (2.27)) hold too.
The equations for the first row of unknown control points derived from the above theorem will be developed in Section 4. 3.2. Conditions for geometric continuity of second order The equation (3.14) (N, ‘Y) = (N, K) follows from condition (2.33) for a GC2-joint between the surface patches x and y by inner multiplication by the normal vector N = N(u) and it is equivalent to this condition-(K = K(c) was introduced in (2.34X) Analogously to the case of the CCL-joint, a system of inhomogeneous linear equations for the vector polynomial 2y (i.e., for the second row of the unknown control points of the patch y) results from (3.14). The corresponding homogeneous system is again provided by system (3.5). In general, the solutions of (3.5) are well known. Now, a special solution of (3.14) has to be found. The vector polynomial K = K(U) does not represent such a solution because its degree in u is equal to n + 2. Let ;h”( u) =
P2.0 + PZ.IL’,
= a2.0 + @2,,U + u2.2u2
fdu) Lj)
i.,,(
= T2,0 + T2,,U
+
q,3u3,
+ T2,2u2
(PZ.i*
u2.jy
‘2.k
E ‘).
(3.15)
Then, the following conditions result by comparing the coefficients of v”+’ and u”+~ on the left- and right-hand side of equation (2.33): P2.1 = n”2.3 (n
-
2nP,,ou,.29
+ ?2.2 =
-n(n
-
1)u2,3
+ ?2,2 =
nu2.2
+ 72,l
=
-2n(n
-
w1.2)2,
-(n -
-
l)(n
w7,,,~,,2.
-
w,z)29
(3.16)
B. Jiirtler,
154
P. Wassum/ Geometric
continuity
The coefficients pz.,, u2.3 and T~_~are determined by these conditions. One of the two coefficients u2,2 and TV,, as well as the coefficients pz.O, gzVO,c2,,, ~~,rr can be chosen arbitrarily. These five free form-parameters correspond to the five solutions (3.11) of the homogeneous system. Therefore, the following theorem holds: Theorem. If the highest degree in c of the normal rector N is really 3n - 2 and N is irreducible, then the surface patches x and y hate contact of second order (i.e., a GC2-joint) along their common boundary curce r = u = 0, if and only if they hare contact of first order there and ‘y can be written as 2y( c) = F#,“( Lq’ x ( L3) + s1.,,,( c)“.+( c) + i.,J
c)“x( L’) + (r.,(c))’
+ (s,,( Ls))‘Of( c) + 2r.,( c)s.“( c)‘i(
2x(c) (3.17)
/I),
where the coefficients satisfy (3.12), (3.13), (3.15) and (3.16).
The conditions (2.201, (3.12) and (3.17) for geometric introducing the strip element in the next section.
continuity will be summarized
by
3.3. The strip element
The conditions (2.20), (3.12) and (3.17) for geometric continuity of 0-th, first and second order can be summarized as follows: 1
OY ly
=
i., + s.,
L, +f.,,D,. Ill2Y where r.” = P,,~,
D, +stDL?
q,2L’2,
i.,
TI.IL’,
r*,o
+
(9
0
r ‘u
0
?.,, + 2r.,s.,D,
s., = (T,,0 + @,.IL’+ =
0
rt II
“‘X
-
(3.18)
‘x 2X
1
(b)
A = P2.0 A Shu = a20, + U2,IC + u2.2u2 r’ltu
i ‘UU = 72.0
+ 729 (9
(a) (b) (c) and the
+ P2.1~3 + u2.,v3 7 -I- 72,2v2.
(6)
(6)
: All parameters can be chosen arbitrarily. : One of these parameters can be chosen arbitrarily. : All parameters are determined.
conditions (3.13) and (3.16) hold.
The hypervectors (Ox, ‘x, 2x,. . . IT and (‘y, ‘y, ‘y,. . . IT can be called the strip elements of the surface patches x and y along the boundary curves r = u = 0, analogously to the curve elements of a curve in a point. Then, equation (3.18) describes a transformation of these strip elements. Conditions for geometric continuity of higher than second order can be developed by continuing the studies of Sections 2 and 3.
155
B. Jiittler, P. Wassum / Geometric continuity
4. Construction
of a Ccl-joint
between two BCzier surface patches
The conditions for the control points of the patch y resulting from (3.12) will be developed in this chapter. If x is written as BCzier surface patch, then the five sohttions (3.11) of (3.5) read
‘yr( ti)
=‘X(
L’) =
5
*n(b,,j
B,!‘( L’)
- b,,j),
j=O ‘Y~(G)=~~(L’)=
i B,!‘(C) j-0
.(("-j)(bo,j+,-bO,j)+i(bO.j-bo,j-,)),
CBi”(~).j(bo,j-bo,j-,),
‘Y~(L’)=u~~(u)=
j=O
‘Yq(U) =OX( L’) = t By( L’) * bo,j, j=O
kBy(c)*jbo,j_,.
‘Y~(L’)=L’(~‘x(u)-u’~(u))=
(4.1)
j-0
The O-th and the first derivative with respect to u of y are (cf. (1.11)) I3;( ~1)* bcj
“y( L!) = i
and
j=O 'Y(~) =
~
Bi"(U)
.~*(b~j-b~j).
(4.2)
j=O
r= u = 0
The identity of the control points of x and y at the boundary bcj = bo,j
(j = 0,. . . , n)
(4.3) directly follows from the Co-condition (2.20). (If (1.8) does not hold, then (4.3) and (4.5) are the equations for the control points of y’. The control points of y follow from division by CL.) From (3.12) resp. from ry=
&JYj
(fc;ER)
(4.4)
i-0
the representation bcj=bo,j
+o(b,,j-b0.j)
(j=O,...,
n; a, P,
+Pbo,j+ Y,
6,
YLbo,j+‘Lbo.j-l n n
+’ =bo.j+ n
1
(4.5)
E E q.
for the first row of unknown control points of y is obtained with some calculations. A geometric interpretation of (4.5) can be given in R 4. The equation (4.5) takes the inhomogeneous form n-j j j W;“,j= WO,j+ a( -Wo,j+ 1 and n n n Wl,j
_b,,j= -!-
-
wO,j)
+
pwO,j
+
YswO,i
+
SvwO,j_*
+
E
wO,jbO,j + "(wl.j_bl,j- wO,j_bO,j)+ PwO,j-bO.j+ YLwO.j_bO,j
n
w;".j
+S'WO,j-lbO,j-1 n (j=O,...,
n-j + E pwO,j+ n
n; a, P, Y, 6, E E W with control points denoted likewise (1.3).
dO,j+
1
1
(4.6)
156
B. Jiittler, P. Wassum /
Geometric
continuity
In (4.51, only CY< 0 make sense. Patch y is not regular along the boundary II = 0 for (Y= 0 and for (Y> 0 patch x will be continued by patch y in the undesired direction.
Conclusion
In this paper, explicit necessary and sufficient conditions for geometric continuity between two adjacent rational surface patches with the same degree along their common boundary curve have been developed. These conditions yield a description of the set of all patches joining geometric continuously with a given patch. Out of this set, a particular patch can be selected with respect to background-information (e.g., to conditions provided by requirements from an approximation or interpolation problem). The conditions obtained in this paper can be generalized to the case of the geometric continuous joint between two patches with different degrees along their common boundary curve. Then the degrees of the coefficients in (3.12) and in (3.17) have to be adapted. Generally, the system of linear equations (3.5) resulting from the necessary and sufficient conditions has exactly 5 + 3h linear independent solutions, where the highest degree in c of ‘y (resp. 2y> is n + h. These solutions read ci ‘x(u) (i = 0,. . . , h), ~‘j ‘x(c) (j = 0,. . . , h), tek ‘i(u>(k=O ,..., h+l)anduhf’ (n’x(c> - u”i(.(o)). Of course, in this case the homogeneous coordinates of the common boundary curve can differ by linear factors. The strip elements of triangular rational surface patches are different from the strip elements of the tensor-product rational surface patches: the degrees of the vector polynomials ‘x resp. jy (for a triangular rational surface patch x resp. y) are n -i resp. n -i. The methods developed in this paper can be adapted to this strip elements. The solutions resulting from a reducible normal vector are studied in [Wassum ‘91aI.
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