Curvature computations for degenerate surface patches
Computer
Aided
Geometric
Design
241
9 (1902) 211-270
North-Holland
COMAID
277
Curvature computations surface patches
for degenerate
Franz-Erich Wolter and S6amus T. Tuohy IMIT. Drparrment of Ocean Engineering, Design Laboratory, Cambridge. Received
August
MA 02139-9910,
USA
1991
Abstracl Wolfer,
F.-E.
Geometric This
and
Design
paper
S.T.
Tuohy.
9 (1997)
presents
a method
representation
defined
representation
is degenerate
curvatures height function computed
plane
pairwise
by using
using derivatives Under function unique
plane
order
partial
partial
tangent
directions.
derivatives
derivatives
curvatures
of the surface
Computer
Aided
curvatures
of the approximated
a locally point
of the corresponding
differential
surface
geometry;
C’-smooth
height only
a quadratic
approximating
in the contact
for
point
of the height
on S which
end
S at 4 are then we
4 can be computed a definition
degenerate
function
well defined
4. One
with
a local
h is defined
II at 4. For this purpose
4. We also present
Q, nor a locally surface
of
at a point
appropriate
of the point
of the degenerate
degenerate
of a surface from
derivatives curves
defined
employs
function
entities
the surface
h at 4. We also show how the
function
assumptions
a degenerate
and well
method
partial
of the function
order
plane
surfaces
curvature
from
4 in S where
the height
order
The
emanating
weaker
in a neighborhood
approximation
surfaces:
curves
at the degenerate
4 where
of the height
up to nth
under
S obtained
at a point
of any three
of any order
one does not require
plane
at 4. The surface
Keywords. Degenerate
patches,
Our computation
the second
derivatives
derivatives
patch
curvatures
of the point
order
n of n + 1 surface
surface
order
surface
the point sef S has a tangent
that
4. In our method
independent
assumptions
representation
of a surface
are to be computed.
by using second
linearly
up to order
local second
tangent
of S at the point
over the tangent
the curvatures
has to assume
I) how the partial
those weaker
the surface
degenerate
To compute
of S in a neighborhood
to compute
of generalized
for
curvatures
domain.
the curvatures
the second
is extended
show (in Theorem concept
4 where
/I at 4 are computed
up in (I with method
one
representation
over the tangent
computations
to compute
over a rectangular
at the point
function
Curvature
111-270.
representation
(single-valued)
needs
defined
surface
of
height
the existence
function
quadratic
of a
surfaces.
of a
over the
then define
Q.
curvature.
1. Introduction
In geometric modeling with rational B-spline surface patches defined on a rectangular domain we frequently encounter situations where the related rectangular surface patch collapses to a triangular patch. In such a case, one boundary edge may collapse into a boundary vertex of the patch. Therefore, for the given parametric surface representation the Jacobian matrix corresponding to that vertex point will generally not have maximal rank. Nevertheless, it is quite possible that the point set defined by the collapsed triangular patch can be represented by another regular surface map defined over a triangular domain and may therefore have well defined surface curvatures everywhere. Clearly, the original parametrizaCorrespondme 02139-9910.
IO: F.-E.
Walter,
h_lIT,
Department
of Ocean
Engineering,
Design
USA.
0167~8396/92/SO5.00
D 1992 - El sevier
Science
Publishers
B.V.
All
rights
resewed
Laboratory,
Cambridge,
MA
F.-E. Walter, S. 7. T1coh.v/ Curvature for degenerate sutjaces
242
tion over the rectangular patch cannot be used in the classical way to compute the Gauss curvature or the principal curvatures of the collapsed patch at points which are too close to the boundary edge collapsed into the vertex. The classical curvature computation is not possible here because such a curvature computation requires a division by the determinant of the matrix defined by the first fundamental form related to the degenerate representation. As this determinant is zero, the classical method is not applicable. We want to present a method in this paper which allows fast and accurate curvature computations for the above described surface patch. Our paper will focus on the case where a surface patch is created by collapsing precisely one boundary edge in exactly one corner vertex. We assume that the original surface representation is regular everywhere except at that boundary edge which is collapsed in that vertex. Although we focus on this special case, the main ideas presented in this paper can also be applied in other situations. The cases discussed in the example section are taken from the class of rational B-spline surfaces as those are the most common ones in geometric modeling. However, the methods presented here are valid also for more general classes of surfaces as those methods require only that continuous second order partial derivatives exist and can be computed. We make the assumptions that the patch as a point set has a well defined surface normal n and curr’atures at the degenerate point p. Therefore, we assume that the patch has locally a nondegenerate C2-smooth representation cia the height function z = g(x, y) ocer an area ’ in the tangent plane at the point p where the z-axis points into the direction of the surface normal n at p. The key idea of our approach is to derice a nonsingular system of linear equations to compute the partial dericatices of the ftmction g(x, y). In this context we deri6.e Theorem 1 which shows how all partial deril’atices of g(x, y> up to order n can be computed using dericatices up to order n (at p) of n + 1 curL’es where those curces are assumed to hate pairwise linearly independent tangents at p. The curr*ature r*alues at p can then be expressed in terms of dericatices of g(x, y) up to second order. This solL*esthe cun’ature computation problem. We also present a concept to define generalized surface curvatures under weaker assumptions appropriate for degenerate surfaces. Under those weaker assumptions one does not require a local C2-smooth height function representation of the surface ocer the tangent plane at the degenerate point p. One does not eL.en require a locally well defined (single-calued) height function representation of the surface in a neighborhood of the point p. One only needs the existence of a unique local second order approximation of the degenerate surface with a quadratic frrnction defined ocer the tangent plane at p (cf. the Key Remark in Section 3.2). The surface curvatures of the corresponding approximating quadratic surface define then the cun’atures of the approximated degenerate surface in the contact point p. The main theoretical contributions of this paper are probably this generalized surface curvature concept and Theorem 1 explaining a general method to determine the higher order partial derivatives of g(x, y) by employing derivatives of curves. Those higher order derivatives can e.g. be used to determine a higher order Taylor development at p for the Christoffel symbols at p to approximate the governing differential equations of geodesics in a neighborhood of p. Those higher order derivatives are also useful to compute tensors relevant in structural shell properties which use higher than second order derivatives. The paper is structured as follows: In Section 2 we discuss related work of other authors. Section 3 is the main part of this article and contains a derivation and discussion of the method. Subsection 3.2 contains the Key Remark explaining the subtleties concerning the well definedness of the local height function representation of the degenerate surface. This Key
’
We are relaxing
the
requirement“existenceof havinga
later in the Key Remark
in Section 3.2.
C’-smooth
height function
representation
for the patch”
Remark explains also the generalized surface curvature concept which requires only the existence of the approximating quadratic surface. Section 3 demonstrates the application of the method in several examples. Section 5 contains conclusions and discusses further applications. The Appendix contains the proof of a technical assertion and the derivation of some differential geometric formulas which we include to make the paper self-contained.
2. Survey of previous related work In free-form surface geometric modeling one works usually with spline surface patches described by a function f(~i, r) defined over a rectangular domain in the u, Lv-parameter space. In general, the Jacobian matrix of the surface representation f(u, r*>is assumed to be of rank 2. Under this assumption, the classical interrogation techniques (e.g. curvature computations) for the geometry of the surface patch are well understood, see [Faux & Pratt ‘791, [Farin ‘SS]. The situation where a rectangular surface patch degenerates to a triangular patch because one boundary edge degenerates (collapses) into a point is different from the preceding case and is not as well understood. Nevertheless, this degenerate case is considered to be of interest for design problems e .g. because it allows a possibility to create triangular patches defined by a parametrization over a rectangular domain. Triangular patches are found to be advantageous for various applications, see [Farin ‘881. Therefore, the situation where a rectangular surface patch degenerates to a triangular patch has been addressed in the well known books [Faux & Pratt ‘79, pp. 235-2383 and [Farin ‘88, pp. 211-2141. In the spline surface case this degenerate situation arises in case several control points are chosen to coincide see e.g. [F arin ‘881. Conditions for the existence of a surface normal on the triangular point set ’ given by the degenerate spline patch have been discussed in the aforementioned references of Faux & Pratt and Farin. In the considered triangular B-spline patch case, those conditions imply that the (degenerate) vertex (control point) and the adjacent control points are coplanar. Faux and Pratt discuss a method to compute a surface normal in the degenerate case based on a Taylor development for the partial derivative a,,f(u, c). They give a necessary and sufficient condition for the existence of the surface normal using the requirement that the vectors a,.f(u, O), a,,a,.jYu, 0) are all in the plane tangent to the patch point set at f(0, 0). For Bezier patches this condition implies that the singular vertex (control point) and the four adjacent control points must be coplanar. Although the literature discusses methods to compute the surface normal of the degenerate patch, it does not address the harder problem of computing curvature (or higher order differential geometric) entities at the degenerate vertex of the patch. This paper intends to fill this gap. It has turned out that this paper could employ ideas which were introduced in [Pegna & Wolter ‘89, ‘921 with different intentions and in a different context, namely in the area of blend surfaces. The Three-Tangent theorem introduced in [Pegna & Wolter ‘89, ‘921 to give a criterion guaranteeing second order (curvature continuous) contact of two surfaces at a contact point is generalized in this paper (cf. Corollary 1, i.e., the ‘n + 1 Tangents Theorem’ in Section 3.5) and used as a tool to determine the curvature entities of the degenerate patch. The ‘Three Tangent Theorem’ of [Pegna & Wolter ‘89, ‘921 says: “If two regular C2-smooth surface patches S, and S, share the tangent plane at a common point p, then S, and Sz share all curvature entities 3 at the point p if and only if the normal curvatures of S, and S, agree at p for three pairwise linearly independent tangent directions.” ’ This means 3 This
includes
the existence
of the surface
e.g. Gauss curvature,
normal
at the degenerate
mean curvature
and principal
vertex. curvature
directions.
F.-E. Wolfer. S. T Tuohy / Curcarrrre for degenerate surfaces
211
3. The problem and the solution
method
3.1. Formulation of the problem
Let f(u, P):[O, 11X [O, 11 + W’ be a surface patch. We assume that f(u, L’) is degenerate at a corner point (~(a, r,) = (0, 0) corresponding to f(0, 0) on the surface. This means we assume that a,,f(O, O), aCf(O, 0) are linearly dependent. We discuss here a particular case where the surface collapses to a triangular patch at the point f(0, 0). Let us assume that f(rr, 0) =f(O, 0) for ail u E [0, 11. Let us assume further that all vectors in the set V= R,J(u, 0) Iu E LO,111are contained in the plane spanned by a,,f(l, O), a,,f(O, 0) and that V contains at least three pairwise linearly independent vectors, see Fig. 1. We also require that f(u, c) is not degenerate for all L’ with 0 < L’< E for some positive number E and we require that a,.f(lc, u) # 0 for all u, c E [O, 11. In this situation the curvatures of the surface patch at the point f(0, 0) cannot be computed using the first fundamental tensor gl, associated with the surface representation f(u, V>where gij = a,j(o,
0) - a,f(o,
0)
with a, = a,,, a, = a,.
because this tensor has det(gij) = 0. Therefore, the matrix (gij> is not invertible which makes standard operations to compute Gauss and mean curvature ill-defined because those operations require the division by the determinant det(gij). When we say we mean to compute curvatures of the surface patch at f(0, 0) =p then we think of computing geometric entities associated with the point set S = (f(u, c>l(u, c> E [O, 11X [O, 111in a neighborhood of f(0, 0). To be more specific we rephrase the problem as follows. 3.2. Reformulation of the problem Jntroduce a new coordinate system with center (0, 0, 0) =f(O, 0). Note that we had required that the tangent plane S, of S at p =f(O, 0) is well defined. This means that in our particular situation the tangent vectors of all surface curves at the points of the edge (f(u, 0)l u E [0, 11) are contained in the tangent plane S, spanned by a,.f(l, O), a,,f(O, 0). We choose the z-axis of the coordinate system pointing into a direction normal to the tangent plane S,, thus this direction may be given by the cross product of the two vectors a,.f(O, Oh a&l, 0). For the x-direction of the new coordinate system we may take the vector a,.f(O, 0). The y-direction is chosen to be normal to the x- and to the z-direction. Let (X(U, L’), Y(L1, P), Z(U, C)) =f(n,
L’).
(1)
I t f(u,v)
Fig. 1. Collapsing
patch.
F. -15. Wolfer, S. T Tuohy /
C~crc~~rctre
for degenerate surfaces
215
On a set
O<E, (x(n17 L’,), Y(U,, u,)) = (-&9 then z(ur, cl) =z(u2,
L’Z)! Y(U,, 4))
L.*).
(5)
Note that implication (5) must hold if the function value z =g(x, y) is required to be unique and depends only on the coordinates (x, y>. Obviously implication (5) may in general not be valid. Moreover, it is quite possible that there exists no height representation of the patch over any chosen (x, y)-plane. In our examples given in the example section, implication (5) holds true because one can verify in all those examples directly that the function ccl(u, c) = (X(& L’), Y(Ll, L’))
(6)
is injective on the domain ’ D={(u,c)Io~u~1,o = (0, 0) although more precisely one should t-other use the clumsy
F.-E. Wolfer, S. T Atohy / Curc~ature for degenerate surfaces
117
terminology of ‘second order Taylor coefficients in the approximation given in (12)‘. The existence of those coefficients (although sufficient to define surface cun.atures) does not er’en require C’-smoothness of the function g(x, y) in any neighborhood of (x, y) = (0, 0) nor does it require the well definedness of the height function representation of the surface patch. All geometrical examples in our example section have well defined height function
representations but the theoretical concepts presented here are applicable under weaker assumptions. It is easy to construct examples which do not have a well defined height function. The example section contains examples (107), (108), which have representations with well defined height functions g(x, y), g(x) respectively. Both height functions are C”-smooth on Q!J( D,), 0 < E < 1 or on lx IO <x < 1) respectively. However, these height functions do not have a second order Taylor approximation in a neighborhood of the points (x, y) = (0, O), x = 0 respectively and the conditions (12), (11) are not valid for both examples. 3.3. Solution method for the problem The key concept of our approach consists now in computing the above second order derivatives by using the curvature vectors (or second derivatives) of isoparameter curces o,(c)
=f(u,,
r,),
L’Q E, II, a number in [0, 11.
(14)
We shall need the curvature vector k, (or the second derivative) of the curve a,(c) at L’= 0. Let n be the unit normal vector of the patch at the point (0, 0, g(0, 0)). Thus n points into the z-direction of the coordinate system. Then, with wn = acf(Un, 0) the expression nc(w,) =n.k,,
(15)
yields the normal curcature of the surface in the direction w,, at the point p = (0, 0, g(0, 0)). By assumption w, is a vector in the x, y-plane. Let W,,= w,J I w,, ( and let X,, y,, be the (x, y) coordinates of Zn. Then we have n*k,=nc(F,)
=g,,(0,0)f~+2gx,(0,0)~~,~,+g,,(0,
O)y,‘.
(16)
This follows from differential geometry, see Assertion 2 and Assertion 3 in the Appendix The right side of equation (16) is the normal curvature for the direction W,,of the collapsed patch and the left side is derived from the curvature vector of the surface curve (Y,(U). Therefore, both sides of equation (16) are geometric entities defined independently of the parametrization of the involved geometric object. The right side of equation (16) contains also the unknowns a,a,g(O, 0) which must be computed. It is however possible to determine the right side of (16) also simply by using the second derivative of a,,(~‘), namely by using the equation n
- a”(O)
= g,,(O,
0)x,’
+ 2&,(0,
O)X,Y,
+ g,,(O,
O)Yi
(17)
now with (xn, y,,) being the (x, y) coordinates of the vector wn. The proof of equation (17) is contained in the proof of Assertion 3 in the Appendix. In equation (17) the left side is not a parametrization independent geometric entity and as (xn, y,)’ = aJ(cl,, 0) in general does not have unit length, the right side of (17) in general does not describe the normal curvature related to the direction given by (xn, y,)‘. Therefore, from a differential geometric (parametrization invariant) view point, relation (17) may be considered to be less satisfying than the relation in equation (16). Let now w, =(x,, y,), w2 = (x,, yZ), wj = (x,, y3) (or W, = (x,, Yl), W,(x,, Yz >, W3 = (x3, y,)) be three vectors (or three unit vectors respectively) in the X, y-plane such that every pair of those three vectors is linearly independent. Then (16) or (17) respectively yield the following system of equations which we give in matrix form: Lu=b
(18)
F.-E. Walter, S. T Tuohy i Cure amre for degenerate surfaces
218
with
b = (nc(iG,),
nc(Z,),
nc(iC,))’
or
b = (n *(Y;‘(O), n *cy;I(O), n *cr;(O))’
respectively
and the matrix L defined by L=(Lij),
l 0, then the point
250
F.-E. U’olter. S. T Ttcohy / Ctrrr afurr
fur degenerate surfims
Let I E (1, 21. If (36) does not hoId and if t g,,r - ‘Cl 1+ 1g.rv 1 2 1&TX? 1+ 1s!Tyy- K/ I
(37)
then the maximal principal cuu’ature direction e, or the minimal principal curvature direction ez are given by
0)’ v;kJ2+ (gxx- KA2 ( -g.c, g.,.r7
et=(*)
Kt 7
(38)
respectively. If none of (36), (37) holds, then (gYY-K/t
e,=(k)
, d( g,, - d2
-gay, 0)’ (39) + (&J2
.
Note that a principal curvature direction is defined without specifying the orientation of the direction; this explains why the sign is not specified in (38) or (39), i.e., both signs + or are allowed. 1. It is well known that the maximal and minimal principal curvatures K,, are two eigenvalues of the matrix (wij) described in (30) and (32). Thus K,, K* are the two roots of the characteristic polynomial P(K) of the matrix (w,~) where Proof of Assertion K*
P(K)
=
det( wlj -
with ai, =
1 0
~6~~)
=
0
if i =j, otherwise.
(40) -
Therefore, with (wii> being a (2, 2)-matrix K(, ~~ are solution of the quadratic equation = 0. It is easily seen that those solutions K,, K2 have the form given in the right side of equation (35). This proves the claim of equation (35). The Gauss curvature is defined as the product of the two principal CUrWtUreS K,, K*. Therefore by using (35), a straightforward computation shows (33). The mean curvature is defined as the mean value of the principal curvatures therefore using again (35) then equation (34) is an obvious consequence. We just showed (33) and (34) by employing (35). However, it is not necessary to employ (35) for this purpose. One can also argue more elegantly from an invariant linear algebra viewpoint. Namely, one can use that:
P(K)
Determinant
and trace of any quadratic matrix agree with the
product and sum (respectively)
of the eigenvalues of that matrix.
(41)
The right side of equation (33) is obviously the determinant of matrix (wij>. Therefore, by (41) this right side is equal to the product of the eigenvalues of matrix (wij). This proves (33) as well. The right side of equation (34) is the mean value of the trace elements of matrix (w,,). Therefore, by (41) the right side of equation (34) equals the mean value of eigenvalues of matrix (wij). This proves the claim of equation (34). An umbilical point is defined by the condition that at that point the maximal and the minimal principal curvatures agree. This requirement is obviously equivalent with the condition that the radicand in equation (35) must be zero. This radicand being zero is equivalent to the conditions given in (36). This proves that in our situation the criterion given in (36) characterizes umbilical points. We show now our claims given in (37), (38) and (39) concerning the description of the principal curvature directions. Recall that the principal curvature directions correspond to the eigenvectors of the linear map represented by the matrix (wij> with respect to the two basis
F.-E. Wolrer. S. T Trcoh?: / Curt atwe for degenerate surfaces
‘51 -_
vectors a,h(O, O), C$h(O,0) being the partial derivatives of the parametric representation h(x, y) of the surface patch. Therefore. using the coordinates A’,, A! of an eigenvector of the matrix (w,~) as coefficients in a linear combination with those basis (tangent) vectors, i.e., defining pj = A’,@h(O, 0)) + A+,,h(O,
O)),
(42)
yields a principal curvature direction p ‘. This means that, in order to determine principal directions one has to compute first eigenvectors of the matrix CM.;,)with respect to the natural basis (1, O)‘, (0, 1)’ of R2. Thus we must compute vectors (A’,, A’,), 1 E (1, 21, with ( wij - /Qij)( A’,, A’*)’= 0.
(43)
As we exclude (36), the considered point is not an umbilical point and the solutions of (43) build a one-dimensional vector space. Therefore either the first or the second row of matrix A’ = ( wi, - A$,,)
(44)
contains nonzero elements. To determine
the solution vector
d, = ( A’,, 4)
(45)
of (43) we employ that row vector of A’ which has the larger norm, where this norm is defined by the sum of the absolute values of the vector coordinates. This choice of the preferred row vector is reflected by the condition stated in inequality (37) and by the negation of (37) stated in conjunction with (39). The solution vector of (43) is chosen to be orthogonal on the preferred row vector, i.e.. we determine the solution vector d, by rotating the preferred row vector by 90 degrees. This solution vector d, must also be orthogonal on the other row vector of the matrix A’ because this (non-preferred) row vector is linearly dependent on the preferred row vector. This linear dependence holds because the matrix A’ must have rank less than 2 as its determinant det(A’) = 0 by (40). Thus we find a solution vector d, of (43). Inserting the coordinates of d, as coefficients in the linear combination defined in (42) where now %h(O, 0) = (1, 0, O)‘,
$h(O, 0) = (0, 1, O)‘,
gives a principal curvature direction pI which after normalization yields the vector e described in (38) and (39) respectively. Those considerations show the validity of the claims concerning the description of the principal curvature directions stated in (37), (38) and (39). This completes the proof of Assertion 1. q 3.5. Computation of higher order dericatires at the degenerate certex Key to computation of the curvature entities of the degenerate surface patch was the computation of the second order partial derivatives of the non-degenerate representation h(x, y) = (x, y, g(x, y>) at MO, 0) of the surface patch. It is possible to extend the method described in Section 3.3 to compute higher order derivatives of the representation h(x, y) at the vertex MO, 0). We explain this extension by computing third order partial derivatives aikjg(O, 0) at the vertex point. It will then be obvious how this method generalizes to compute further higher order derivatives. Let (~(5) =h(a(s)) be a curve in the degenerate
patch. We use the notation
u(s) = (LI,(J), 111(s)) = (x(s).
y(s)),
(46)
F.-E. Walter, S. T. Tuohy / Curawe
22
for degenerate surfaces
i.e., a(s) is the projection of the curve (Y(S) onto the X, y-plane or synonymously called II,, LL~-plane.We shall use the following notation hi, =
h, = ‘lrih( ‘1, ‘2) 7 hikj
= aLL,au,aLL,h(u,,
LL2),
aLr,aLLkh(LL,, i,
k, j E
LL2),
(1, 2}.
(47)
Successive application of the chain rule yields the second derivatives expression (Y”(S) = i
&u:‘(s) +
i=l
i
h,,u;(s)LL;(s)
(48)
i,k = 1
derived in the Appendix in (128). To compute the third order derivatives we differentiate a”(s) with respect to s and get (49)
Thus
i.k =‘I
This yields at the vertex point MO, 0)
k
+
i,k=
hik( uI’L4; +
u;LL;).
(51)
1
Note that (51) follows from (49) because h&O, 0) = 0 by (4). The second order partial derivatives hik(O, 0) have already been determined in (20) and the derivatives u:)(s) can be computed directly by evaluating u:(s)
=(Y”(s) *ei
with e, = (1, 0, 0), e2= (0, 1, 0).
Therefore the right side of (51) contains only known entities. The left side of (51) can be viewed as a homogeneous polynomial in the variables u;, u; if we assume that the third order partial derivatives hlkj are permutation invariant with respect to the indices i, k, j. This is justified by Schwartz theorem if the function is C3-smooth, see [Nikolsky ‘851. Using this index permutation invariance, the left side of (51) can be represented in a simplified way by the left side of the subsequent equation h,ll(u;)3
+ 3hltZ(u;>2r~; + 3h221u;(u;)2+h222(u;)3=4(U;,
U;)
where d(u,, u2) abbreviates the right side of (51). As in the case of the second derivatives we want to use a matrix equation Mx=c to determine
(52)
the unknown third order derivatives contained in the vector
I= (hlrr7 3h,r2, 3h22r7 h222).
(53)
F.-E. Woltrr. S. 7: Tu0h.v / Curvaturefor degenerate surfaces
The matrix icl in equation (52) is built by 4 rows r,, vector ,u = (,u;. , ui)’ and the rows are defined by r, :=
(
(pi)‘,
( ju;)2ju>
, ,u;(,Llz)2,
1 <j G 3, where
753
each row is related
( jL()i).
to a
(9
We can solve the matrix equation (52) if the matrix Lemma 1 contains a necessary and sufficient condition matrix M. We shall use the notation
M is non-singular. The subsequent for the non-singularity of the above
det( ju’ ju’)
(55)
where the expression (55) denotes the determinant defined by the two vectors 1~‘, ju’.
of the matrix
containing
the two columns
Lemma 1. Let rhe ser A = Iju’ = Cju;, j ~1;)’ j 0 < j < n} contain II + 1 rectors from R8?. Let M be a quadratic (n, n)-mafrir defined by n rows rj where rj=
i
(ju;)“,
(ju;)“-‘ju;,
(ju~)n-2(ju~)2,...,(jui)‘*),
O<jgn.
(56)
Therl n
det( M) =
n
det( ;u’ jll’) =
( ;u; ju; -j~l;
,u:).
(57)
O
Aided
Geometric
Design
241
9 (1902) 211-270
North-Holland
COMAID
277
Curvature computations surface patches
for degenerate
Franz-Erich Wolter and S6amus T. Tuohy IMIT. Drparrment of Ocean Engineering, Design Laboratory, Cambridge. Received
August
MA 02139-9910,
USA
1991
Abstracl Wolfer,
F.-E.
Geometric This
and
Design
paper
S.T.
Tuohy.
9 (1997)
presents
a method
representation
defined
representation
is degenerate
curvatures height function computed
plane
pairwise
by using
using derivatives Under function unique
plane
order
partial
partial
tangent
directions.
derivatives
derivatives
curvatures
of the surface
Computer
Aided
curvatures
of the approximated
a locally point
of the corresponding
differential
surface
geometry;
C’-smooth
height only
a quadratic
approximating
in the contact
for
point
of the height
on S which
end
S at 4 are then we
4 can be computed a definition
degenerate
function
well defined
4. One
with
a local
h is defined
II at 4. For this purpose
4. We also present
Q, nor a locally surface
of
at a point
appropriate
of the point
of the degenerate
degenerate
of a surface from
derivatives curves
defined
employs
function
entities
the surface
h at 4. We also show how the
function
assumptions
a degenerate
and well
method
partial
of the function
order
plane
surfaces
curvature
from
4 in S where
the height
order
The
emanating
weaker
in a neighborhood
approximation
surfaces:
curves
at the degenerate
4 where
of the height
up to nth
under
S obtained
at a point
of any three
of any order
one does not require
plane
at 4. The surface
Keywords. Degenerate
patches,
Our computation
the second
derivatives
derivatives
patch
curvatures
of the point
order
n of n + 1 surface
surface
order
surface
the point sef S has a tangent
that
4. In our method
independent
assumptions
representation
of a surface
are to be computed.
by using second
linearly
up to order
local second
tangent
of S at the point
over the tangent
the curvatures
has to assume
I) how the partial
those weaker
the surface
degenerate
To compute
of S in a neighborhood
to compute
of generalized
for
curvatures
domain.
the curvatures
the second
is extended
show (in Theorem concept
4 where
/I at 4 are computed
up in (I with method
one
representation
over the tangent
computations
to compute
over a rectangular
at the point
function
Curvature
111-270.
representation
(single-valued)
needs
defined
surface
of
height
the existence
function
quadratic
of a
surfaces.
of a
over the
then define
Q.
curvature.
1. Introduction
In geometric modeling with rational B-spline surface patches defined on a rectangular domain we frequently encounter situations where the related rectangular surface patch collapses to a triangular patch. In such a case, one boundary edge may collapse into a boundary vertex of the patch. Therefore, for the given parametric surface representation the Jacobian matrix corresponding to that vertex point will generally not have maximal rank. Nevertheless, it is quite possible that the point set defined by the collapsed triangular patch can be represented by another regular surface map defined over a triangular domain and may therefore have well defined surface curvatures everywhere. Clearly, the original parametrizaCorrespondme 02139-9910.
IO: F.-E.
Walter,
h_lIT,
Department
of Ocean
Engineering,
Design
USA.
0167~8396/92/SO5.00
D 1992 - El sevier
Science
Publishers
B.V.
All
rights
resewed
Laboratory,
Cambridge,
MA
F.-E. Walter, S. 7. T1coh.v/ Curvature for degenerate sutjaces
242
tion over the rectangular patch cannot be used in the classical way to compute the Gauss curvature or the principal curvatures of the collapsed patch at points which are too close to the boundary edge collapsed into the vertex. The classical curvature computation is not possible here because such a curvature computation requires a division by the determinant of the matrix defined by the first fundamental form related to the degenerate representation. As this determinant is zero, the classical method is not applicable. We want to present a method in this paper which allows fast and accurate curvature computations for the above described surface patch. Our paper will focus on the case where a surface patch is created by collapsing precisely one boundary edge in exactly one corner vertex. We assume that the original surface representation is regular everywhere except at that boundary edge which is collapsed in that vertex. Although we focus on this special case, the main ideas presented in this paper can also be applied in other situations. The cases discussed in the example section are taken from the class of rational B-spline surfaces as those are the most common ones in geometric modeling. However, the methods presented here are valid also for more general classes of surfaces as those methods require only that continuous second order partial derivatives exist and can be computed. We make the assumptions that the patch as a point set has a well defined surface normal n and curr’atures at the degenerate point p. Therefore, we assume that the patch has locally a nondegenerate C2-smooth representation cia the height function z = g(x, y) ocer an area ’ in the tangent plane at the point p where the z-axis points into the direction of the surface normal n at p. The key idea of our approach is to derice a nonsingular system of linear equations to compute the partial dericatices of the ftmction g(x, y). In this context we deri6.e Theorem 1 which shows how all partial deril’atices of g(x, y> up to order n can be computed using dericatices up to order n (at p) of n + 1 curL’es where those curces are assumed to hate pairwise linearly independent tangents at p. The curr*ature r*alues at p can then be expressed in terms of dericatices of g(x, y) up to second order. This solL*esthe cun’ature computation problem. We also present a concept to define generalized surface curvatures under weaker assumptions appropriate for degenerate surfaces. Under those weaker assumptions one does not require a local C2-smooth height function representation of the surface ocer the tangent plane at the degenerate point p. One does not eL.en require a locally well defined (single-calued) height function representation of the surface in a neighborhood of the point p. One only needs the existence of a unique local second order approximation of the degenerate surface with a quadratic frrnction defined ocer the tangent plane at p (cf. the Key Remark in Section 3.2). The surface curvatures of the corresponding approximating quadratic surface define then the cun’atures of the approximated degenerate surface in the contact point p. The main theoretical contributions of this paper are probably this generalized surface curvature concept and Theorem 1 explaining a general method to determine the higher order partial derivatives of g(x, y) by employing derivatives of curves. Those higher order derivatives can e.g. be used to determine a higher order Taylor development at p for the Christoffel symbols at p to approximate the governing differential equations of geodesics in a neighborhood of p. Those higher order derivatives are also useful to compute tensors relevant in structural shell properties which use higher than second order derivatives. The paper is structured as follows: In Section 2 we discuss related work of other authors. Section 3 is the main part of this article and contains a derivation and discussion of the method. Subsection 3.2 contains the Key Remark explaining the subtleties concerning the well definedness of the local height function representation of the degenerate surface. This Key
’
We are relaxing
the
requirement“existenceof havinga
later in the Key Remark
in Section 3.2.
C’-smooth
height function
representation
for the patch”
Remark explains also the generalized surface curvature concept which requires only the existence of the approximating quadratic surface. Section 3 demonstrates the application of the method in several examples. Section 5 contains conclusions and discusses further applications. The Appendix contains the proof of a technical assertion and the derivation of some differential geometric formulas which we include to make the paper self-contained.
2. Survey of previous related work In free-form surface geometric modeling one works usually with spline surface patches described by a function f(~i, r) defined over a rectangular domain in the u, Lv-parameter space. In general, the Jacobian matrix of the surface representation f(u, r*>is assumed to be of rank 2. Under this assumption, the classical interrogation techniques (e.g. curvature computations) for the geometry of the surface patch are well understood, see [Faux & Pratt ‘791, [Farin ‘SS]. The situation where a rectangular surface patch degenerates to a triangular patch because one boundary edge degenerates (collapses) into a point is different from the preceding case and is not as well understood. Nevertheless, this degenerate case is considered to be of interest for design problems e .g. because it allows a possibility to create triangular patches defined by a parametrization over a rectangular domain. Triangular patches are found to be advantageous for various applications, see [Farin ‘881. Therefore, the situation where a rectangular surface patch degenerates to a triangular patch has been addressed in the well known books [Faux & Pratt ‘79, pp. 235-2383 and [Farin ‘88, pp. 211-2141. In the spline surface case this degenerate situation arises in case several control points are chosen to coincide see e.g. [F arin ‘881. Conditions for the existence of a surface normal on the triangular point set ’ given by the degenerate spline patch have been discussed in the aforementioned references of Faux & Pratt and Farin. In the considered triangular B-spline patch case, those conditions imply that the (degenerate) vertex (control point) and the adjacent control points are coplanar. Faux and Pratt discuss a method to compute a surface normal in the degenerate case based on a Taylor development for the partial derivative a,,f(u, c). They give a necessary and sufficient condition for the existence of the surface normal using the requirement that the vectors a,.f(u, O), a,,a,.jYu, 0) are all in the plane tangent to the patch point set at f(0, 0). For Bezier patches this condition implies that the singular vertex (control point) and the four adjacent control points must be coplanar. Although the literature discusses methods to compute the surface normal of the degenerate patch, it does not address the harder problem of computing curvature (or higher order differential geometric) entities at the degenerate vertex of the patch. This paper intends to fill this gap. It has turned out that this paper could employ ideas which were introduced in [Pegna & Wolter ‘89, ‘921 with different intentions and in a different context, namely in the area of blend surfaces. The Three-Tangent theorem introduced in [Pegna & Wolter ‘89, ‘921 to give a criterion guaranteeing second order (curvature continuous) contact of two surfaces at a contact point is generalized in this paper (cf. Corollary 1, i.e., the ‘n + 1 Tangents Theorem’ in Section 3.5) and used as a tool to determine the curvature entities of the degenerate patch. The ‘Three Tangent Theorem’ of [Pegna & Wolter ‘89, ‘921 says: “If two regular C2-smooth surface patches S, and S, share the tangent plane at a common point p, then S, and Sz share all curvature entities 3 at the point p if and only if the normal curvatures of S, and S, agree at p for three pairwise linearly independent tangent directions.” ’ This means 3 This
includes
the existence
of the surface
e.g. Gauss curvature,
normal
at the degenerate
mean curvature
and principal
vertex. curvature
directions.
F.-E. Wolfer. S. T Tuohy / Curcarrrre for degenerate surfaces
211
3. The problem and the solution
method
3.1. Formulation of the problem
Let f(u, P):[O, 11X [O, 11 + W’ be a surface patch. We assume that f(u, L’) is degenerate at a corner point (~(a, r,) = (0, 0) corresponding to f(0, 0) on the surface. This means we assume that a,,f(O, O), aCf(O, 0) are linearly dependent. We discuss here a particular case where the surface collapses to a triangular patch at the point f(0, 0). Let us assume that f(rr, 0) =f(O, 0) for ail u E [0, 11. Let us assume further that all vectors in the set V= R,J(u, 0) Iu E LO,111are contained in the plane spanned by a,,f(l, O), a,,f(O, 0) and that V contains at least three pairwise linearly independent vectors, see Fig. 1. We also require that f(u, c) is not degenerate for all L’ with 0 < L’< E for some positive number E and we require that a,.f(lc, u) # 0 for all u, c E [O, 11. In this situation the curvatures of the surface patch at the point f(0, 0) cannot be computed using the first fundamental tensor gl, associated with the surface representation f(u, V>where gij = a,j(o,
0) - a,f(o,
0)
with a, = a,,, a, = a,.
because this tensor has det(gij) = 0. Therefore, the matrix (gij> is not invertible which makes standard operations to compute Gauss and mean curvature ill-defined because those operations require the division by the determinant det(gij). When we say we mean to compute curvatures of the surface patch at f(0, 0) =p then we think of computing geometric entities associated with the point set S = (f(u, c>l(u, c> E [O, 11X [O, 111in a neighborhood of f(0, 0). To be more specific we rephrase the problem as follows. 3.2. Reformulation of the problem Jntroduce a new coordinate system with center (0, 0, 0) =f(O, 0). Note that we had required that the tangent plane S, of S at p =f(O, 0) is well defined. This means that in our particular situation the tangent vectors of all surface curves at the points of the edge (f(u, 0)l u E [0, 11) are contained in the tangent plane S, spanned by a,.f(l, O), a,,f(O, 0). We choose the z-axis of the coordinate system pointing into a direction normal to the tangent plane S,, thus this direction may be given by the cross product of the two vectors a,.f(O, Oh a&l, 0). For the x-direction of the new coordinate system we may take the vector a,.f(O, 0). The y-direction is chosen to be normal to the x- and to the z-direction. Let (X(U, L’), Y(L1, P), Z(U, C)) =f(n,
L’).
(1)
I t f(u,v)
Fig. 1. Collapsing
patch.
F. -15. Wolfer, S. T Tuohy /
C~crc~~rctre
for degenerate surfaces
215
On a set
O<E, (x(n17 L’,), Y(U,, u,)) = (-&9 then z(ur, cl) =z(u2,
L’Z)! Y(U,, 4))
L.*).
(5)
Note that implication (5) must hold if the function value z =g(x, y) is required to be unique and depends only on the coordinates (x, y>. Obviously implication (5) may in general not be valid. Moreover, it is quite possible that there exists no height representation of the patch over any chosen (x, y)-plane. In our examples given in the example section, implication (5) holds true because one can verify in all those examples directly that the function ccl(u, c) = (X(& L’), Y(Ll, L’))
(6)
is injective on the domain ’ D={(u,c)Io~u~1,o = (0, 0) although more precisely one should t-other use the clumsy
F.-E. Wolfer, S. T Atohy / Curc~ature for degenerate surfaces
117
terminology of ‘second order Taylor coefficients in the approximation given in (12)‘. The existence of those coefficients (although sufficient to define surface cun.atures) does not er’en require C’-smoothness of the function g(x, y) in any neighborhood of (x, y) = (0, 0) nor does it require the well definedness of the height function representation of the surface patch. All geometrical examples in our example section have well defined height function
representations but the theoretical concepts presented here are applicable under weaker assumptions. It is easy to construct examples which do not have a well defined height function. The example section contains examples (107), (108), which have representations with well defined height functions g(x, y), g(x) respectively. Both height functions are C”-smooth on Q!J( D,), 0 < E < 1 or on lx IO <x < 1) respectively. However, these height functions do not have a second order Taylor approximation in a neighborhood of the points (x, y) = (0, O), x = 0 respectively and the conditions (12), (11) are not valid for both examples. 3.3. Solution method for the problem The key concept of our approach consists now in computing the above second order derivatives by using the curvature vectors (or second derivatives) of isoparameter curces o,(c)
=f(u,,
r,),
L’Q E, II, a number in [0, 11.
(14)
We shall need the curvature vector k, (or the second derivative) of the curve a,(c) at L’= 0. Let n be the unit normal vector of the patch at the point (0, 0, g(0, 0)). Thus n points into the z-direction of the coordinate system. Then, with wn = acf(Un, 0) the expression nc(w,) =n.k,,
(15)
yields the normal curcature of the surface in the direction w,, at the point p = (0, 0, g(0, 0)). By assumption w, is a vector in the x, y-plane. Let W,,= w,J I w,, ( and let X,, y,, be the (x, y) coordinates of Zn. Then we have n*k,=nc(F,)
=g,,(0,0)f~+2gx,(0,0)~~,~,+g,,(0,
O)y,‘.
(16)
This follows from differential geometry, see Assertion 2 and Assertion 3 in the Appendix The right side of equation (16) is the normal curvature for the direction W,,of the collapsed patch and the left side is derived from the curvature vector of the surface curve (Y,(U). Therefore, both sides of equation (16) are geometric entities defined independently of the parametrization of the involved geometric object. The right side of equation (16) contains also the unknowns a,a,g(O, 0) which must be computed. It is however possible to determine the right side of (16) also simply by using the second derivative of a,,(~‘), namely by using the equation n
- a”(O)
= g,,(O,
0)x,’
+ 2&,(0,
O)X,Y,
+ g,,(O,
O)Yi
(17)
now with (xn, y,,) being the (x, y) coordinates of the vector wn. The proof of equation (17) is contained in the proof of Assertion 3 in the Appendix. In equation (17) the left side is not a parametrization independent geometric entity and as (xn, y,)’ = aJ(cl,, 0) in general does not have unit length, the right side of (17) in general does not describe the normal curvature related to the direction given by (xn, y,)‘. Therefore, from a differential geometric (parametrization invariant) view point, relation (17) may be considered to be less satisfying than the relation in equation (16). Let now w, =(x,, y,), w2 = (x,, yZ), wj = (x,, y3) (or W, = (x,, Yl), W,(x,, Yz >, W3 = (x3, y,)) be three vectors (or three unit vectors respectively) in the X, y-plane such that every pair of those three vectors is linearly independent. Then (16) or (17) respectively yield the following system of equations which we give in matrix form: Lu=b
(18)
F.-E. Walter, S. T Tuohy i Cure amre for degenerate surfaces
218
with
b = (nc(iG,),
nc(Z,),
nc(iC,))’
or
b = (n *(Y;‘(O), n *cy;I(O), n *cr;(O))’
respectively
and the matrix L defined by L=(Lij),
l 0, then the point
250
F.-E. U’olter. S. T Ttcohy / Ctrrr afurr
fur degenerate surfims
Let I E (1, 21. If (36) does not hoId and if t g,,r - ‘Cl 1+ 1g.rv 1 2 1&TX? 1+ 1s!Tyy- K/ I
(37)
then the maximal principal cuu’ature direction e, or the minimal principal curvature direction ez are given by
0)’ v;kJ2+ (gxx- KA2 ( -g.c, g.,.r7
et=(*)
Kt 7
(38)
respectively. If none of (36), (37) holds, then (gYY-K/t
e,=(k)
, d( g,, - d2
-gay, 0)’ (39) + (&J2
.
Note that a principal curvature direction is defined without specifying the orientation of the direction; this explains why the sign is not specified in (38) or (39), i.e., both signs + or are allowed. 1. It is well known that the maximal and minimal principal curvatures K,, are two eigenvalues of the matrix (wij) described in (30) and (32). Thus K,, K* are the two roots of the characteristic polynomial P(K) of the matrix (w,~) where Proof of Assertion K*
P(K)
=
det( wlj -
with ai, =
1 0
~6~~)
=
0
if i =j, otherwise.
(40) -
Therefore, with (wii> being a (2, 2)-matrix K(, ~~ are solution of the quadratic equation = 0. It is easily seen that those solutions K,, K2 have the form given in the right side of equation (35). This proves the claim of equation (35). The Gauss curvature is defined as the product of the two principal CUrWtUreS K,, K*. Therefore by using (35), a straightforward computation shows (33). The mean curvature is defined as the mean value of the principal curvatures therefore using again (35) then equation (34) is an obvious consequence. We just showed (33) and (34) by employing (35). However, it is not necessary to employ (35) for this purpose. One can also argue more elegantly from an invariant linear algebra viewpoint. Namely, one can use that:
P(K)
Determinant
and trace of any quadratic matrix agree with the
product and sum (respectively)
of the eigenvalues of that matrix.
(41)
The right side of equation (33) is obviously the determinant of matrix (wij>. Therefore, by (41) this right side is equal to the product of the eigenvalues of matrix (wij). This proves (33) as well. The right side of equation (34) is the mean value of the trace elements of matrix (w,,). Therefore, by (41) the right side of equation (34) equals the mean value of eigenvalues of matrix (wij). This proves the claim of equation (34). An umbilical point is defined by the condition that at that point the maximal and the minimal principal curvatures agree. This requirement is obviously equivalent with the condition that the radicand in equation (35) must be zero. This radicand being zero is equivalent to the conditions given in (36). This proves that in our situation the criterion given in (36) characterizes umbilical points. We show now our claims given in (37), (38) and (39) concerning the description of the principal curvature directions. Recall that the principal curvature directions correspond to the eigenvectors of the linear map represented by the matrix (wij> with respect to the two basis
F.-E. Wolrer. S. T Trcoh?: / Curt atwe for degenerate surfaces
‘51 -_
vectors a,h(O, O), C$h(O,0) being the partial derivatives of the parametric representation h(x, y) of the surface patch. Therefore. using the coordinates A’,, A! of an eigenvector of the matrix (w,~) as coefficients in a linear combination with those basis (tangent) vectors, i.e., defining pj = A’,@h(O, 0)) + A+,,h(O,
O)),
(42)
yields a principal curvature direction p ‘. This means that, in order to determine principal directions one has to compute first eigenvectors of the matrix CM.;,)with respect to the natural basis (1, O)‘, (0, 1)’ of R2. Thus we must compute vectors (A’,, A’,), 1 E (1, 21, with ( wij - /Qij)( A’,, A’*)’= 0.
(43)
As we exclude (36), the considered point is not an umbilical point and the solutions of (43) build a one-dimensional vector space. Therefore either the first or the second row of matrix A’ = ( wi, - A$,,)
(44)
contains nonzero elements. To determine
the solution vector
d, = ( A’,, 4)
(45)
of (43) we employ that row vector of A’ which has the larger norm, where this norm is defined by the sum of the absolute values of the vector coordinates. This choice of the preferred row vector is reflected by the condition stated in inequality (37) and by the negation of (37) stated in conjunction with (39). The solution vector of (43) is chosen to be orthogonal on the preferred row vector, i.e.. we determine the solution vector d, by rotating the preferred row vector by 90 degrees. This solution vector d, must also be orthogonal on the other row vector of the matrix A’ because this (non-preferred) row vector is linearly dependent on the preferred row vector. This linear dependence holds because the matrix A’ must have rank less than 2 as its determinant det(A’) = 0 by (40). Thus we find a solution vector d, of (43). Inserting the coordinates of d, as coefficients in the linear combination defined in (42) where now %h(O, 0) = (1, 0, O)‘,
$h(O, 0) = (0, 1, O)‘,
gives a principal curvature direction pI which after normalization yields the vector e described in (38) and (39) respectively. Those considerations show the validity of the claims concerning the description of the principal curvature directions stated in (37), (38) and (39). This completes the proof of Assertion 1. q 3.5. Computation of higher order dericatires at the degenerate certex Key to computation of the curvature entities of the degenerate surface patch was the computation of the second order partial derivatives of the non-degenerate representation h(x, y) = (x, y, g(x, y>) at MO, 0) of the surface patch. It is possible to extend the method described in Section 3.3 to compute higher order derivatives of the representation h(x, y) at the vertex MO, 0). We explain this extension by computing third order partial derivatives aikjg(O, 0) at the vertex point. It will then be obvious how this method generalizes to compute further higher order derivatives. Let (~(5) =h(a(s)) be a curve in the degenerate
patch. We use the notation
u(s) = (LI,(J), 111(s)) = (x(s).
y(s)),
(46)
F.-E. Walter, S. T. Tuohy / Curawe
22
for degenerate surfaces
i.e., a(s) is the projection of the curve (Y(S) onto the X, y-plane or synonymously called II,, LL~-plane.We shall use the following notation hi, =
h, = ‘lrih( ‘1, ‘2) 7 hikj
= aLL,au,aLL,h(u,,
LL2),
aLr,aLLkh(LL,, i,
k, j E
LL2),
(1, 2}.
(47)
Successive application of the chain rule yields the second derivatives expression (Y”(S) = i
&u:‘(s) +
i=l
i
h,,u;(s)LL;(s)
(48)
i,k = 1
derived in the Appendix in (128). To compute the third order derivatives we differentiate a”(s) with respect to s and get (49)
Thus
i.k =‘I
This yields at the vertex point MO, 0)
k
+
i,k=
hik( uI’L4; +
u;LL;).
(51)
1
Note that (51) follows from (49) because h&O, 0) = 0 by (4). The second order partial derivatives hik(O, 0) have already been determined in (20) and the derivatives u:)(s) can be computed directly by evaluating u:(s)
=(Y”(s) *ei
with e, = (1, 0, 0), e2= (0, 1, 0).
Therefore the right side of (51) contains only known entities. The left side of (51) can be viewed as a homogeneous polynomial in the variables u;, u; if we assume that the third order partial derivatives hlkj are permutation invariant with respect to the indices i, k, j. This is justified by Schwartz theorem if the function is C3-smooth, see [Nikolsky ‘851. Using this index permutation invariance, the left side of (51) can be represented in a simplified way by the left side of the subsequent equation h,ll(u;)3
+ 3hltZ(u;>2r~; + 3h221u;(u;)2+h222(u;)3=4(U;,
U;)
where d(u,, u2) abbreviates the right side of (51). As in the case of the second derivatives we want to use a matrix equation Mx=c to determine
(52)
the unknown third order derivatives contained in the vector
I= (hlrr7 3h,r2, 3h22r7 h222).
(53)
F.-E. Woltrr. S. 7: Tu0h.v / Curvaturefor degenerate surfaces
The matrix icl in equation (52) is built by 4 rows r,, vector ,u = (,u;. , ui)’ and the rows are defined by r, :=
(
(pi)‘,
( ju;)2ju>
, ,u;(,Llz)2,
1 <j G 3, where
753
each row is related
( jL()i).
to a
(9
We can solve the matrix equation (52) if the matrix Lemma 1 contains a necessary and sufficient condition matrix M. We shall use the notation
M is non-singular. The subsequent for the non-singularity of the above
det( ju’ ju’)
(55)
where the expression (55) denotes the determinant defined by the two vectors 1~‘, ju’.
of the matrix
containing
the two columns
Lemma 1. Let rhe ser A = Iju’ = Cju;, j ~1;)’ j 0 < j < n} contain II + 1 rectors from R8?. Let M be a quadratic (n, n)-mafrir defined by n rows rj where rj=
i
(ju;)“,
(ju;)“-‘ju;,
(ju~)n-2(ju~)2,...,(jui)‘*),
O<jgn.
(56)
Therl n
det( M) =
n
det( ;u’ jll’) =
( ;u; ju; -j~l;
,u:).
(57)
O
