Some Characterizations of Families of Surfaces ... - Semantic Scholar
Some Characterizations of Families of Surfaces Using Functional Equations ENRIQUE CASTILLO and ANDRE´S IGLESIAS University of Cantabria
In this article functional equations are used to characterize some families of surfaces. First, the most general surfaces in implicit form f( x, y, z) 5 0, such that any arbitrary intersection with the planes z 5 z 0 , y 5 y 0 , and x 5 x 0 are linear combinations of sets of functions of the other two variables, are characterized. It is shown that only linear combinations of tensor products of univariate functions are possible for f( x, y, z). Second, we obtain the most general families of surfaces in explicit form such that their intersections with planes parallel to the planes y 5 0 and x 5 0 belong to two, not necessarily equal, parametric families of curves. Finally, functional equations are used to analyze the uniqueness of representation of GordonCoons surfaces. Some practical examples are used to illustrate the theoretical results. Categories and Subject Descriptors: 1.3.5 [Computer Graphics]: Computational Geometry and Object Modeling General Terms: Design Additional Key Words and Phrases: Explicit equations, functional equations, geometric modeling, implicit equations, parametric representation, surfaces, surface representations, tensor product surface
1. INTRODUCTION Properties of cross-sections of surfaces or intersections with planes parallel to the coordinate planes are very important in many applied fields. For example, in CAGD, many objects are usually designed by defining a number of cross-sections and the shape of the surface between them must be determined by using some interpolation scheme, generally through one or more longitudinal cross-sections or profiles (see Faux and Pratt [1990] and references therein). On the other hand, some methods of representing an algebraic curve require the intersection of a surface z 5 F( x, y) with the plane z 5 0 (see, e.g., Hoschek and Lasser [1993, p. 496] and references therein).
Authors’ address: Department of Applied Mathematics and Computational Sciences, University of Cantabria, Avenida de los Castros, s/n, 39005 Santander, Spain. Permission to make digital / hard copy of part or all of this work for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and / or a fee. © 1997 ACM 0730-0301/97/0700 –0296 $03.50 ACM Transactions on Graphics, Vol. 16, No. 3, July 1997, Pages 296 –318.
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Bathymetry (contour lines) of the Santa Marina beach in Spain.
As shown by Farin [1987, p. 304], in scientific computing the contour lines (obtained through intersections between a number of parallel planes and a given surface) are often of great importance. For example, Figure 1 shows the bathymetry (contour lines) of the Santa Marina beach (Spain). Some algorithms for obtaining such contour lines have been studied by Petersen [1983] for the case of three- and four-dimensional surfaces and even generalized to the case of an adaptive contouring of a trivariate interpolant. Efficient procedures for completely tracing the intersection of a plane with rational parametric surfaces have been analyzed by Chandru and Kochar [Farin 1987, pp. 305–318], and in the same line, Farouki [1986] considers a generalization of this problem using algebraic surfaces of low degree instead of a plane. Other computational methods for sectioning can be found in Hoitsma and Roche [1983] and Lee and Fredericks [1984]. Similarly, in the medical area it has been shown that it is possible to reconstruct and display the external surface of the organ under investigation from a set of parallel slices corresponding to different levels (see Ekoule et al. [1991] and references). Surface reconstruction from planar cross-sections is also analyzed by Boissonat [1985] with implications for pattern recognition and computer vision. In general, computing planar ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Fig. 2.
Bathymetry (perspective) of the Santa Marina beach in Spain.
intersections of geometric objects is a very important capability of CAD/ CAM and many other geometric-modeling systems (see Mortenson [1985]). In addition to this, in several areas of engineering, contouring techniques are used to represent a three-dimensional surface in two dimensions (see Anand [1993]). Contour plots are created as intersections of the surface with planes z 5 z 0 for different values of z 0 . Similarly, for nonparametric (i.e., implicit or explicit) surfaces this is usually done by drawing its intersections with planes x 5 x 0 and y 5 y 0 , for selected values of x 0 and y 0 . From these intersections, areas or volumes can be easily approximated. As a first example, Figure 2 shows a perspective of the Santa Marina beach sea bottom, corresponding to the contour lines in Figure 1, obtained by this method. Figure 3 shows a similar representation of an object made by six cylinders. In this context, surfaces such that the preceding intersections belong to the same families are very convenient, since all drawings reduce to the graphical representation of a single family of curves. In CAGD surfaces are dealt with in several forms: parametric, explicit, and implicit equations. The most common representation in the commercial software and research fields are parametric equations. This representation allows a quick computation of the coordinates of all points on a curve or surface, and its popularity depends, in part, on whether the design problem in surface modeling systems has been successfully addressed. Moreover, ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Fig. 3. Surface represented by drawing the intersections of all its components with planes x 5 x 0 and y 5 y 0 .
the parametric form can be used to define a curve segment or surface patch constraining the parameters to intervals. Because curves and surfaces are usually bound in computer graphics, this characteristic is of considerable importance. Nevertheless, in the last few years, explicit and implicit representations are being used more frequently in CAGD, allowing a better treatment of several problems. As one example, the point classification problem is easily solved with the implicit representation: it consists of a simple evaluation of the implicit functions. This is useful in many applications, such as solid modeling, for example, where points must be defined inside or outside the boundaries of an object (Hoffmann [1989]). Through implicit representation, the problem is reduced to a trivial sign test. Furthermore, the implicit representation offers surfaces of desired smoothness with the lowest possible degree. Finally, when we restrict ourselves to polynomial functions, the implicit representation is more general than the parametric representation (but probably not if we allow arbitrarily complicated functions) and several methods have been described in order to solve the problem of implicitizaACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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tion, that is, obtaining the implicit representation of rational surfaces (see, e.g., Sederberg [1983], Farin [1990], or Hoffmann [1989]). Although this conversion is always possible, difficulties arise when base points are present, but these problems are not insurmountable (see Chionh and Goldman [1992]). From the foregoing considerations, it is clear that identification of the most general families of surfaces with arbitrary planar cross-sections parallel to the coordinate axes belonging to given parametric and nonparametric families is an important problem. Our aim in this article is twofold: to obtain such characterizations and to show the power of functional equations in this process of characterization. The article is structured as follows. We start in Section 2 by giving some elementary notions of functional equations and some tools to be used later in the article. In Section 3 we apply functional equations to characterize the most general surfaces in implicit form, such that their intersections with the planes z 5 z 0 , y 5 y 0 , and x 5 x 0 are linear combinations of sets of given functions of the other two variables. In Section 4 we find the most general surfaces in explicit form such that their intersections with planes parallel to the planes y 5 0 and x 5 0 belong to given parametric families of curves. Finally, in Section 5 we use functional equations to analyze the uniqueness of representation of Gordon-Coons surfaces. 2. SOME BASIC CONCEPTS OF FUNCTIONAL EQUATIONS In this section, for those not familiar with this type of equation, we give a collection of basic results of functional equations to be used later. For a detailed study of functional equations the reader is referred to Acze´l [1966] or Castillo and Ruiz-Cobo [1992]. A functional equation can be considered as an equation in which the unknowns are functions. To distinguish functional equations from differential or integral equations, which also have functions as unknowns, we exclude equations containing infinitesimal operations. As an illustrative example we consider Pexider’s main equation
f~ x 1 y! 5 g~ x! 1 h~ y!;
x, y [ R,
(1)
where f(.), g(.), and h(.) are the unknown functions, which are assumed to belong to the class of continuous functions, and the equation is considered to hold in R (domain of the functional equation). The most general system of solutions of (1) is
f ~ x ! 5 Ax 1 B 1 C;
g ~ x ! 5 Ax 1 B;
h ~ x ! 5 Ax 1 C,
(2)
where A, B, and C are arbitrary constants. No other functions satisfy the preceding equation. As a second example, the following theorem gives the general solution of n equations of the form ( k51 f k ( x) g k ( y) 5 0, which are used in Sections 3, 4, and 5. ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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THEOREM 1 [Acze´l [1966, p. 160] or Castillo and Ruiz-Cobo [1992, p. 52]]. All solutions of the equation
O f ~ x ! g ~ y ! 5 0, n
f~ x! z g~ y! 5
k
k
(3)
k51
where f( x) 5 ( f 1 ( x), . . . , f n ( x)), g( y) 5 ( g 1 ( y), . . . , g n ( y)), and z is used to denote the dot product of two vectors, can be written in the form
f~ x! 5 w~ x!A g ~ y ! 5 c ~ y ! B,
(4)
where w( x) 5 ( w 1 ( x), . . . , w r ( x)), c( y) 5 ( c r11 ( y), . . . , c n ( y)), r is an integer between 0 and n, { w 1 ( x), . . . , w r ( x)} and { c r11 ( x), . . . , c n ( x)} are two arbitrary systems of linearly independent functions, and A and B are constant matrices, which satisfy
AB T 5 0.
(5)
COROLLARY 1. Let {u 1 ( x), . . . , u I ( x)} and {v 1 ( y), . . . , v J ( y)} be two linearly independent sets of known functions; then the solution of the functional equation
a~ y! z u~ x! 5 b~ x! z v~ y!,
(6)
where {a1( y), . . . , aI( y)} and {b1( x), . . . , bJ( x)} are the unknown functions, is
a ~ y ! 5 v ~ y ! D;
b ~ x ! 5 u ~ x ! D T,
(7)
where D is an arbitrary constant matrix. PROOF.
Expression (6) is equivalent to
~ a ~ y !u 2 v ~ y !! z ~ u ~ x !u b ~ x !! 5 0. Thus, according to Theorem 1, we have
~ u ~ x !u b ~ x !! 5 u ~ x !~ I Iu C ! ~ a ~ y !u 2 v ~ y !! 5 v ~ y !~ D u 2 I J! , where C and D are I 3 J and J 3 I constant matrices, respectively, such that
~ I Iu C !~ D u 2 I J! T 5 0 N D T 5 C.
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It is worthwhile mentioning some characteristic features of functional equations. (1) The general solution of a functional equation is strongly dependent on the domain of its definition (C, R, Q, etc.) and the class of functions (arbitrary, monotonic, continuous, differentiable, etc.) being considered. (2) In addition to arbitrary constants, as in Equation (2), arbitrary functions, as in Equation (4), can appear in the general solution. (3) Unlike any other kind of equation, from a single equation we can determine several unknown functions (see, e.g., Equations (1) and (2), or (3) and (4) and (5)). The following theorem is used later. THEOREM 2 [Castillo and Galambos [1987]]. The general nonnegative and integrable solution of the functional equation
z ~ x, y ! 5 ~ a 1~ y ! x 1 a 2~ y !! a3~ y! 5 ~ b 1~ x ! y 1 b 2~ x !! b3~ x!,
(8)
where a 1 (.), a 2 (.), a 3 (.) and b 1 (.), b 2 (.), b 3 (.) are unknown functions, is
z ~ x, y ! 5 @ C ~ x 2 A !~ y 2 B ! 1 D # E z ~ x, y ! 5 E ~ x 2 A ! C~ y 2 B ! D exp @ M log ~ x 2 A ! log ~ y 2 B !# , which depend on five and six parameters, respectively. For the proof see the preceding reference. 3. FAMILIES OF IMPLICIT SURFACES In this section we deal with the case of surfaces in implicit form. We look for the most general surfaces in implicit form such that their planar cross-sections parallel to the coordinate planes satisfy some conditions. This allows the designer to choose families of surfaces with cross-sections at his or her convenience. The following theorem gives these conditions in a precise form and the corresponding solution. THEOREM 3. The most general family of implicit surfaces f( x, y, z) 5 0, such that their intersections with the planes z 5 z 0 , y 5 y 0 , and x 5 x 0 are linear combinations of sets 8 5 {u 1 ( y, z), u 2 ( y, z), . . . , u I ( y, z)}, 9 5 {v 1 ( z, x), v 2 ( z, x), . . . , v J ( z, x)} and 0 5 {w 1 ( x, y), w 2 ( x, y), . . . , w K ( x, y)}, of functions of the other two variables, is of the form
O O O @C I
f ~ x, y, z ! 5
J
K
ijk
a i~ x ! b j~ y ! g k~ z !# ,
(9)
i51 j51 k51
where a( x), b( y) and g( z) are vectors of arbitrary functions and C ijk are the elements of an arbitrary constant matrix C. ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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In addition, the 8, 9, and 0 functions cannot be arbitrary, but of the form
O O @C J
u i~ y, z ! 5
K
ijk
b j~ y ! g k~ z !# ;
ijk
ai ~ x!gk ~ z!#;
j 5 1, . . . , J,
(11)
ai ~ x!bj ~ y!#;
k 5 1, . . . , K.
(12)
i 5 1, . . . , I,
(10)
j51 k51
O O @C I
vj ~ z, x! 5
K
i51 k51
O O @C I
wk ~ x, y! 5
J
ijk
i51 j51
PROOF. According to the preceding assumptions, we look for surfaces f( x, y, z) 5 0 such that they satisfy the system of functional equations
O a ~ x ! u ~ y, z ! 5 O b ~ y ! v ~ z, x ! I
f ~ x, y, z ! ;
J
i
i
j
i51
O b ~ y!v ~ z, x! 5 O g ~ z!w ~ x, y!, J
f~ x, y, z! ;
j
j51
K
j
j
k
j51
k
(13)
k51
where the sets { a i ( x); i 5 1, . . . , I}, { b j ( y); j 5 1, . . . , J}, and { g k ( z); k 5 1, . . . , K} can be assumed without loss of generality as sets of linearly independent functions. Note that if they are not, we can rewrite equations in (13) in the same form but with linearly independent sets. The system of equations (13) states the conditions for the sets 8, 9, and 0 to be compatible with the sets { a i ( x); i 5 1, 2, . . . , I}, { b j ( y); j 5 1, 2, . . . , J}, and { g k ( z); k 5 1, 2, . . . , K}. For any fixed z, the first equation in (13) is of the form of Equation (6) and, according to Corollary 1 we have (see (7))
u ~ y, z ! 5 b ~ y ! A T~ z !
(14)
v ~ z, x ! 5 a ~ x ! A ~ z ! ,
(15)
where A( z) is a matrix whose elements are functions of z. Replacing Equation (14) or (15) in (13) we get
O O A ~ z!a ~ x!b ~ y!. I
f ~ x, y, z ! 5 a ~ x ! A ~ z ! b ~ y ! 5 T
J
ij
i
j
i51 j51
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Similarly, for any fixed x, the second equation in (13) leads to
v ~ z, x ! 5 g ~ z ! B T~ x ! w ~ x, y ! 5 b ~ y ! B ~ x ! .
(16)
Now from Equations (15) and (16) we obtain for each j 5 1, . . . , J,
O A ~ z!a ~ x! 5 O B ~ x!g ~ z!, I
v j~ z, x ! 5
K
ij
i
jk
i51
k
k51
which is also of the form given in Corollary 1, and then we can write
OOC I
v j~ z, x ! 5
K
~2! ijk
a i~ x ! g k~ z ! ,
i51 k51
(2) where C ijk are the elements of a constant matrix. It is clear that similar expressions can be obtained for u i ( y, z) and w k ( x, y). Now replacing this into Equation (13), we get
OOO I
f ~ x, y, z ! 5
J
K
i51 j51 k51
O O O @C I
f ~ x, y, z ! 5
J
O O O @C
~3! ijk
a i~ x ! b j~ y ! g k~ z !#
O O O @C
~1! ijk
a i~ x ! b j~ y ! g k~ z !#
I
~2! @ C ijk a i~ x ! b j~ y ! g k~ z !# 5
J
K
i51 j51 k51
K
I
~3! ijk
a i~ x ! b j~ y ! g k~ z !# 5
i51 j51 k51
J
K
i51 j51 k51
(17) but, taking into account that the preceding sets of functions are linearly independent, we get ~1! ~2! ~3! C ijk 5 C ijk 5 C ijk 5 C ijk ,
which together with (17) leads to (9).
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Remarks: (1) Note that no constraints have been imposed on the f( x, y, z) functions. Thus an arbitrary class of functions has been assumed. (2) The implicit equation of the surface (9) is a linear combination of the tensor product of the sets of functions in a( x), b( y), and g( z). (3) Algebraic surfaces are particular cases of this family. ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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(4) The only functions u i ( y, z), v j ( z, x), and w k ( x, y) satisfying condition (13) are of the form (10) through (12), where a(.), b(.), and g(.) are the function coefficients in (13). (5) Note that the functional equations are imposed not only on a fixed number of given cross-sections, but on an arbitrary planar section parallel to the coordinate planes. (6) Functional equations allow characterizing the functional form of the solution from a simple compatibility condition. 4. FAMILIES OF EXPLICIT SURFACES In this section we look for the most general surface in explicit form, z 5 z( x, y), such that their intersections with planes parallel to the planes y 5 0 and x 5 0 belong to given (not necessarily equal) parametric families of curves, that is,
z 5 z ~ x, y ! 5 h ~ x, a 1~ y ! , a 2~ y ! , . . . , a k~ y !! z 5 z ~ x, y ! 5 r ~ y, b 1~ x ! , b 2~ x ! , . . . , b m~ x !! .
(18)
This leads to the functional equation
h ~ x, a 1~ y ! , a 2~ y ! , . . . , a k~ y !! 5 r ~ y, b 1~ x ! , b 2~ x ! , . . . , b m~ x !! ,
(19)
where we assume that a 1 (.), a 2 (.), . . . , a k (.) and b 1 (.), b 2 (.), . . . , b m (.) are unknown functions to be determined and h(.) and r(.) are known functions, which are selected in order to have convenient cross-sections. Note that a given surface is completely defined by one of the two equations in (18). For two different representations (the two equations in (18)) to correspond to the same surface they must satisfy some conditions. Thus the functional equation (19) plays the role of a compatibility condition. In the following sections we analyze different cases of h(.) and r(.) parametric families of curves. n 4.1 Explicit Surfaces of the Form z 5 s(( i51 p i (x)q i (y))
In this section we take h( x, a 1 , a 2 , . . . , a k ) 5 ( ki51 a i u i ( x) and r( y, b 1 , b 2, . . . , b m) 5 ( m i51 b i v i ( y), and we solve the associated problem by means of the following theorem. THEOREM 4. The most general surface in explicit form, z 5 z( x, y), such that all sections with planes parallel to the planes y 5 0 and x 5 0 are linear combinations of given sets of linearly independent functions 8 5 {u 1 (.), u 2 (.), . . . , u k (.)} and 9 5 {v 1 (.), v 2 (.), . . . , v m (.)}, respectively, is
z 5 z ~ x, y ! 5 v ~ y ! Au T~ x ! ,
(20)
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where A is an arbitrary constant matrix. Thus we obtain a linear combination of the tensor products of the vectors of functions u( x) and v( y). PROOF.
According to the preceding assumptions, we must have
z ~ x, y ! 5 a ~ y ! z u ~ x ! z ~ x, y ! 5 b ~ x ! z v ~ y ! ,
(21)
where u( x) 5 (u 1 ( x), . . . , u k ( x)) and v( y) 5 (v 1 ( y), . . . , v m ( y)) are known and a( y) 5 ( a 1 ( y), . . . , a k ( y)) and b( x) 5 ( b 1 ( x), . . . , b m ( x)) are to be determined. Expressions (21) imply
a~ y! z u~ x! 5 b~ x! z v~ y!, which is a functional equation of the form given in Corollary 1. Thus its general solution is
a~ y! 5 v~ y!A b ~ x ! 5 u ~ x ! A T, where A is an arbitrary (m 3 k) constant matrix. Thus the explicit equation of the parametric family of surfaces becomes (20). e Since any explicit surface z 5 z( x, y) can be written as an equivalent implicit surface f( x, y, z) 5 z( x, y) 2 z, the preceding theorem can be obtained as a particular case of Theorem 4, as follows. According to (9) we can write
O O O @C I
f ~ x, y, z ! 5
J
K
ijk
a i~ x ! b j~ y ! g k~ z !# 5 z ~ x, y ! 2 z,
i51 j51 k51
and making z 5 z 0 we obtain
O O a ~ x!b ~ y! O @C I
J
K
i
i51 j51
j
ijk
g k~ z 0!# 5 z ~ x, y ! 2 z 0 ,
k51
and then we get
O O C9 a ~ x ! b ~ y ! , I
z ~ x, y ! 5 z 0 1
J
ij
i51 j51
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i
j
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where
O @C K
C9ij 5
ijk
g k~ z 0!# ,
k51
which is equivalent to (20). COROLLARY 2.
If, instead of (21), we consider
z ~ x, y ! 5 s ~ a ~ y ! z u ~ x !! z ~ x, y ! 5 s ~ b ~ x ! z v ~ y !! , where s(.) is an invertible function, the general solution for z( x, y) becomes
z ~ x, y ! 5 s ~ v ~ y ! Au T~ x !! .
(22)
This corollary can be directly proved by applying Theorem 4 to the explicit function z 5 s 21 ( z( x, y)). So, given u( x) and v( y), Equation (22) defines its associated parametric family of surfaces, and any surface from this family has an associated matrix A. The intersections of two surfaces of this family, with parameters defined by matrices A1 and A2, have as projection on the plane z 5 0 the curve
v ~ y !~ A 1 2 A 2! u T~ x ! 5 0,
(23)
which together with (22) allows the drawing of the curve. The intersections of these surfaces with planes z 5 z 0 have as projections on the plane z 5 0, the curve
v ~ y ! Au T~ x ! 5 z 0 .
(24)
Example 1 (Nonparametric Bicubic Tensor Product Surfaces). If the surface z 5 z( x, y) is such that all sections with planes parallel to the planes y 5 0 and x 5 0 are third-degree polynomials, we have s(w) 5 w,
~ u 1~ x ! , u 2~ x ! , u 3~ x ! , u 4~ x !! 5 ~ 1, x, x 2, x 3!
and
~ v 1~ x ! , v 2~ x ! , v 3~ x ! , v 4~ x !! 5 ~ 1, y, y 2, y 3! . Then, according to (22), the explicit equation of the surface becomes
z 5 z ~ x, y ! 5 ~ 1, y, y 2, y 3! A ~ 1, x, x 2, x 3! T.
(25)
It is obvious that intersections of the preceding surface by planes z 5 z 0 are algebraic curves (see (24)). Similarly, (23) shows that the projection on ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Fig. 4.
Set of points defining the surface.
the plane z 5 0 of the intersections of two surfaces of this family are algebraic curves too. The family (25) depends on 16 parameters and then one surface can be forced to pass through 16 given points. Figure 4 shows an example where the 16 points have been selected in such a way that certain sets of 4 points belonging to the planes x 5 i and y 5 j with i 5 1, . . . , 4; j 5 1, . . . , 4. The figure also shows the polynomial curves passing through every 4 points in each of these sets. Figure 5 shows the interpolating surface. Now we force all vertical plane intersections to be third-degree polynomials in X or Y. Making y 5 px 1 q or x 5 ry 1 t in (25) and canceling coefficients of x and y of degree larger than 3, we get
a 44 5 a 43 5 a 34 5 a 42 5 a 33 5 a 24 5 0. Thus the new family becomes
z 5 z ~ x, y ! 5 ~ 1, y, y 2, y 3!
1
a 11 a 12 a 13 a 14 a 21 a 22 a 23 0 a 31 a 32 0 0 a 41 0 0 0
21 2 1 x x2 x3
.
Example 2 (Some Exponential Families). Now we consider the case s(w) 5 exp(w), (u 1 ( x), u 2 ( x), u 3 ( x)) 5 (1, x, log( x)) and (v 1 ( y), v 2 ( y), v 3 ( y)) 5 (1, y, y 2 ). Then we get the family of surfaces
z ~ x, y ! 5 exp @~ 1, y, y 2! A ~ 1, x, log ~ x !! T# , which depends on 9 parameters. The intersections of two surfaces of this family, with associated parameters A1 and A2, have the following projection on the plane z 5 0
~ 1, y, y 2!~ A 1 2 A 2!~ 1, x, log ~ x !! T 5 0. ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Interpolating surface.
As one example, in Figure 6 we show the surface with equation
z ~ x, y ! 5 exp @ 2x ~ 1 1 y 1 y 2! 1 log ~ x !~ 1 1 y/ 2 !# . 4.2 Some Other Surfaces In this section we illustrate the power of the method giving two more particular examples of (18). We take
h ~ x, a 1 , a 2 , a 3! 5 s ~~ a 1x 1 a 2! a3! r ~ y, b 1 , b 2 , b 3! 5 s ~~ b 1y 1 b 2! b3!; that is,
z ~ x, y ! 5 s @~ a 1~ y ! x 1 a 2~ y !! a3~ y!# z ~ x, y ! 5 s @~ b 1~ x ! y 1 b 2~ x !! b3~ x!#, which leads to the functional equation (8). Theorem 2 gives the two families of solutions of (8) and Figures 7 and 8 show the two surfaces associated with the explicit equations z 5 1 2 exp(2( xy 2 1) 2 ) and z 5 exp(2( xy 2 1) 2 ), respectively. Their crossplanar sections by planes of the form x 5 x 0 and y 5 y 0 are obvious. It is worthwhile pointing out that in spite of the general form of the functional equation (8) the only feasible solutions are parametric families; ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Fig. 6.
One surface from the exponential family.
that is, the functional form of the solutions is determined by the compatibility condition (8).
5. GORDON-COONS SURFACES Functional equations are equally useful for proving certain properties of surfaces in parametric form. In this section we prove a uniqueness theorem for this type of surface. We start by introducing them and the basic notation using the Gordon approach (see Gordon [1993, pp. 117–134]). Consider the following problem. Assume two families of parametric curves
$ g i~ t !u i 5 1, 2, . . . , M %
and
$ f j~ s !u j 5 1, 2, . . . , N % ,
where
g i~ t ! 5
1
g i~1!~ t ! g i~2!~ t ! g ~i3!~ t !
2
;
f j~ s ! 5
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f j~1!~ s ! f j~2!~ s ! f ~j3!~ s !
2
,
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Surface z 5 1 2 exp (2( xy 2 1) 2).
and the superindices 1, 2, and 3 of the g and f functions refer to the x, y, and z components, respectively. All functions gi (t) and fj (s) have to be defined on common parameter intervals [t# 0 , t# 1 ] and [s# 0 , s# 1 ], respectively. These curves intersect in a set of space points whose coordinates are obtained for some values of the parameters t 1 , t 2 . . . , t N and s 1 , s 2 , . . . , s M . For these two families to define a surface they must satisfy the following compatibility conditions,
u ij 5 g i~ t j! 5 f j~ s i! . Then it is possible to build a surface v(s, t) interpolating the M 1 N given curves, that is, satisfying the system of vector equations
v ~ s i , t ! 5 g i~ t ! ,
~ i 5 1, 2, . . . , M !
v ~ s, t j! 5 f j~ s ! ,
~ j 5 1, 2, . . . , N ! . ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Fig. 8.
Surface z 5 exp (2( xy 2 1) 2).
Thus the problem of interpolating a surface through an intersecting skeletal network of three-dimensional curves can be reduced to the following scalar-value problem. Construct a bivariate function v(s, t) that interpolates the M 1 N univariate functions {gi (t)ui 5 1, . . . , M}, {fj (s)u j 5 1, . . . , N }. To solve this problem, Gordon, in the end of the 1960s and 1970s, proposed the solution given by the following theorem. THEOREM 5. If { f i (s)ui 5 1, 2, . . . , M} and { c j (t)u j 5 1, 2, . . . , N } are any two sets of functions such that they satisfy the conditions
f i~ s k! 5 d ik
(26)
c j~ t k! 5 d jk ,
(27)
where d ij is the Kronecker d , and if {gi (t)ui 5 1, 2, . . . , M} and {fj (s)u j 5 1, 2, . . . , N } are any two sets of functions such that the compatibility conditions are satisfied:
g i~ t j! 5 f j~ s i! , ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
i, j.
(28)
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Then the bivariate function
O g ~t!f ~s! 1 O f ~s!c ~t! 2 O O u f ~s!c ~t! M
v ~ s, t ! 5
N
i
i
i51
M
j
N
j
ij
j51
i
j
(29)
i51 j51
is one solution of the interpolation problem
v ~ s k , t ! 5 g k~ t ! ,
~ k 5 1, 2, . . . , M !
v ~ s, t p! 5 f p~ t ! ,
~ p 5 1, 2, . . . , N ! .
Note that (29) gives v(s, t) as a function of the three surfaces
O g ~t!f ~s!, M
v 1~ s, t ! 5
i
i
i51
which interpolates the family of curves {gi (t)ui 5 1, 2, . . . , M},
O f ~s!c ~t!, N
v 2~ s, t ! 5
j
j
j51
which interpolates the family of curves {fj (s)u j 5 1, 2, . . . , N }, and
O O u f ~s!c ~t!, M
v 3~ s, t ! 5
N
ij
i
j
i51 j51
which interpolates the set of points {uij ui 5 1, 2, . . . , M; j 5 1, 2, . . . , N }. Note that it is again a tensor product. Gordon-Coons surfaces (see Gordon [1993] and Coons [1964]) are frequently referred to as transfinite interpolants [Farin 1990, p. 381; Hoschek and Lasser 1993, p. 371], since they interpolate continuous data, that is, all points along the prescribed boundary curves. 5.1 Uniqueness Theorem For the surface family defined by Equation (29), the uniqueness of representation is analyzed; that is, we try to find out whether there is only one representation for such a surface. To be more precise we give the following theorem. THEOREM 6.
Given the sets of linearly independent functions
$ f i~ s !u i 5 1, 2, . . . , M %
and
$ c j~ t !u j 5 1, 2, . . . , N % ,
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•
there exist unique sets of functions
$ g i~ t !u i 5 1, 2, . . . , M %
$ f j~ s !u j 5 1, 2, . . . , N %
and
satisfying the compatibility conditions
u ij 5 g i~ t j! 5 f j~ s i! , such that the surface v(s, t) is given by (29). PROOF. We use functional equations for this proof. Let {gi (t), fj (s)} and {g*i (t), f *j (s)} be two such representations satisfying the same compatibility conditions
u ij 5 g i~ t j! 5 f j~ s i! 5 g *i~ t j! 5 f *j~ s i! ;
(30)
then we have
O g ~t!f ~s! 1 O f ~s!c ~t! 5 O M
N
i
M
i
j
i51
j
j51
O f *~ s ! c ~ t ! ; N
g *i~ t ! f i~ s ! 1
i51
j
j
j51
that is, the following functional equation holds
O @ g ~ t ! 2 g *~ t !# f ~ s ! 1 O @ f ~ s ! 2 f *~ s !# c ~ t ! ; 0. M
N
i
i
i
j
i51
j
j
j51
Its solution is given by (see Corollary 1)
O A c ~ t ! f g ~ t ! 5 O A c ~ t ! 1 g *~ t ! ; N
g i~ t ! 2 g *i~ t ! 5
N
ij
j
i
ij
j51
i
O A f ~ s ! f f ~ s ! 5 2 O A f ~ s ! 1 f *~ s ! ; M
f j~ s ! 2 f *j~ s ! 5 2
j
i
j51
M
ij
i
j
i51
ij
i
j
j,
i51
where A is a matrix of constants. Returning now to the compatibility conditions (30) and using definition (27), we get
O A c ~t ! 1 g*~t ! f u N
uij 5 gi ~tj ! 5
ik
k
j
i
j
ij
5 Aij cj ~tj ! 1 uij 5 Aij 1 uij ;
K51
which is the desired uniqueness.
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5.2 A Particular Case In this section we analyze the particular case of surfaces in which the sets of functions {gi (t)} and {fj (s)} are linear combinations of the sets { c j (t)} and { f i (s)}, respectively. In this case, the surface v(s, t) appears as a very simple form, as indicated by the following theorem. THEOREM 7. If the conditions of the preceding theorem hold and the two sets of functions
$ g i~ t !u i 5 1, 2, . . . , M %
and
$ f j~ s !u j 5 1, 2, . . . , N %
are linear combinations of the sets of functions
$ c j~ t !u j 5 1, 2, . . . , N %
and
$ f i~ s !u i 5 1, 2, . . . , M % ,
that is,
g 5 Ac;
f 5 Bf,
(31)
then A 5 BT 5 U 5 {uij ui 5 1, . . . , M; j 5 1, . . . , N }. PROOF. form as
The surface v(s, t) satisfies (29), which can be written in matrix
v 5 f Tg 1 f Tc 2 f TU c .
(32)
Substitution of (31) in (32) leads to
v 5 f TA c 1 f TB Tc 2 f TU c 5 f T~ A 1 B T 2 U ! c and then
v ~ s i , t j! 5 g i~ t j! 5 f j~ t i! 5 f T~ s i!~ A 1 B T 2 U ! c ~ t j! .
(33)
Now using the definitions (26) and (27) we get
u ij 5 v ~ s i , t j! 5 A ij 1 B ji 2 U ij .
(34)
Applying the same definitions (26) and (27) to Equations (31) we get
g i~ t j! 5 ~ g ~ t j!! i 5 ~ A c ! i 5 A ij
(35)
f j~ s i! 5 ~ f ~ s i!! j 5 ~ B f ! j 5 B ji .
(36)
and
Now, inserting (34), (35), and (36) into the compatibility conditions (28), we immediately have
U ij 5 A ij 5 B ji , ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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which proves the theorem.
e
Note that in this case we have not only uniqueness of representation that holds under more general conditions, but we know the form of the matrix U. Note also that the form of the surface v(s, t) is extremely simple. In fact, for this particular case, the three terms in expression (29) coincide, that is,
v ~ s, t ! 5 v 1~ s, t ! 5 v 2~ s, t ! 5 v 3~ s, t ! . These particular surfaces are usually called tensor product surfaces in CAGD. Consequently, for interpolating the surface v(s, t) it is enough to interpolate any one of the families of curves {gi (t)} or {fj (s)}, or to interpolate the set of points {uij }. Adequate selection of the sets of functions {gi (t)} and {fj (s)} can simplify the problem of determining the intersection curves of two different surfaces, as illustrated in the following example. Example 3 (Parametric Form). In this example we build four parametric surfaces in parametric form. For the first three surfaces (spheres) we have used the base curves
g i~ t ! 5
1
a sin ~ t ! cos ~ s i! a sin ~ t ! sin ~ s i! a cos ~ t ! 1 b
2
;
f i~ s ! 5
1
a sin ~ t j! cos ~ s ! a sin ~ t j! sin ~ s ! a cos ~ t j! 1 b
2
with a 5 1, b 5 0, a 5 0.75, b 5 0.75 and a 5 0.5, b 5 1.5, respectively. For the fourth surface (cylinder) we have taken the base curves
g i~ t ! 5
1
cos ~ s i! / 2 sin ~ s i! / 2 t21
2
;
f i~ s ! 5
1
cos ~ s ! / 2 sin ~ s ! / 2 tj 2 1
2
.
The parameters s and t belong to the intervals [0, 3p/2] and [0, p], respectively. The surfaces and their intersections are shown in Figure 9. Note that intersections of these surfaces can be easily determined, since the sets {fj (s)} coincide for all of them and they correspond to t 5 constant. 6. SUMMARY AND CONCLUDING REMARKS In this article we have applied functional equations to solve some problems of characterization, existence, and uniqueness of families of surfaces. We have characterized the most general surfaces in implicit form such that their intersections with the planes z 5 z 0 , y 5 y 0 , and x 5 x 0 are linear combinations of sets of given functions of the other two variables. It turned out that they are linear combinations of the tensor products of functions of each of the variables alone. Later, we have found the most general surfaces in explicit form such that their intersections with planes parallel to the ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Four parametric surfaces and their intersections.
planes y 5 0 and x 5 0 belong to some given families of curves. Finally, we have used functional equations to analyze the uniqueness of representation of Gordon-Coons surfaces. Thus functional equations have been shown to be very powerful and elegant tools for solving these types of problems. ACKNOWLEDGMENTS
We thank the University of Cantabria and Direccio´n General de Investigacio´n Cientı´fica y Te´cnica (DGICYT) (project PB92-0504) for partial support of this work. We also thank the three referees and the Editors for some helpful discussions that improved the article. REFERENCES ´ , J. 1966. Lectures on functional equations and their applications, Vol. 19. MathematACZEL ics in Science and Engineering. Academic Press, New York, NY. ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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ANAND, V. B. 1993. Computer Graphics and Geometric Modeling for Engineers, Chap. 13. Wiley, New York, NY. BOISSONAT, J. D. 1985. Surface reconstruction from planar cross-sections. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (June), 393–397. CASTILLO, E. AND GALAMBOS, J. 1987. Bivariate distributions with normal conditionals. In Proceedings of the IASTED International Symposium on Simulation, Modeling and Development SMD-87 (Cairo, Egypt), 59 – 62. CASTILLO, E. AND RUIZ-COBO, R. 1992. Functional Equations in Science and Engineering. Marcel Dekker, New York, NY. CHIONH, E. W. AND GOLDMAN, R. N. 1992. Implicitizing rational surfaces with base points by applying perturbations and the factors of zero theorem. In Mathematical Methods in Computer Aided Geometric Design. T. Lynche, L. L. Schumaker, eds., Academic Press, New York, NY, 101–110. COONS, S. A. 1964. Surfaces for computer-aided design. MIT, Mechanical Engineering Dept., Design Div. EKOULE, A. B., PEYRIN, F. C., AND ODET, C. L. 1991. A triangulation algorithm from arbitrary shaped multiple planar contours. ACM Trans. Graph. 10, 182–199. FARIN, G. E. 1987. Geometric modeling: Algorithms and new trends. In Proceedings of the Conference on Geometric Modeling and Robotics SIAM (Philadelphia, PA). FARIN, G. E. 1990. Curves and Surfaces for Computer Aided Geometric Design. second ed. Academic Press, New York, NY. FAROUKI, R. T. 1986. The characterization of parametric surface sections. Comput. Vision, Graph. Image Process. 33, 209 –236. FAUX, I. D. AND PRATT, M. J. 1990. Computational Geometry for Design and Manufacture. Chap. 8. Ellis Horwood, Chichester, UK. GORDON, W. J. 1993. Sculptured Surface Definition via Blending-Function Methods: Fundamental Developments of Computer-Aided Geometric Modeling. Academic Press, New York, NY, 117–134. HOFFMANN, C. M. 1989. Geometric and Solid Modeling: An Introduction, Morgan-Kaufmann, San Mateo, CA. HOITSMA, D. H. AND ROCHE, M. 1983. The computation of all plane-surface intersections for CAD/CAM applications. Comput. Aided Geom. Model. NASA C.P. 2272, 15–18. HOSCHEK, J. AND LASSER D. 1993. Fundamentals of CAGD. A. K. Peters, Wellesley, MA. LEE, R. B. AND FREDERICKS, D. A. 1984. Intersection of parametric surfaces and a plane. IEEE Comput. Graph. Appl. 4, 48 –51. MORTENSON, M. E. 1985. Geometric Modeling. Chap. 7, Wiley, New York, NY. PETERSEN, C. S. 1983. Contours of three and four-dimensional surfaces. Masters thesis, Dept. Mathematics, Univ. Utah, Salt Lake City, UT. SEDERBERG, T. W. 1983. Implicit and parametric curves and surfaces for computer aided geometric design. Ph.D. thesis, Purdue University, West Lafayette, IN. Received September 1995; revised April 1996; accepted February 1997
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In this article functional equations are used to characterize some families of surfaces. First, the most general surfaces in implicit form f( x, y, z) 5 0, such that any arbitrary intersection with the planes z 5 z 0 , y 5 y 0 , and x 5 x 0 are linear combinations of sets of functions of the other two variables, are characterized. It is shown that only linear combinations of tensor products of univariate functions are possible for f( x, y, z). Second, we obtain the most general families of surfaces in explicit form such that their intersections with planes parallel to the planes y 5 0 and x 5 0 belong to two, not necessarily equal, parametric families of curves. Finally, functional equations are used to analyze the uniqueness of representation of GordonCoons surfaces. Some practical examples are used to illustrate the theoretical results. Categories and Subject Descriptors: 1.3.5 [Computer Graphics]: Computational Geometry and Object Modeling General Terms: Design Additional Key Words and Phrases: Explicit equations, functional equations, geometric modeling, implicit equations, parametric representation, surfaces, surface representations, tensor product surface
1. INTRODUCTION Properties of cross-sections of surfaces or intersections with planes parallel to the coordinate planes are very important in many applied fields. For example, in CAGD, many objects are usually designed by defining a number of cross-sections and the shape of the surface between them must be determined by using some interpolation scheme, generally through one or more longitudinal cross-sections or profiles (see Faux and Pratt [1990] and references therein). On the other hand, some methods of representing an algebraic curve require the intersection of a surface z 5 F( x, y) with the plane z 5 0 (see, e.g., Hoschek and Lasser [1993, p. 496] and references therein).
Authors’ address: Department of Applied Mathematics and Computational Sciences, University of Cantabria, Avenida de los Castros, s/n, 39005 Santander, Spain. Permission to make digital / hard copy of part or all of this work for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and / or a fee. © 1997 ACM 0730-0301/97/0700 –0296 $03.50 ACM Transactions on Graphics, Vol. 16, No. 3, July 1997, Pages 296 –318.
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Bathymetry (contour lines) of the Santa Marina beach in Spain.
As shown by Farin [1987, p. 304], in scientific computing the contour lines (obtained through intersections between a number of parallel planes and a given surface) are often of great importance. For example, Figure 1 shows the bathymetry (contour lines) of the Santa Marina beach (Spain). Some algorithms for obtaining such contour lines have been studied by Petersen [1983] for the case of three- and four-dimensional surfaces and even generalized to the case of an adaptive contouring of a trivariate interpolant. Efficient procedures for completely tracing the intersection of a plane with rational parametric surfaces have been analyzed by Chandru and Kochar [Farin 1987, pp. 305–318], and in the same line, Farouki [1986] considers a generalization of this problem using algebraic surfaces of low degree instead of a plane. Other computational methods for sectioning can be found in Hoitsma and Roche [1983] and Lee and Fredericks [1984]. Similarly, in the medical area it has been shown that it is possible to reconstruct and display the external surface of the organ under investigation from a set of parallel slices corresponding to different levels (see Ekoule et al. [1991] and references). Surface reconstruction from planar cross-sections is also analyzed by Boissonat [1985] with implications for pattern recognition and computer vision. In general, computing planar ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Fig. 2.
Bathymetry (perspective) of the Santa Marina beach in Spain.
intersections of geometric objects is a very important capability of CAD/ CAM and many other geometric-modeling systems (see Mortenson [1985]). In addition to this, in several areas of engineering, contouring techniques are used to represent a three-dimensional surface in two dimensions (see Anand [1993]). Contour plots are created as intersections of the surface with planes z 5 z 0 for different values of z 0 . Similarly, for nonparametric (i.e., implicit or explicit) surfaces this is usually done by drawing its intersections with planes x 5 x 0 and y 5 y 0 , for selected values of x 0 and y 0 . From these intersections, areas or volumes can be easily approximated. As a first example, Figure 2 shows a perspective of the Santa Marina beach sea bottom, corresponding to the contour lines in Figure 1, obtained by this method. Figure 3 shows a similar representation of an object made by six cylinders. In this context, surfaces such that the preceding intersections belong to the same families are very convenient, since all drawings reduce to the graphical representation of a single family of curves. In CAGD surfaces are dealt with in several forms: parametric, explicit, and implicit equations. The most common representation in the commercial software and research fields are parametric equations. This representation allows a quick computation of the coordinates of all points on a curve or surface, and its popularity depends, in part, on whether the design problem in surface modeling systems has been successfully addressed. Moreover, ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Fig. 3. Surface represented by drawing the intersections of all its components with planes x 5 x 0 and y 5 y 0 .
the parametric form can be used to define a curve segment or surface patch constraining the parameters to intervals. Because curves and surfaces are usually bound in computer graphics, this characteristic is of considerable importance. Nevertheless, in the last few years, explicit and implicit representations are being used more frequently in CAGD, allowing a better treatment of several problems. As one example, the point classification problem is easily solved with the implicit representation: it consists of a simple evaluation of the implicit functions. This is useful in many applications, such as solid modeling, for example, where points must be defined inside or outside the boundaries of an object (Hoffmann [1989]). Through implicit representation, the problem is reduced to a trivial sign test. Furthermore, the implicit representation offers surfaces of desired smoothness with the lowest possible degree. Finally, when we restrict ourselves to polynomial functions, the implicit representation is more general than the parametric representation (but probably not if we allow arbitrarily complicated functions) and several methods have been described in order to solve the problem of implicitizaACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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tion, that is, obtaining the implicit representation of rational surfaces (see, e.g., Sederberg [1983], Farin [1990], or Hoffmann [1989]). Although this conversion is always possible, difficulties arise when base points are present, but these problems are not insurmountable (see Chionh and Goldman [1992]). From the foregoing considerations, it is clear that identification of the most general families of surfaces with arbitrary planar cross-sections parallel to the coordinate axes belonging to given parametric and nonparametric families is an important problem. Our aim in this article is twofold: to obtain such characterizations and to show the power of functional equations in this process of characterization. The article is structured as follows. We start in Section 2 by giving some elementary notions of functional equations and some tools to be used later in the article. In Section 3 we apply functional equations to characterize the most general surfaces in implicit form, such that their intersections with the planes z 5 z 0 , y 5 y 0 , and x 5 x 0 are linear combinations of sets of given functions of the other two variables. In Section 4 we find the most general surfaces in explicit form such that their intersections with planes parallel to the planes y 5 0 and x 5 0 belong to given parametric families of curves. Finally, in Section 5 we use functional equations to analyze the uniqueness of representation of Gordon-Coons surfaces. 2. SOME BASIC CONCEPTS OF FUNCTIONAL EQUATIONS In this section, for those not familiar with this type of equation, we give a collection of basic results of functional equations to be used later. For a detailed study of functional equations the reader is referred to Acze´l [1966] or Castillo and Ruiz-Cobo [1992]. A functional equation can be considered as an equation in which the unknowns are functions. To distinguish functional equations from differential or integral equations, which also have functions as unknowns, we exclude equations containing infinitesimal operations. As an illustrative example we consider Pexider’s main equation
f~ x 1 y! 5 g~ x! 1 h~ y!;
x, y [ R,
(1)
where f(.), g(.), and h(.) are the unknown functions, which are assumed to belong to the class of continuous functions, and the equation is considered to hold in R (domain of the functional equation). The most general system of solutions of (1) is
f ~ x ! 5 Ax 1 B 1 C;
g ~ x ! 5 Ax 1 B;
h ~ x ! 5 Ax 1 C,
(2)
where A, B, and C are arbitrary constants. No other functions satisfy the preceding equation. As a second example, the following theorem gives the general solution of n equations of the form ( k51 f k ( x) g k ( y) 5 0, which are used in Sections 3, 4, and 5. ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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THEOREM 1 [Acze´l [1966, p. 160] or Castillo and Ruiz-Cobo [1992, p. 52]]. All solutions of the equation
O f ~ x ! g ~ y ! 5 0, n
f~ x! z g~ y! 5
k
k
(3)
k51
where f( x) 5 ( f 1 ( x), . . . , f n ( x)), g( y) 5 ( g 1 ( y), . . . , g n ( y)), and z is used to denote the dot product of two vectors, can be written in the form
f~ x! 5 w~ x!A g ~ y ! 5 c ~ y ! B,
(4)
where w( x) 5 ( w 1 ( x), . . . , w r ( x)), c( y) 5 ( c r11 ( y), . . . , c n ( y)), r is an integer between 0 and n, { w 1 ( x), . . . , w r ( x)} and { c r11 ( x), . . . , c n ( x)} are two arbitrary systems of linearly independent functions, and A and B are constant matrices, which satisfy
AB T 5 0.
(5)
COROLLARY 1. Let {u 1 ( x), . . . , u I ( x)} and {v 1 ( y), . . . , v J ( y)} be two linearly independent sets of known functions; then the solution of the functional equation
a~ y! z u~ x! 5 b~ x! z v~ y!,
(6)
where {a1( y), . . . , aI( y)} and {b1( x), . . . , bJ( x)} are the unknown functions, is
a ~ y ! 5 v ~ y ! D;
b ~ x ! 5 u ~ x ! D T,
(7)
where D is an arbitrary constant matrix. PROOF.
Expression (6) is equivalent to
~ a ~ y !u 2 v ~ y !! z ~ u ~ x !u b ~ x !! 5 0. Thus, according to Theorem 1, we have
~ u ~ x !u b ~ x !! 5 u ~ x !~ I Iu C ! ~ a ~ y !u 2 v ~ y !! 5 v ~ y !~ D u 2 I J! , where C and D are I 3 J and J 3 I constant matrices, respectively, such that
~ I Iu C !~ D u 2 I J! T 5 0 N D T 5 C.
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It is worthwhile mentioning some characteristic features of functional equations. (1) The general solution of a functional equation is strongly dependent on the domain of its definition (C, R, Q, etc.) and the class of functions (arbitrary, monotonic, continuous, differentiable, etc.) being considered. (2) In addition to arbitrary constants, as in Equation (2), arbitrary functions, as in Equation (4), can appear in the general solution. (3) Unlike any other kind of equation, from a single equation we can determine several unknown functions (see, e.g., Equations (1) and (2), or (3) and (4) and (5)). The following theorem is used later. THEOREM 2 [Castillo and Galambos [1987]]. The general nonnegative and integrable solution of the functional equation
z ~ x, y ! 5 ~ a 1~ y ! x 1 a 2~ y !! a3~ y! 5 ~ b 1~ x ! y 1 b 2~ x !! b3~ x!,
(8)
where a 1 (.), a 2 (.), a 3 (.) and b 1 (.), b 2 (.), b 3 (.) are unknown functions, is
z ~ x, y ! 5 @ C ~ x 2 A !~ y 2 B ! 1 D # E z ~ x, y ! 5 E ~ x 2 A ! C~ y 2 B ! D exp @ M log ~ x 2 A ! log ~ y 2 B !# , which depend on five and six parameters, respectively. For the proof see the preceding reference. 3. FAMILIES OF IMPLICIT SURFACES In this section we deal with the case of surfaces in implicit form. We look for the most general surfaces in implicit form such that their planar cross-sections parallel to the coordinate planes satisfy some conditions. This allows the designer to choose families of surfaces with cross-sections at his or her convenience. The following theorem gives these conditions in a precise form and the corresponding solution. THEOREM 3. The most general family of implicit surfaces f( x, y, z) 5 0, such that their intersections with the planes z 5 z 0 , y 5 y 0 , and x 5 x 0 are linear combinations of sets 8 5 {u 1 ( y, z), u 2 ( y, z), . . . , u I ( y, z)}, 9 5 {v 1 ( z, x), v 2 ( z, x), . . . , v J ( z, x)} and 0 5 {w 1 ( x, y), w 2 ( x, y), . . . , w K ( x, y)}, of functions of the other two variables, is of the form
O O O @C I
f ~ x, y, z ! 5
J
K
ijk
a i~ x ! b j~ y ! g k~ z !# ,
(9)
i51 j51 k51
where a( x), b( y) and g( z) are vectors of arbitrary functions and C ijk are the elements of an arbitrary constant matrix C. ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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In addition, the 8, 9, and 0 functions cannot be arbitrary, but of the form
O O @C J
u i~ y, z ! 5
K
ijk
b j~ y ! g k~ z !# ;
ijk
ai ~ x!gk ~ z!#;
j 5 1, . . . , J,
(11)
ai ~ x!bj ~ y!#;
k 5 1, . . . , K.
(12)
i 5 1, . . . , I,
(10)
j51 k51
O O @C I
vj ~ z, x! 5
K
i51 k51
O O @C I
wk ~ x, y! 5
J
ijk
i51 j51
PROOF. According to the preceding assumptions, we look for surfaces f( x, y, z) 5 0 such that they satisfy the system of functional equations
O a ~ x ! u ~ y, z ! 5 O b ~ y ! v ~ z, x ! I
f ~ x, y, z ! ;
J
i
i
j
i51
O b ~ y!v ~ z, x! 5 O g ~ z!w ~ x, y!, J
f~ x, y, z! ;
j
j51
K
j
j
k
j51
k
(13)
k51
where the sets { a i ( x); i 5 1, . . . , I}, { b j ( y); j 5 1, . . . , J}, and { g k ( z); k 5 1, . . . , K} can be assumed without loss of generality as sets of linearly independent functions. Note that if they are not, we can rewrite equations in (13) in the same form but with linearly independent sets. The system of equations (13) states the conditions for the sets 8, 9, and 0 to be compatible with the sets { a i ( x); i 5 1, 2, . . . , I}, { b j ( y); j 5 1, 2, . . . , J}, and { g k ( z); k 5 1, 2, . . . , K}. For any fixed z, the first equation in (13) is of the form of Equation (6) and, according to Corollary 1 we have (see (7))
u ~ y, z ! 5 b ~ y ! A T~ z !
(14)
v ~ z, x ! 5 a ~ x ! A ~ z ! ,
(15)
where A( z) is a matrix whose elements are functions of z. Replacing Equation (14) or (15) in (13) we get
O O A ~ z!a ~ x!b ~ y!. I
f ~ x, y, z ! 5 a ~ x ! A ~ z ! b ~ y ! 5 T
J
ij
i
j
i51 j51
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Similarly, for any fixed x, the second equation in (13) leads to
v ~ z, x ! 5 g ~ z ! B T~ x ! w ~ x, y ! 5 b ~ y ! B ~ x ! .
(16)
Now from Equations (15) and (16) we obtain for each j 5 1, . . . , J,
O A ~ z!a ~ x! 5 O B ~ x!g ~ z!, I
v j~ z, x ! 5
K
ij
i
jk
i51
k
k51
which is also of the form given in Corollary 1, and then we can write
OOC I
v j~ z, x ! 5
K
~2! ijk
a i~ x ! g k~ z ! ,
i51 k51
(2) where C ijk are the elements of a constant matrix. It is clear that similar expressions can be obtained for u i ( y, z) and w k ( x, y). Now replacing this into Equation (13), we get
OOO I
f ~ x, y, z ! 5
J
K
i51 j51 k51
O O O @C I
f ~ x, y, z ! 5
J
O O O @C
~3! ijk
a i~ x ! b j~ y ! g k~ z !#
O O O @C
~1! ijk
a i~ x ! b j~ y ! g k~ z !#
I
~2! @ C ijk a i~ x ! b j~ y ! g k~ z !# 5
J
K
i51 j51 k51
K
I
~3! ijk
a i~ x ! b j~ y ! g k~ z !# 5
i51 j51 k51
J
K
i51 j51 k51
(17) but, taking into account that the preceding sets of functions are linearly independent, we get ~1! ~2! ~3! C ijk 5 C ijk 5 C ijk 5 C ijk ,
which together with (17) leads to (9).
e
Remarks: (1) Note that no constraints have been imposed on the f( x, y, z) functions. Thus an arbitrary class of functions has been assumed. (2) The implicit equation of the surface (9) is a linear combination of the tensor product of the sets of functions in a( x), b( y), and g( z). (3) Algebraic surfaces are particular cases of this family. ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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(4) The only functions u i ( y, z), v j ( z, x), and w k ( x, y) satisfying condition (13) are of the form (10) through (12), where a(.), b(.), and g(.) are the function coefficients in (13). (5) Note that the functional equations are imposed not only on a fixed number of given cross-sections, but on an arbitrary planar section parallel to the coordinate planes. (6) Functional equations allow characterizing the functional form of the solution from a simple compatibility condition. 4. FAMILIES OF EXPLICIT SURFACES In this section we look for the most general surface in explicit form, z 5 z( x, y), such that their intersections with planes parallel to the planes y 5 0 and x 5 0 belong to given (not necessarily equal) parametric families of curves, that is,
z 5 z ~ x, y ! 5 h ~ x, a 1~ y ! , a 2~ y ! , . . . , a k~ y !! z 5 z ~ x, y ! 5 r ~ y, b 1~ x ! , b 2~ x ! , . . . , b m~ x !! .
(18)
This leads to the functional equation
h ~ x, a 1~ y ! , a 2~ y ! , . . . , a k~ y !! 5 r ~ y, b 1~ x ! , b 2~ x ! , . . . , b m~ x !! ,
(19)
where we assume that a 1 (.), a 2 (.), . . . , a k (.) and b 1 (.), b 2 (.), . . . , b m (.) are unknown functions to be determined and h(.) and r(.) are known functions, which are selected in order to have convenient cross-sections. Note that a given surface is completely defined by one of the two equations in (18). For two different representations (the two equations in (18)) to correspond to the same surface they must satisfy some conditions. Thus the functional equation (19) plays the role of a compatibility condition. In the following sections we analyze different cases of h(.) and r(.) parametric families of curves. n 4.1 Explicit Surfaces of the Form z 5 s(( i51 p i (x)q i (y))
In this section we take h( x, a 1 , a 2 , . . . , a k ) 5 ( ki51 a i u i ( x) and r( y, b 1 , b 2, . . . , b m) 5 ( m i51 b i v i ( y), and we solve the associated problem by means of the following theorem. THEOREM 4. The most general surface in explicit form, z 5 z( x, y), such that all sections with planes parallel to the planes y 5 0 and x 5 0 are linear combinations of given sets of linearly independent functions 8 5 {u 1 (.), u 2 (.), . . . , u k (.)} and 9 5 {v 1 (.), v 2 (.), . . . , v m (.)}, respectively, is
z 5 z ~ x, y ! 5 v ~ y ! Au T~ x ! ,
(20)
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where A is an arbitrary constant matrix. Thus we obtain a linear combination of the tensor products of the vectors of functions u( x) and v( y). PROOF.
According to the preceding assumptions, we must have
z ~ x, y ! 5 a ~ y ! z u ~ x ! z ~ x, y ! 5 b ~ x ! z v ~ y ! ,
(21)
where u( x) 5 (u 1 ( x), . . . , u k ( x)) and v( y) 5 (v 1 ( y), . . . , v m ( y)) are known and a( y) 5 ( a 1 ( y), . . . , a k ( y)) and b( x) 5 ( b 1 ( x), . . . , b m ( x)) are to be determined. Expressions (21) imply
a~ y! z u~ x! 5 b~ x! z v~ y!, which is a functional equation of the form given in Corollary 1. Thus its general solution is
a~ y! 5 v~ y!A b ~ x ! 5 u ~ x ! A T, where A is an arbitrary (m 3 k) constant matrix. Thus the explicit equation of the parametric family of surfaces becomes (20). e Since any explicit surface z 5 z( x, y) can be written as an equivalent implicit surface f( x, y, z) 5 z( x, y) 2 z, the preceding theorem can be obtained as a particular case of Theorem 4, as follows. According to (9) we can write
O O O @C I
f ~ x, y, z ! 5
J
K
ijk
a i~ x ! b j~ y ! g k~ z !# 5 z ~ x, y ! 2 z,
i51 j51 k51
and making z 5 z 0 we obtain
O O a ~ x!b ~ y! O @C I
J
K
i
i51 j51
j
ijk
g k~ z 0!# 5 z ~ x, y ! 2 z 0 ,
k51
and then we get
O O C9 a ~ x ! b ~ y ! , I
z ~ x, y ! 5 z 0 1
J
ij
i51 j51
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where
O @C K
C9ij 5
ijk
g k~ z 0!# ,
k51
which is equivalent to (20). COROLLARY 2.
If, instead of (21), we consider
z ~ x, y ! 5 s ~ a ~ y ! z u ~ x !! z ~ x, y ! 5 s ~ b ~ x ! z v ~ y !! , where s(.) is an invertible function, the general solution for z( x, y) becomes
z ~ x, y ! 5 s ~ v ~ y ! Au T~ x !! .
(22)
This corollary can be directly proved by applying Theorem 4 to the explicit function z 5 s 21 ( z( x, y)). So, given u( x) and v( y), Equation (22) defines its associated parametric family of surfaces, and any surface from this family has an associated matrix A. The intersections of two surfaces of this family, with parameters defined by matrices A1 and A2, have as projection on the plane z 5 0 the curve
v ~ y !~ A 1 2 A 2! u T~ x ! 5 0,
(23)
which together with (22) allows the drawing of the curve. The intersections of these surfaces with planes z 5 z 0 have as projections on the plane z 5 0, the curve
v ~ y ! Au T~ x ! 5 z 0 .
(24)
Example 1 (Nonparametric Bicubic Tensor Product Surfaces). If the surface z 5 z( x, y) is such that all sections with planes parallel to the planes y 5 0 and x 5 0 are third-degree polynomials, we have s(w) 5 w,
~ u 1~ x ! , u 2~ x ! , u 3~ x ! , u 4~ x !! 5 ~ 1, x, x 2, x 3!
and
~ v 1~ x ! , v 2~ x ! , v 3~ x ! , v 4~ x !! 5 ~ 1, y, y 2, y 3! . Then, according to (22), the explicit equation of the surface becomes
z 5 z ~ x, y ! 5 ~ 1, y, y 2, y 3! A ~ 1, x, x 2, x 3! T.
(25)
It is obvious that intersections of the preceding surface by planes z 5 z 0 are algebraic curves (see (24)). Similarly, (23) shows that the projection on ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Fig. 4.
Set of points defining the surface.
the plane z 5 0 of the intersections of two surfaces of this family are algebraic curves too. The family (25) depends on 16 parameters and then one surface can be forced to pass through 16 given points. Figure 4 shows an example where the 16 points have been selected in such a way that certain sets of 4 points belonging to the planes x 5 i and y 5 j with i 5 1, . . . , 4; j 5 1, . . . , 4. The figure also shows the polynomial curves passing through every 4 points in each of these sets. Figure 5 shows the interpolating surface. Now we force all vertical plane intersections to be third-degree polynomials in X or Y. Making y 5 px 1 q or x 5 ry 1 t in (25) and canceling coefficients of x and y of degree larger than 3, we get
a 44 5 a 43 5 a 34 5 a 42 5 a 33 5 a 24 5 0. Thus the new family becomes
z 5 z ~ x, y ! 5 ~ 1, y, y 2, y 3!
1
a 11 a 12 a 13 a 14 a 21 a 22 a 23 0 a 31 a 32 0 0 a 41 0 0 0
21 2 1 x x2 x3
.
Example 2 (Some Exponential Families). Now we consider the case s(w) 5 exp(w), (u 1 ( x), u 2 ( x), u 3 ( x)) 5 (1, x, log( x)) and (v 1 ( y), v 2 ( y), v 3 ( y)) 5 (1, y, y 2 ). Then we get the family of surfaces
z ~ x, y ! 5 exp @~ 1, y, y 2! A ~ 1, x, log ~ x !! T# , which depends on 9 parameters. The intersections of two surfaces of this family, with associated parameters A1 and A2, have the following projection on the plane z 5 0
~ 1, y, y 2!~ A 1 2 A 2!~ 1, x, log ~ x !! T 5 0. ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Interpolating surface.
As one example, in Figure 6 we show the surface with equation
z ~ x, y ! 5 exp @ 2x ~ 1 1 y 1 y 2! 1 log ~ x !~ 1 1 y/ 2 !# . 4.2 Some Other Surfaces In this section we illustrate the power of the method giving two more particular examples of (18). We take
h ~ x, a 1 , a 2 , a 3! 5 s ~~ a 1x 1 a 2! a3! r ~ y, b 1 , b 2 , b 3! 5 s ~~ b 1y 1 b 2! b3!; that is,
z ~ x, y ! 5 s @~ a 1~ y ! x 1 a 2~ y !! a3~ y!# z ~ x, y ! 5 s @~ b 1~ x ! y 1 b 2~ x !! b3~ x!#, which leads to the functional equation (8). Theorem 2 gives the two families of solutions of (8) and Figures 7 and 8 show the two surfaces associated with the explicit equations z 5 1 2 exp(2( xy 2 1) 2 ) and z 5 exp(2( xy 2 1) 2 ), respectively. Their crossplanar sections by planes of the form x 5 x 0 and y 5 y 0 are obvious. It is worthwhile pointing out that in spite of the general form of the functional equation (8) the only feasible solutions are parametric families; ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Fig. 6.
One surface from the exponential family.
that is, the functional form of the solutions is determined by the compatibility condition (8).
5. GORDON-COONS SURFACES Functional equations are equally useful for proving certain properties of surfaces in parametric form. In this section we prove a uniqueness theorem for this type of surface. We start by introducing them and the basic notation using the Gordon approach (see Gordon [1993, pp. 117–134]). Consider the following problem. Assume two families of parametric curves
$ g i~ t !u i 5 1, 2, . . . , M %
and
$ f j~ s !u j 5 1, 2, . . . , N % ,
where
g i~ t ! 5
1
g i~1!~ t ! g i~2!~ t ! g ~i3!~ t !
2
;
f j~ s ! 5
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f j~1!~ s ! f j~2!~ s ! f ~j3!~ s !
2
,
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Fig. 7.
•
311
Surface z 5 1 2 exp (2( xy 2 1) 2).
and the superindices 1, 2, and 3 of the g and f functions refer to the x, y, and z components, respectively. All functions gi (t) and fj (s) have to be defined on common parameter intervals [t# 0 , t# 1 ] and [s# 0 , s# 1 ], respectively. These curves intersect in a set of space points whose coordinates are obtained for some values of the parameters t 1 , t 2 . . . , t N and s 1 , s 2 , . . . , s M . For these two families to define a surface they must satisfy the following compatibility conditions,
u ij 5 g i~ t j! 5 f j~ s i! . Then it is possible to build a surface v(s, t) interpolating the M 1 N given curves, that is, satisfying the system of vector equations
v ~ s i , t ! 5 g i~ t ! ,
~ i 5 1, 2, . . . , M !
v ~ s, t j! 5 f j~ s ! ,
~ j 5 1, 2, . . . , N ! . ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Fig. 8.
Surface z 5 exp (2( xy 2 1) 2).
Thus the problem of interpolating a surface through an intersecting skeletal network of three-dimensional curves can be reduced to the following scalar-value problem. Construct a bivariate function v(s, t) that interpolates the M 1 N univariate functions {gi (t)ui 5 1, . . . , M}, {fj (s)u j 5 1, . . . , N }. To solve this problem, Gordon, in the end of the 1960s and 1970s, proposed the solution given by the following theorem. THEOREM 5. If { f i (s)ui 5 1, 2, . . . , M} and { c j (t)u j 5 1, 2, . . . , N } are any two sets of functions such that they satisfy the conditions
f i~ s k! 5 d ik
(26)
c j~ t k! 5 d jk ,
(27)
where d ij is the Kronecker d , and if {gi (t)ui 5 1, 2, . . . , M} and {fj (s)u j 5 1, 2, . . . , N } are any two sets of functions such that the compatibility conditions are satisfied:
g i~ t j! 5 f j~ s i! , ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
i, j.
(28)
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Then the bivariate function
O g ~t!f ~s! 1 O f ~s!c ~t! 2 O O u f ~s!c ~t! M
v ~ s, t ! 5
N
i
i
i51
M
j
N
j
ij
j51
i
j
(29)
i51 j51
is one solution of the interpolation problem
v ~ s k , t ! 5 g k~ t ! ,
~ k 5 1, 2, . . . , M !
v ~ s, t p! 5 f p~ t ! ,
~ p 5 1, 2, . . . , N ! .
Note that (29) gives v(s, t) as a function of the three surfaces
O g ~t!f ~s!, M
v 1~ s, t ! 5
i
i
i51
which interpolates the family of curves {gi (t)ui 5 1, 2, . . . , M},
O f ~s!c ~t!, N
v 2~ s, t ! 5
j
j
j51
which interpolates the family of curves {fj (s)u j 5 1, 2, . . . , N }, and
O O u f ~s!c ~t!, M
v 3~ s, t ! 5
N
ij
i
j
i51 j51
which interpolates the set of points {uij ui 5 1, 2, . . . , M; j 5 1, 2, . . . , N }. Note that it is again a tensor product. Gordon-Coons surfaces (see Gordon [1993] and Coons [1964]) are frequently referred to as transfinite interpolants [Farin 1990, p. 381; Hoschek and Lasser 1993, p. 371], since they interpolate continuous data, that is, all points along the prescribed boundary curves. 5.1 Uniqueness Theorem For the surface family defined by Equation (29), the uniqueness of representation is analyzed; that is, we try to find out whether there is only one representation for such a surface. To be more precise we give the following theorem. THEOREM 6.
Given the sets of linearly independent functions
$ f i~ s !u i 5 1, 2, . . . , M %
and
$ c j~ t !u j 5 1, 2, . . . , N % ,
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there exist unique sets of functions
$ g i~ t !u i 5 1, 2, . . . , M %
$ f j~ s !u j 5 1, 2, . . . , N %
and
satisfying the compatibility conditions
u ij 5 g i~ t j! 5 f j~ s i! , such that the surface v(s, t) is given by (29). PROOF. We use functional equations for this proof. Let {gi (t), fj (s)} and {g*i (t), f *j (s)} be two such representations satisfying the same compatibility conditions
u ij 5 g i~ t j! 5 f j~ s i! 5 g *i~ t j! 5 f *j~ s i! ;
(30)
then we have
O g ~t!f ~s! 1 O f ~s!c ~t! 5 O M
N
i
M
i
j
i51
j
j51
O f *~ s ! c ~ t ! ; N
g *i~ t ! f i~ s ! 1
i51
j
j
j51
that is, the following functional equation holds
O @ g ~ t ! 2 g *~ t !# f ~ s ! 1 O @ f ~ s ! 2 f *~ s !# c ~ t ! ; 0. M
N
i
i
i
j
i51
j
j
j51
Its solution is given by (see Corollary 1)
O A c ~ t ! f g ~ t ! 5 O A c ~ t ! 1 g *~ t ! ; N
g i~ t ! 2 g *i~ t ! 5
N
ij
j
i
ij
j51
i
O A f ~ s ! f f ~ s ! 5 2 O A f ~ s ! 1 f *~ s ! ; M
f j~ s ! 2 f *j~ s ! 5 2
j
i
j51
M
ij
i
j
i51
ij
i
j
j,
i51
where A is a matrix of constants. Returning now to the compatibility conditions (30) and using definition (27), we get
O A c ~t ! 1 g*~t ! f u N
uij 5 gi ~tj ! 5
ik
k
j
i
j
ij
5 Aij cj ~tj ! 1 uij 5 Aij 1 uij ;
K51
which is the desired uniqueness.
e
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5.2 A Particular Case In this section we analyze the particular case of surfaces in which the sets of functions {gi (t)} and {fj (s)} are linear combinations of the sets { c j (t)} and { f i (s)}, respectively. In this case, the surface v(s, t) appears as a very simple form, as indicated by the following theorem. THEOREM 7. If the conditions of the preceding theorem hold and the two sets of functions
$ g i~ t !u i 5 1, 2, . . . , M %
and
$ f j~ s !u j 5 1, 2, . . . , N %
are linear combinations of the sets of functions
$ c j~ t !u j 5 1, 2, . . . , N %
and
$ f i~ s !u i 5 1, 2, . . . , M % ,
that is,
g 5 Ac;
f 5 Bf,
(31)
then A 5 BT 5 U 5 {uij ui 5 1, . . . , M; j 5 1, . . . , N }. PROOF. form as
The surface v(s, t) satisfies (29), which can be written in matrix
v 5 f Tg 1 f Tc 2 f TU c .
(32)
Substitution of (31) in (32) leads to
v 5 f TA c 1 f TB Tc 2 f TU c 5 f T~ A 1 B T 2 U ! c and then
v ~ s i , t j! 5 g i~ t j! 5 f j~ t i! 5 f T~ s i!~ A 1 B T 2 U ! c ~ t j! .
(33)
Now using the definitions (26) and (27) we get
u ij 5 v ~ s i , t j! 5 A ij 1 B ji 2 U ij .
(34)
Applying the same definitions (26) and (27) to Equations (31) we get
g i~ t j! 5 ~ g ~ t j!! i 5 ~ A c ! i 5 A ij
(35)
f j~ s i! 5 ~ f ~ s i!! j 5 ~ B f ! j 5 B ji .
(36)
and
Now, inserting (34), (35), and (36) into the compatibility conditions (28), we immediately have
U ij 5 A ij 5 B ji , ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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which proves the theorem.
e
Note that in this case we have not only uniqueness of representation that holds under more general conditions, but we know the form of the matrix U. Note also that the form of the surface v(s, t) is extremely simple. In fact, for this particular case, the three terms in expression (29) coincide, that is,
v ~ s, t ! 5 v 1~ s, t ! 5 v 2~ s, t ! 5 v 3~ s, t ! . These particular surfaces are usually called tensor product surfaces in CAGD. Consequently, for interpolating the surface v(s, t) it is enough to interpolate any one of the families of curves {gi (t)} or {fj (s)}, or to interpolate the set of points {uij }. Adequate selection of the sets of functions {gi (t)} and {fj (s)} can simplify the problem of determining the intersection curves of two different surfaces, as illustrated in the following example. Example 3 (Parametric Form). In this example we build four parametric surfaces in parametric form. For the first three surfaces (spheres) we have used the base curves
g i~ t ! 5
1
a sin ~ t ! cos ~ s i! a sin ~ t ! sin ~ s i! a cos ~ t ! 1 b
2
;
f i~ s ! 5
1
a sin ~ t j! cos ~ s ! a sin ~ t j! sin ~ s ! a cos ~ t j! 1 b
2
with a 5 1, b 5 0, a 5 0.75, b 5 0.75 and a 5 0.5, b 5 1.5, respectively. For the fourth surface (cylinder) we have taken the base curves
g i~ t ! 5
1
cos ~ s i! / 2 sin ~ s i! / 2 t21
2
;
f i~ s ! 5
1
cos ~ s ! / 2 sin ~ s ! / 2 tj 2 1
2
.
The parameters s and t belong to the intervals [0, 3p/2] and [0, p], respectively. The surfaces and their intersections are shown in Figure 9. Note that intersections of these surfaces can be easily determined, since the sets {fj (s)} coincide for all of them and they correspond to t 5 constant. 6. SUMMARY AND CONCLUDING REMARKS In this article we have applied functional equations to solve some problems of characterization, existence, and uniqueness of families of surfaces. We have characterized the most general surfaces in implicit form such that their intersections with the planes z 5 z 0 , y 5 y 0 , and x 5 x 0 are linear combinations of sets of given functions of the other two variables. It turned out that they are linear combinations of the tensor products of functions of each of the variables alone. Later, we have found the most general surfaces in explicit form such that their intersections with planes parallel to the ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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Four parametric surfaces and their intersections.
planes y 5 0 and x 5 0 belong to some given families of curves. Finally, we have used functional equations to analyze the uniqueness of representation of Gordon-Coons surfaces. Thus functional equations have been shown to be very powerful and elegant tools for solving these types of problems. ACKNOWLEDGMENTS
We thank the University of Cantabria and Direccio´n General de Investigacio´n Cientı´fica y Te´cnica (DGICYT) (project PB92-0504) for partial support of this work. We also thank the three referees and the Editors for some helpful discussions that improved the article. REFERENCES ´ , J. 1966. Lectures on functional equations and their applications, Vol. 19. MathematACZEL ics in Science and Engineering. Academic Press, New York, NY. ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
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ACM Transactions on Graphics, Vol. 16, No. 3, July 1997.
