A C1 surface scheme is described which interpolates points defined on an arbitrary mesh network. The scheme involves the blending of strip ‘functions’ developed from a rational spline method. The rational spline provides interval and point tension weights which can be used to control the shape of the surface scheme

Gregory, JA

Yuen, PK

English

Brunel University Research Archive

                 TR/14/91 October 1991 An Arbitrary Mesh Network Scheme using Rational Splines  by  John A. Gregory and P. K. Yuen     Submitted for publication in: Mathematical Methods in CAGD and Image Processing Tom Lyche and L L Schumaker (eds.)               (Per)      QA   1  B 78     z157747xAn Arbitrary Mesh Network Scheme using Rational Splines   John A. Gregory and P.K. Yuen Abstract. A C1 surface scheme is described which interpolates                points defined on an arbitrary mesh network. The scheme involves               the blending of strip ‘functions’ developed from a rational spline    method. The rational spline provides interval and point tension       weights which can be used to control the shape of the surface scheme.     §1. Introduction Most ‘free-form’ or ‘sculptured’ modelling systems for Computer Aided Design           are based on the representation of the surface as a complex of ‘rectangular            patches’. In this paper, however, we wish to give a progress report on some           research into the construction of a C1 surface which interpolates points defined               on an arbitrary 3-D mesh network. The construction is based on the blending                   of ‘strip functions’ defined on the mesh, which here are developed from a               rational spline method. The rational spline provides ‘interval’ and ‘point tension’ parameters to           control the shape of the strip functions and hence of the blended surface.                      The strip functions contain position and tangent plane information and are             blended together to form polygonal patches appropriate to the topological             structure of the mesh. We begin, in Section 2, by considering the topology and data structure                     of the mesh network, where for the purposes of this paper some simplifying    assumptions are made. The rational spline strip functions are developed in              Section 3 and the blending of these strip functions is considered in Section                       4. The paper is completed in Section 5 with some examples and concluding           remarks concerning the future direction of the research. Mathematical Methods in CAGD and Image Processing 1 Tom Lyche and L. L. Schumaker (eds.). 2 John A. Gregory and P.K. Yuen §2. The Mesh Network The mesh network is described by a set of 3-D directed polygonal lines whose            vertices axe a given set of interpolation points. The polygonal lines are as-                     sumed to intersect at certain of the vertices in such a way as to define a mesh              network of faces. The faces are topologically equivalent to either triangular,          rectangular or, in general, polygonal domains. Each edge of a face is defined                       by a signed subset of one of the polygonal lines, where the sign indicates                    whether it has the positive or negative direction of that line. The mesh network is assumed equivalent to an orientable surface (thus                  having an inside and outside) and the edges of the faces, together with their                 vertices, are labelled in an anti-clockwise order when viewed from the outside.                 The data structure thus consists of: • Polygonal Lines Ei,   i =  1, . . . , n, each  defined  by an ordered set of ver-                tices denoted by Pi,j ∈ IR3, j = 1, . . . , ni. • Faces Mi , i = l , . . . ,m ,  defined  by their anti-clockwise ordered edges,                 namely, signed subsets of the polygonal lines. It can be observed that the vertices divide into the two classes of face                   vertices and internal edge vertices. Here, we make the simplifying assumptions                 that the face and internal edge vertices form mutually exclusive sets and that                exactly two of the polygonal lines meet or intersect at any face vertex. (Thus                  every face vertex is labelled exactly twice with the above notation.) This                       allows the consistent definition of the strip functions of the following section,                  with respect to an appropriate local rectangular cartesian coordinate system.                         In particular, a consistent tangent plane will be defined at each face vertex.    Figure 2.1. Example of a mesh network. As a concrete example. consider the mesh network shown in Figure 2.1.                  Here, there are three directed polygonal lines Ei, i = 1,..., 6, through a                         total of fifteen vertices. The vertices decompose into a set of six face vertices            Arbitrary Mesh Network Scheme 3 and nine internal edge vertices, giving a set of four triangular faces Mi, i =                         1 , . . . ,4. (The edges of the face hidden by the persective view of the mesh                 network are shown with broken lines as an aid to understanding the three              dimensional figure.) §3. The Rational Spline Strip Functions We begin by constructing a set of rational spline curves, each associated with a               given polygonal line Ei, and interpolating the ordered vertices which define Ei.                  This will provide the basic design tool for the construction of the arbitrary                       mesh network surface, since the surface will be constrained to contain the                       spline network. The ordered vertices of each polygonal line decompose into face vertices                    and interior edge vertices. The ordered face vertices are associated with in-                 creasing integer knots and we choose to distribute the knots associated with                         the interior edge vertices between these integers, in proportion to their num-                        ber between the subsequent face vertices. This gives a knot vector which may                     not be appropriate for standard parametric cubic spline interpolation and we                      thus wish to provide shape control parameters which allow easy manipulation                      of the spline curve. This is achieved here by the use of a C2 rational cubic                       spline method, see [6], which provides interval and point tension control of the              curve. (An alternative choice would be the C1 weighted υ -spline described in                 Foley [2], which also provides interval and point tension control.) Given a set of values P, ∈  IR.3, i = 0,..., N, defined at knots si, i =                         0,..., N, with interval spacing hi := si+1 — si, > 0, then a parametric C1                                   piecewise rational cubic function p : [s0, sn ]→IR,3 is defined by  =+ :)(| ]1,[ sP ii ss .)1()1())(1()()1()1(2213223iiiiiiiiiiiii PWVPβθγθθαθβθβγθθαγθθαθ+−+−++−++−+− +  Here,  (3.1)  ,/)(:| ]1,[ iiss hssii −=+θ  (3.2) and  ,:,: 11 ++ +−++= iiiiiiiiiiiiii DhPWDhPV βγβαγα (3.3) where the scalar weights iα , iβ , iγ  are assumed positive and Di, i = 0,...,N                      denote first derivative values defined at the knots. Thus p has the Hermite            interpolation properties that  p(si) = Pi, and p(1)(si) = Di, i = 0,... ,N. (3.4) Suppose the ‘end conditions’ D0 and DN are given. Then the unknowns                         Di, i = 1,..., N can be found by imposing the C2 consistency constraints  P(2) (si+) = P(2)( ), i = l , . . . ,N – l .  (3.5) −is4 John A. Gregory and P.K. Yuen These constraints lead to a tridiagonal system of equations in the unknowns,                   which is diagonally dominant if γi , >αi, and γi, > βi, i = 1,..., N – 1. We                       thus have a sufficient condition for the existence of a unique, easily computed             solution to the consistency constraints. The scalar weights of the piecewise rational form provide shape control           parameters for the C2 curve which results from this interpolatory process, see                     [6], where a local support basis representation is also discussed. In particular,                      we choose  βi-1  = αi  : =l/ λi, (3.6) giving weights λi, i = 0,. . . ,N ,  and γi, i = 0,... ,N – 1, for the rational                         spline scheme, where the diagonal dominance condition is now γi, iλ  > 1, i =                   l,...,N – 2. The weight iλ  is called a point tension parameter, associated with the                        knot si, since a tangent discontinuity is introduced into the spline curve at Pi                        as iλ ∞→ . The weight γi is called an interval tension parameter, associated                     with the interval [si,,si+i], since the curve is tightened to the line segment                       joining Pi, Pi+1 as γi, . These properties are illustrated in Figure 3.1,                        ∞→where the cubic spline interpolant of the first curve is given by the use of the                 default parameters iλ := 1, i = 0,..., N, and γi := 2, i = 0,..., N – 1.  Figure 3.1. Tension behaviour of the rational spline.. Application of the rational spline method to the vertex data along each              polygonal line Ei of the arbitrary mesh network, using the previously defined                    knot distribution, results in a network of spline curves. These curves define a                    wire frame model of a surface which has the same topological structure as the                  mesh network and the shape of the wire frame model can be manipulated by                varying the control parameters which govern the shape of each curve. At each                   face vertex of the mesh network there is a unique tangent plane for the wire                    frame surface, defined by the derivative parameters (i.e. tangent vectors) of                        the two spline curves which meet or intersect at that vertex. In particular, atArbitrary Mesh Network Scheme 5 any integer knot of a rational spline curve, a ‘cross boundary’ tangent vector                         is defined. We now define a bivariate ‘strip function’ extension of the rational spline            method on [s0,sN] × [ε ,ε ], for someε  > 0, by the Taylor form  Pext(s,t):=p(s) + tpt,(s), (3.6) where pt(s) denotes the piecewise cubic Hermite interpolant to a set of cross              boundary tangent and ‘twist’ vectors, defined on a subset of the rational                         spline knots. Here, the cross boundary tangent denotes a derivative along the                        t direction and the twist vector denotes a (1,1) derivative parameter along the                        s and t directions. The twist vectors can be considered as derivative unknowns                      to be determined by imposing C2 constraints across the knot subset. In this                        case pt (s) is the cubic spline interpolant to the cross boundary tangent vectors,               where we assume that appropriate end conditions have been imposed. It can                        be observed that  pext(s,0) = p(s) and ∂ t pext(s,0) = pt(s), (3.7) and hence the strip function extension contains position and tangent plane             information about the rational spline curve. The rational spline extension method is considered along each curve of the                wire frame model, where the cross boundary tangents are given on integer sub-                  sets of the spline knots. Since these cross boundary tangents are defined with                respect to the local parameterizations of their corresponding spline curves, it                        is necessary to change certain of their signs for them to be consistent with                         the t parameterization on the strip function domain. This sign information                         can be obtained from the face table data structure. The twist vectors for                         the rational spline extensions are initially determined by imposing C2 spline            constraints on the piecewise cubic Hermite cross boundary tangent function,                       see above. This will give incompatible twist vector values at each face vertex,   corresponding to the two rational spline extensions which meet or intersect at                     that vertex. This incompatibility must be allowed for in the construction of                         the surface scheme of the following section or, alternatively, the strip functions                  can be redefined by taking the average of the two twist vector parameters at                      each face vertex. In either case, the surface model is now one of rational spline                 strip functions which can be blended to form a C1 surface. §4. Blending the Strip Functions on Polygonal Domains Consider a face of the arbitrary mesh network. At a face vertex, two edges                        meet and along each edge a piece of strip function is defined, consistent with                         a C1 surface. More precisely, our definition and parameterization of the strip             functions implies that the two strip function pieces can be written in terms                         of a (u, v) rectangular cartesian coordinate system about each vertex as  p(0,v) +upu(0,v), 0 u 1,   and p(u,0) + vp≤ ≤ v(u,0), 0≤  v ≤  l, (4.1) 6 John A. Gregory and P.K. Yuen cf (3.6). Here, after appropriate simple affine transformations, p(0, v)                         and p(u,0) represent the pieces of the rational spline curves, and pu(0,v) and                  pu(u,0) represent the cross boundary tangent functions, between the face                      vertices of each edge. Moreover, these Taylor functions are consistent with a                       C1 surface p(u, v). In particular, a C1 extension of the two strip functions                         is defined for (u,v)∈  [0, l]2 by the ‘rationally corrected’ Boolean sum Taylor         interpolant, see Gregory [3],  p(u, v) := p(0, v) + upu (0, v) + p(u, 0) + vpv (u, 0)   (4.2) ,1),()0,0()0,0()0,0(]1[⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡−vvutpppuuuwhere  ,)0,0()0,0(:),( ,,vuvPuPvut vuuv ++=  (4.3) and pu,v(0,0) := pvu∂∂ (0,0), pv,u (0,0) := pvu∂∂ (0,0) denote the twist vec-                         tors at the vertex. If pu,v(0,0) p≠ u,v (0,0), the rational term introduces                     singularities which are removable to first order at the vertex. If pu,v (0,0) =                 pu,v(0,0), then  t(u,v) := pu,v (0,0), (4.3) giving a standard Boolean sum form. Now each face of the mesh network is topologically equivalent to a polyg-                onal domain and the final surface is constructed by blending the Boolean sum               Taylor interpolants on each of these polygons in the following manner. This                follows the approach of Charrot and Gregory [1], see also [3] and [5]. Let , denote a generic regular polygon with vertices XΩ i, i = 0,..., N – 1.        Associated with each vertex Xi there is a Boolean sum Taylor interpolant                         Pi(u, v) say, which must be reparameterized onto the polygonal domain. This                        is achieved by ‘central projections’. Thus let  (1 -ui )Xi + ui Xi+1, and (1 – vi)Xi + vi Xi-1 (4.4) define the points of intersection of the two central projections through X Ω∈                     with the sides of the polygon as shown in Figure 4.1. Then  ,,:).(:)(2111 ⎟⎟⎠⎞⎜⎜⎝⎛++== −+−−iiiiiiiii ddddddruXφ  (4.5) defines the central projection coordinate chart about Xi, where di(X) denotes                        the perpendicular distance from X to the side XiXi+1. The blended polygonal surface patch is now defined by   (4.6) ∑−=•=10)),(()(:)(Niii XpXwXp φArbitrary Mesh Network Scheme  7  Figure 4.1. The central projection coordinate chart.   Figure 5.1. A single triangular patch. 8 John A. Gregory and P.K.  Yuen where  ∑ ∏∏−= −≠−≠= 102,12,1:)( Nk jkkjjiijiddXw  (4.7) defines a weight function which is constructed such that pi contributes, in                    position and tangent plane, only on those edges where it interpolates a strip                function. §5. Examples and Concluding Remarks We conclude by giving two simple examples of the arbitrary mesh network                 scheme. The first case, shown in Figure 5.1, is that of a single triangular                         patch, defined with zero twists, and illustrates the effect of increasing the                     interval tension parameter between two internal edge vertices.    Figure 5.2. A mesh network of triangular patches.  Arbitrary Mesh Network Scheme 9 The second case, shown in Figure 5.2, is of the mesh network of four                triangular patches, given as the concrete example of Section 2. In the figure,                        the rational spline network is that defined by the special case of cubic spline       interpolation. The figure shows that the blending approach proposed here                         gives a smooth, tangent plane continuous surface. Our current research is directed towards allowing more than two curves                         to meet or intersect at any face vertex, thus removing a major restriction on                         the topology of the network proposed here. This involves modification of the           derivative parameters of the rational spline network at such face vertices, in                     order that a consistent tangent plane is defined there. It is also proposed                         to investigate shape control of the interior of a patch, other than through                         the control of its boundary. Finally, the method can be extended to the                  construction of higher order continous surfaces, although more care is now                   needed in the construction of the blends of the strip functions, as is discussed                        in [4] and [5]. Acknowledgements: This work was supported by the SERC ACME Grant          GR/F80838, in collaboration with Siemens Nixdorf Information Systems Ltd.         References 1. Charrot, P., and J. A. Gregory, A pentagonal surface patch for computer                  aided geometric design, Computer-Aided Geom. Design 1 (1984),   87-94. 2 .  Foley, T. A., Interpolation with interval and point tension controls using            w e i g h t e d  v - s p l i n e s ,  A C M  T r a n s .  M a t h .  S o f t w a r e  1 3  ( 1 9 8 7 ) ,  6 8 - 9 6 .  3. Gregory, J. A., C1 rectangular and non-rectangular surface patches, in                 Surfaces in Computer Aided Geometric Design, R. E. Barnhill and W.                   Boehm (eds.), North Holland, Amsterdam, 1983, 25-33. 4. Gregory, J. A., and J. Hahn, Geometric continuity and convex combina-                        tion patches, Computer-Aided Geom. Design 4 (1987), 79-89. 5. Gregory, J. A., J. Hahn, and V.K.H. Lau, High order continuous polyg-                       onal patches, TR/06/91, Mathematics & Statistics Department, Brunel            University. 6. Gregory, J. A., M. Sarfraz, and P. K. Yuen, A rational spline with interval                  and point tension, in preparation.  John A Gregory and P.K.Yuen  Department of Mathematics and Statistics Brunel University Uxbridge, UB8 3PH, U.K. John.Gregory@ brunel.ac.uk Pak-Kay.Yuen@ brunel.ac.uk           NOT TO BE         REMOVED       FROM THE LIBRARY           

A pentagonal surface patch for computer aided geometric design,

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C 1 rectangular and non-rectangular surface patches,

C1 rectangular and non-rectangular surface patches,

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An arbitrary mesh network scheme using rational splines

An arbitrary mesh network scheme using rational splines

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