Surfaces for irregular regionsby C. G. SingletonCADCentre Limited

Surface modelling is an accepted part of CADCAM, and there is awide range of systems offering the facility. The mathematicsunderlying the surfaces tends to differ from system to system, butthere is an overall similarity in that most surfaces are based on arectangular parametric grid. While this is the most practical way ofcreating surfaces for general use it makes them intrinsically unsuitedto certain commonly occurring types of geometrical configuration.This article describes a collaborative venture between CADCentreLtd. and Loughborough University which has produced a new type ofsurface to cope with these irregular geometries.

Introduction

Because free-form surfaces can becoaxed into the most complex shapes itis easy to assume that any sculpturedregion can be suitably modelled bythem given sufficient time and effort.This is not always the case, however,and this article demonstrates a particu-lar limitation of conventional surfacesand explains the reason for it. It thendescribes a new type of surface whichhas been developed specifically to over-come this problem.

Fig. 1 illustrates a single patch takenfrom a sculptured surface, in this case abicubic surface created in the CAD-Centre Surface Modeller, which is partof the CNC machining suite. Until now,all CNC surfaces have been based on arectangular parametric grid which is

navigated using two parameters,referred to as U and V. Thus any pointon a surface can be specified uniquely,by a U,Vpa\r, and the surface propertiesat that point (position, surface normaletc.) can then be evaluated. The userdefines the surface by setting vectors atthe nodes which lie at the corners of thepatches; these vectors are typically thenode position, U and V tangents and atwist vector.

Although other surface modellershave their own ways of defining sur-faces, the underlying mathematics willtend to be similar, and this is certainlytrue of Bezier and B-spline based sys-tems. Therefore, while all the surfacesused as illustrations in this article aretaken from GNC, the ideas apply muchmore generally.

Returning to the patch in Fig. 1, it can

be seen that the patch boundaries arealigned with a roughly rectangular lat-tice. Assuming that the surface hadbeen sensibly designed, this latticeshould be related to the principal direc-tions of curvature of the part, and nor-mally these would be suited to theimposition of a rectangular parametricgrid. The simplest case which demon-strates where the lines of curvature ofthe part are not suited to a rectangulargrid is shown in Fig. 2. In this case thereare three independent lines of curva-ture which produce a central triangularregion.

There are two ways of modelling thisregion with a four-sided patch. The firstis to make the L/and I/tangents at node(0,0) collinear, which will create thepatch boundaries correctly. However,this leads to a problem in defining thesurface normal at that node, and this inturn causes problems in creating theoffset, surface for machining. The sec-ond way is to make two of the nodescoincident, thus introducing a zero-length span between them. Even whenthe surface mathematics can cope withpatches distorted to this extent, prob-lems are once again likely to manifestthemselves when the offset surface iscomputed.

If, despite these difficulties, the usermanages to define an acceptable sur-

Fig. 1 Bicubic patch

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Fig. 2 Distorted patch

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Fig. 3 Wing section

Fig. 4 Bottle with handle

face to fill the region, he will still findthat the parametric arrangement of thissurface does not correspond to thatrequired to define the surroundingareas. In the first case there will be anextra node in the middle of one spanwhich is not required in the adjacentpatch; in the second case the tangentmagnitudes required to set up the nullspan will not correspond to those of theneighbouring patches. In either case itis likely to be a non-trivial task to ensurethat slope and curvature continuity ismaintained across the boundaries ofthe region.

Given that certain geometrical config-urations exist which are not suitable formodelling by conventional surfaces,how often do these situations occur? Infact, most parts modelled by sculptured

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surfaces would benefit from the inclu-sion of at least one three- or five-sidedpatch. For instance, Fig. 3 represents asurface which might be used in design-ing the wing section of a car, the shadedarea showing where a three-sided patchfits more naturally than a four-sidedone. In Fig. 4 a set of five-sided patcheshave been embedded in the middle ofthe bottle surface to form the blendregion into the handle section.

Many more examples could bequoted to further illustrate the point, allof them taken from common, everydayobjects. It has long been recognisedthat conventional surface mathematicswas inadequate in this respect and anumber of solutions have been put for-ward. The technique which has beenintroduced into the CADCentre Surface

Modeller is a 'recursive sub-division'algorithm, and the development of thistechnique is now described.

Recursive sub-division surfaces

In 1978 Catmull and Clark (Ref. 1) pre-sented an algorithm to perform a recur-sive sub-division process on apolyhedron composed of an arbitraryset of points. Each iteration of thealgorithm introduces extra points fromwhich a new polyhedron is built up. Thefacets which link the points in the poly-hedra become smaller as the iterationproceeds and so each polyhedron is arefinement of the previous one. Cat-mull and Clark proved that if the origi-nal set of points were arranged in arectangular net the limiting polyhedronis a uniform B-spline surface.

For an arbitrary arrangement thealgorithm quickly separates the net ofpoints into a series of regular regions(within which the points lie in a rec-tangular grid) separated by a number of'extraordinary points'. These are pointswhere other than four facet edges cometogether, and they can be thought of asthe central points of local irregularregions. The number of extraordinarypoints is fixed from the first iteration,and the total area of the irregularregions becomes progressively smalleras the surrounding regular regionsencroach upon them. In the limitingsurface B-spline slope and curvaturecontinuity applies everywhere except atthe extraordinary points themselves.

In the same journal as the Catmulland Clark paper, Doo and Sabin (Ref. 2)presented a paper which explored thebehaviour of a recursive sub-divisionsurface in the vicinity of an extraordin-ary point. They identified certainweighting factors within the sub-divi-sion process which influenced the sur-face at this point, and then went on tostudy the effect of these weights on thecontinuity properties of the surface.

Interest in this area of surface model-ling at Loughborough University led in1980 to a research project being startedto investigate the optimisation of thesub-division technique. In 1984, Balland Storry (Ref. 3) presented a paperentitled 'Recursively generated B-splinesurfaces', which showed how optimaln-sided surfaces could be generated bycalculating the eigenvalues of the recur-sive sub-division transformation matrix.That same year a co-operative project,funded by a UK Science and Engineer-ing Research Council grant, was startedbetween Loughborough and CAD-Centre Ltd. to build these surfaces intothe CADCentre Surface Modeller. Theproject has now been successfully com-pleted, and the forthcoming release of

Computer-Aided Engineering Journal August 1986

the modeller will incorporate a new sur-face type called an N-spline, the namebeing derived from 'N-sided B-splinebased surface'.

N-spline surfaces

N-splines are designed to solve theproblems discussed so far for whichconventional surfaces are unsuitable,i.e. filling in the irregular areas of partswhich exhibit complex curvature pat-terns. All N-splines consist of a singlepatch, which can have between threeand six sides, and they are thereforeintended to be used in conjunction withone or more regular surfaces. Theunderlying mathematics ensures thattangent continuity between an N-splineand its neighbours is maintained if theN-spline nodes are set up appropri-ately. The simplest way to achieve this isto set up the N-spline nodes directlyfrom the nodes of the surrounding sur-faces, which then produces a 'natural'blend surface over the irregular region.The user is still left with a way of modify-ing the surface in this situation becausethere is a facility for independentlyadjusting the position of the centre-point of the N-spline; in this way the'fullness' of the surface can be altered

without affecting the slope continuitywith adjacent surfaces.

Since there seems to be a limitednumber of part configurations in whichirregular patches will mostly be used,the modeller has been equipped with anumber of automatic generationoptions to cover them. To use theseoptions the overall geometry is definedby one or more regular surfaces, andthen one of the regular patches is sin-gled out for replacement by anN-spline. The program recognises thepart topology and creates the requiredsurface. These predefined casesinclude common situations such asrounding off corners with three-sidedpatches and creating various types ofpipework blends with five- or six-sidedpatches.

Fig. 5 shows an N-spline which hasbeen created to fill in the triangularregion from the example used in theintroduction. The definition vectors ofthe three nodes of the N-spline are pre-cisely the same as the node vectorsrequired to set up the adjacent regionsof a set of abutting regular surfaces. Theinterior shape of the N-spline is illus-trated by the series of cross-sectioncurves drawn at intervals across it. Fig. 6shows cross-sections taken from the

same surface when offset for a cuttingtool. It can be seen that the offset sur-face is well formed in relation to thebase surface, and this comparesfavourably with offsets produced fromfour-sided patches forced into a tri-angular shape.

Example job

The surface in Fig. 7 was created in CNCto model ah electrical spade-connector,the section illustrated being an enlarge-ment of one side of the part. This sur-face is an adequate model of the partexcept for the region around the shoul-der of the wide section, where it can beseen that the patches are highly dis-torted. In addition, the two parameterlines which define the top of the widesection degenerate into the single curvewhich runs along the top of the stemsection. The problem with this para-metric layout is the difficulty it creates inensuring that the patch interiors in theshoulder region conform to the partspecification. Even if the user generatessets of cross-sections to check theshape, none of the parameter linesremotely corresponds to the cross-sec-tional information in the originaldrawing.

Fig. 7 Spade connector

Computer-Aidecl Engineering Journal August 1986

Fig. 8 Composite surface

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Fig. 9 Shoulder section

Seq 1Oiew 4Sea 8.<»0

Cutter1.80 0.50

Fig. 10 Machining paths

Fig. 8 shows a re-parametrisation "ofthe surface which has parameter lineslying in a much more natural fashion inrelation to the part description. Thisnew model is composed of three sep-arate surfaces: a single regular surfacewhich defines the overall shape of thepart and two N-splines which fill in theshoulder region. For this situation onethree-sided and one five-sided patchare the most natural way of modellingthe irregular region; the two surfacesare shown in Fig. 9, with a net of interiorlines drawn to give an idea of the result-ing shape.

The N-splines were created directlyfrom the definition vectors of the neigh-bouring nodes of the regular surface,although in this case the automatic gen-eration facility was not used. The inter-nal shape of the surfaces is the defaultform generated by the program; i.e. thepositions of the centre-points have notbeen modified. The part of the regular

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surface which lies 'beneath' the twoN-splines has been masked off to makeit invisible (and also out of bounds forthe subsequent machining).

Fig. 10 illustrates the machining pathsfor the composite surface, and thesecan be broken down into two separatesections. The first is an area clearance ofthe side wall using cutter paths gener-ated from the parametric lines of that

part of the regular surface. In this regionthe parameter lines correspond to thenatural machining paths and, beingquick to calculate, they are the simplestway of machining the side wall.

The same approach cannot beapplied on the top of the surfacebecause there is no continuous para-metrisation over the shoulder regionwhere the N-splines have been inser-ted. Since it is still preferable to havecontinuous machining paths over thetop of the part, a set of cross-sectionlines have been used to area clear thisregion. These curves form unbrokenpaths over the composite shape formedby the regular surface and theN-splines.

Conclusion

This article has attempted to demon-strate a limitation of conventionalsculptured surfaces in modellingregions with irregular topological con-figurations. The mathematical workinvolved in creating a new surface math-ematics to overcome this problem hasbeen of a very high order, and theauthor has tried to explain some of theconcepts on which it is based. How-ever, it would be unfortunate if readerswere to be misled by the abstrusenature of some of these concepts intothinking that the work is only relevant tothe production of parts which are insome way'special'. Theaim of smoothlyblending an irregular region is relevantto many practical and commonplacesituations in surface design.

The CADCentre approach to model-ling irregular regions has been to recog-nise that conventional surfacemodelling can be a difficult disciplinefor the non-technical user to acquire,and that a new facility should not makeit more so. In the light of this, anattempt has been made to introduce thenew functionality in a way which con-forms to existing surface design prac-tice within CNC. It is hoped that thisapproach will encourage users to applythe new surfaces to real jobs so that abetter understanding of their relevanceto complex part design can be formed.

References

1 CATMULL, E., and CLARK, J.: 'Recursively generated B-spline surfaces on arbitrary topol-ogy meshes', Computer Aided Design, 1978, 10, pp. 350-355

2 DOO, D., and SABIN, M. A.: 'Behaviour of recursive subdivision surfaces near extraordin-ary points', ibid., 1978, 10, pp. 356-360

3 BALL, A. A.,andSTORRY, D. J.T.: 'Recursively generated B-spline surfaces'. Proceedings ofCAD 84, International Conference on Computers in Design Engineering, Brighton,England, 1984 (Butterworth Scientific, 1984)

C. G. Singleton is with CADCentre Limited, High Cross, Madingley Road, Cambridge CB30HB, England, and has been responsible for liaison with Loughborough University of Tech-nology on the Recursive Sub-division Project under UK Science & Engineering ResearchCouncil grant CR/D/10237.

Computer-Aided Engineering Journal August 1986