1989 Surface Lofting

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Computer Aided Geometric Design 6 (1989) 87-96North-Holland 87

Surface lofting and smoothingwith skeletal-linesS. BED1De~artmenl of Meckanical Engineeri~& Universiry of Waterioo, Ontario, Canada

G.W. VICKERSDepmnent of Meckanical Engineering University of Victoriq B.C., CanahReceived My 1987Revised May 1988Abstmet. A method of surface definition and smoothing using a set of three-dimensional skeletal-lines is given.The skeletal-lines represent the essential features of a surface and can be individually smoothed using the firstdivided difference. The modified skeletal-lines are automatically generated by back substitution from thesmoothed first divided difference lines. In this way a surface can be defined and smoothly adjusted through a setof sparse. irregular, inconsistent data.

Keywo& Surface definition and smoothing, skeletal-line representation, back substitution of fist divideddifference.

The critical surfaces in ship hulls, turbine blades, propellers, aeropume fuselages, etc. aredefined in three orthographic views [Bedi & Vickers 871 showing various horizontal, verticaland oblique cross-sections, as shown in Fig. 1. The drawings are done to a smalI scale and maycontain inherent design errors. The computer models of these surt^acesare often generated byidentifying form parameters, such as sectional area, end angles, elc. [Clements 84, Hattori &Matida 77, Munchmeyer 821, or using a control polyhedron [Bamhill 85, Bezier 72, de Boor

abear Iine

BODY PLAN

Fig. 1. Naval architects design drawing showing station-fines, water-tines, shear-line and profile-line.

0167.83%/89/$3.50 0 1989. Rfsevier Science Publishers B.V. (North-Holland)


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72.1 These approaches require an experienced designer [Hattori & Matida 771, well versed insurface design techniques. In this paper a method is deveioped in which a smooth surface maybe generated directly from a set of sparse irregular data contained in orthographic drawingviews.

The current approach to surface modelling has been adopted as it passes through the pointslying on the surface [Bedi & Vickers 87, Bedi, Chemoff & Vickers 861 and readily produces allstation-line and water-line data as required for shipyard manufacture.

2. Surface designThe traditional representation of surfaces showing a rectangular grid of vertical (station-lines)and horizontal (water-lines) sections does not represent the surface uniformly or completely. As

shown in Fig. 1, the features near the stem and keel are undefined. This ambiguity wasacceptable in manual lofting and fairing procedures where a loftperson interpreted the drawingbased on his experience and judgement. However, this is not suitable for computer modelling.In this work, a set of three-dimensional curves, which lie on the hull surface and represent itsessential features, is defined. The lines, called skeletal-lines, form the skeleton around which thesurface is generated. If this skeleton is smooth, the generated surface is observed to blendsmoothly from bow to stem.The skeletal-lines are generated in the front view of the surface by superimposing it withcurved lines that divide the three-dimensional surface into equal sectors. On a uniformlycurving surface, these skelatal-lines are equally spaced, whereas additional lines may be addedto represent sharp curvatures or other unrepresented features of the surface. The skeletal tines,which are three-dimensional curves, are defined as

Y /(X)9 0)z=h(x). (2)

The complete definition of the surface is generated by taking an intersection of the skeletal-linesat a particular station and fitting it with a curve. The equation of this curve is

y=dz9 (3)where /, g and h are cubic polynomials.

Equations (l), (2) and (3) define every point on the surface and can generate any requiredvertical, horizontal or oblique cross-sectional view of the surface. This method has beenimplemented as a menu-driven, personal computer-based package, which has been used formany industrial applications such as hull design, turbine blade design, heart valve design, etc.The machined prototypes had a faired and smooth surface.A front view of a hull as shown in a naval architects plan, superimposed with a set ofskeletal-lines, shown with broken lines, is given in Fig. 2. The surface generated after minormodifzations to this set of skeletal-lines is shown in Fig. 3. Similarly, the front view of a turgoturbine blade with the generating set of skeletal-lines is shown in Fig. 4. The perspective view ofthe final suf~ce of the turgo turbine Made is shown in Fig. 5.

3. Skeletal line method of surface fair@The skeletal-lines are arbitrarily placed by digitizing the points at which they intersect the

station-lines. The errors inherent in the unlofted and unfaired drawing in addition to errors dueto digitizing small drawings, generate a surface that does not necessarily blend smoothly. A


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Round-Bottom-Seiner89

Fig. 2. A body-plan for a round bottl>m1 I I I I seiner fishing boat with 11 smoothedI 2 3 4 5 skeletal-lines.

typical set of initial surfaces with some oscillations is shown in Fig. 6. The oscillations in thesurface can be seen by looking along the water-lines. Visual inspection is a discriminating wayto identify smoothness, but manual modification by adjusting the data locally is a cumbersomeand tedious task.Many approaches to semi-automatic smoothing of surfaces were attempted. One feature thatbecame evident was that it is better to go back and smooth the skeletal-lines (control pointsj

Round-Bottom-Seiner

2 4 6 IIFig, 3. A perspective view of the hull shape of the round bottom wirier.


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Turgo-Turbine-Blade

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0 2 4 6 8 10Fig. 4. A body-plan for a tutgo urbinewith six smoothed skeletal-lines.

and thereby reform the surface, than to adjust the water-lines and station-iines individually[Hattori & Matida 77, Munchmeyer 82, Munchmeyer 791. It was fiirther determined thatblending neighbouring points changes the shape of the resulting surface as it relaxes the curveat that point [Clements 841.In the most suitable method, the first divided difference of the skeletal-lines is used toamplify and identify sections of irregularities [Renz 821. Furthermore, modifications are made

Turgo-Turbine-Blade

, I2 4 6 ll IO

Fig. 5. A perspectiveview of the developed turgo turbine.


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Round-Bottom-Seiner91

2 4 6

Double-Chine-Boat

Fig. 6. Surface produced rom digitizedunsmooth data, representing a round bot-tom seiner fishing vessel and a double

2 4 6 8 I O chine seiner.

to the divided difference curve rather than the skeletal-line itself, and the modified skeletal-lineis generated by back substitution. As the modified skeletal-line is based on a smooth firstdivided difference, it is observed to blend smoothly.Given a curve y =f(x), the first divided difference is defined as

f (xt+ ) - f (xi )Sfxi)= x #+I -xi *The second divided difference is defined as

h(xi)= Ed .,+1) dx,)Xr+l -x,(4)

Skeletal-lines are three-dimensional curves, so their divided difference is plotted in the x-yand the x-z planes to identify unfair points. These points in the first divided difference plotare re-evaluated by fitting a taut-cubic spline through their neighbours. This smoothed divideddifference is the basis of the modified and faired skeletal-lines that are evaluated by back-sub-stitutions. The smoothing process is continued untii a satisfactory surface is obtained. Thedetails of this method are given below.3.1. Smoot hing using di vi ded diff erences

Let a skeletal-line be represerrted by n control points given by[f(x,), f(%)r...rf(X .-l)r /kJ-

The f&o divided difference for these control points is then given by[g(x,), dx2L.dx"- , )~ X(% l )] .


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The divided difference is viewed on a graphics terminal and visually inspected to identifypoints that deviate from the norm. Let point m , with a divided difference of g(x,), be in error.It is modified based on its neighbours, g(x,_Z)r g(x,_,) and 8(x,,,+,) and g(x,+2). Ataut-cubic spline is passed through these points and its value gI( x,) at x,, is the modified andsmoothed divided difference. The modified divided difference, g,(x,), can also be written as

gr(x,) = g(x,) + Ag(xJ. (6)The modified control points are obtained from f(xt) and [x,, x2,. . . , x,] and are given by

f,(Xi+l)=(Xi+*-X*)g(xi)-tf(xi), (7)

wheref,(x,) =f(xt). 63)

The modified control points are then given by the following points[f(x,). I(x~),...rf(x,), f(x,+r) + (x,+1 --xmMg(xm),

f(x,+z) + (x,+1 -x,)Ag(x,)..4x,-,) +(x,+1 -x&k(x,), f(x,) + (x,+1 -x,Mg(x,)].

As is apparent, all control points after the altered point have shifted by the same amount.Hence, it is r~~~ssary to restrict the curve modification to a reg& to prevent it propagatingbeyond the altered points. As only one point is altered, the region of alteration starts atg(x,,,_t) and extends to g(x,+, ). The points on the original curve corresponding to thesedivided differences are f(x,_,) and f(x,+,).Let the divided difference obtained by using equation (7) be

Mxt), fi(x2)r...Jt(x-l)r ft(xn)l.Then the final modified control points are given by the following formula

fbm-1) f(xm+Jf*txi) = Q(x,_,) +o-wf~(x,+,) ifxm-l~xl=m+q~1 (9)

\I \&I! otherwisewhere

h=L_ xi-xm-l .Xm+l -X,-l

In the above derivation, only one point of the divided difference curve was modified, but it ispossible that a range of points needs to be altered in the divided difference curve. The methodof altering a range of points is a minor modification of the above described method.

If the divided difference curve is modified in the range p to q, the curve is obtained usingthe following equationf(x,)_+(* -x)f,(xp ), I ifxpSxi5x,, (10)

otherwisewhere

~+_xl-xpx -x4 P


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S. Bedi, G. W. Vickers / Surface lo fi ing 93

Initialize Display

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IModify Divided Difference

Yes t

Fig. 7. Flowchart for skeletal line smooth-ing using divided difference.

The smoothed control points [fr(x,), fi(xz),...,fi(xn)] are fitted with three-dimensionaltaut-cubic splines or B-splines to obtain the skeletal line.The above method can be extended to modification of the first divided difference based onthe second divided difference. Based on the success of this method. however, it was notnecessary to extend the modification process to second divided difference. The above methodgives very sensitive control in moving the data points. The flowchart of the program implement-ing the above method is given in Fig. 7. This method has been successfully used to identify andmodify inconsistent irregularities on the skeletal-line data for a number of different surfacedesigns. Two of the designs are discussed as examples below.3.2. Chine boat

A set of skeletal-lines is drawn on the body plan of a naval architects plan for adouble-chine boat, according to the approach described earlier. The surface generated fromthese skeletal-lines is shown in Fig. 6. The first divided difference in the x-y and x-z planes offine skeletal-fines is shown in Figs. g and 9, respectively. The skeletal-lines were modified usingthe divided difference approach described above. The modified first divided difference curvesare shown in Figs. 10 and 11. The water-lines and station-lines of the surface generated fromthese modified skeletal-lines are shown in Figs. 12 and 13.3.3. Non-chine boat

The surface generated from the original skeletal-lines digitized from a naval architects planis shown in Fig. 6. These skeletal-lines were modified using the divided difference technique.The station-lines and water-lines generated from the modified skeletal-lines are shov-m in Fig. 3.


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S. Bedi, G. W. Vickers / Surjbce loft ing

clthdg-xy

x F@. 8. The x-y view of the first divideddifference of the unmodified skeletal-lines2 4 6 8 10 of the double-chine boat.

Z4thdgxz

Fii. 9. The x-t view of the first dkidcddifference of the unmodified skeletaMincs2 4 6 II ltl of the double-chine boat.

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0 I I I2 4 6 II IO

Fig. 10. The x-y view of the Tit divideddifference of the modified !skektal-lines ofthe doubleshine boat.


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-I 4thoot4a

Fig. 11. The x-z view of the first divided difference of Fig. 12. The water-lines of the modified surface of thethe modified skeletal-lines of the double-chine boat. double-chine boat.

Double-Chine-Boat

2 4 6 8 IOFig 13. A perspective view of the hull shape of the double-chine fishing boat.

The skeletal-lines represent a surface uniformly and completely. Furthermore, they provide ameans with which the surface can be modified selectively and in a controlled fashion. Theabove methods have now been implemented as a package based on a personal computer. Thepackage is being used at a local shipyard where it has reduced the lofting and fairing time byone-third. The packages versatility is further demonstrated by the variety of applications, suchas kayak design, francis and turgo turbine blade design, hull model design, etc., that it hassuccesfully handled.


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ReferencesBarnhil l, R.E. (1985). Surfaces in CAGD: A survey, Cornfilter Aided Cmmetric Design 2, l-17.Bedi, S. (1984). A study of B-splines and their application to engineering, Masters Thesis, University of BritishColumbia, Vancouver.Bedi, S., W. Chernoff and G.W. Vickers (1986), Computer aided fairing and direct numerically controlled machining of

ship hull hydrodynamic test tq models, J. Ship Research, to appear.Bedi, S., and G.W. Vickers (1937). The generation of smoothed curved surfaces from sparse irregular data, ASME,

computers tn Mechanical Ei@neering. to appear.B&ier, P.E. (1972). Numerical Contro l - hfarhemotics and Appl icat ions. Wiley, New York.Clements. J.C. (1984). Developed plate expansion using geodesics, Marine Technology 21(4).Collae G., and E. Seifert (197,h Interactive fair@ of ship lines - a procedure developed for the model hamburg

basin, Computer-Aided Hull surface Definition Symposium, Annapolis.Coons, S.A. (1976), Surfaces for - flputer aided design of space forms, Massachusetts Institute of Technology, Project

MAC.de Boor, C. (1972). on calculating a& B-splines, J. Approx. Theory6.de Boor, C. (1978), A Guide to S,$ines, Applied Mathematical Sciences, 27, Springer, Berlin.Forrest. A.R. (1%8), Curves as< sttrfaces for computer aided design, Ph.D. Thesis, Cambridge. University of

Cambridge.Hattori, Y., and Y. Matida (1977). isme problems in practical improvement of mathematical fairing, Computer-Aides

tiull Surface Definition Syzqxwun, Annapolis. 3Hoschek, J. (1985). Smoothing of c-es aad surfaces, Computer Aided Geometric Design 2, 97-105.Munchmever,F.C. (1982). Mathematical ship lines and surfaces, Marine Technology 19 (3).Munchmeyer, F.C., C. Shubert anu Ji. Norvacki (1979), Interactive design of fair hull surfaces using B-splines,

Computer Appl icat iun in the Autom ation of Shipyard Opemtions and Ship Design 3, North-Holland, Amsterdam.Munchmeyer, F.C., and G.K.H. Lau (1978), Gn the interactive design of smooth patched surfaces, Inteructioe

Techniques in Compufer-Aided Design, Palazro de, Congressi, Bologna, Italy.Rena, W. (1982). Interactive smoothing of digitized point data, Computer Aided Design 14 (5).Rogers, D.J., S.G. Satterfield and F.A. Rodriguez (1983). Ship hulls, B-splines and CAD/CAM, Computer GraphicAppl. 3 (9).Vickers, G.W., S. Bedi and R Haw (1985). The defntition and manufacture of compound curved surfaces using G-surf,Computers in Industry 6 (3). 173-183.

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1989 Surface Lofting