B-SPLINE SURFACES FOR SHIP HULL DESIGN*

David F. Rogers Steven G. Sat te r f ie ld

United States Naval Academy Annapolis, Maryland 21402

ABSTRACT

The use of true sculptured surface descrip'- tions for design appl icat iof is has been proposed by numerous authors. The actual implementation and use of in teract ive sculptured surface descript ion techniques for design and production has been l im- i ted. The use of such techniques for ship hull design has been even more l imi ted. The present paper describes a prel iminary implementation of such a system for the design of ship hul ls and for the production of towing tank models using numeri- cal control techniques. The present implementa- t ion is based on a Cartesian product B-spline sur- face descript ion. Implementation is on an Evans and Sutherland Picture System supported by a PDP- 11/45 minicomputer.

The B-spline surface is manipulated by i ts associated polygonal net. Both surface and net are three-dimensional. Techniques both good and bad for 3-D picking of a polygon point when the net, i ts associated surface, and the 3-D picking cue in- dependently exist:and can be independently mani- pulated in three space are presented and discussed.

The shape of a B-spline surface of f ixed order is control led by the locat ion of the polygon net points, the number of mul t ip le points at a par- t i cu l a r net point, and the knot vector. Frequently mul t ip le points imply mul t ip le knot vectors. Prac- t i ca l techniques for cont ro l l ing and shaping the surface with and without th is assumption are dis- cussed and the results i l l us t ra ted .

Experience attained by in te rac t i ve ly f i t t i n g a single fourth order B-spline surface patch to the forebody hal f of an actual ship hull described by three dimensional d ig i t i zed points is discussed and the results i l l us t ra ted .

* This work is p a r t i c i a l l y supported by the Naval Ship Engineering Center and the U. S. Coast Guard.

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KEY WORDSAND PHRASES: Computer graphics, Sur- face, Sculptured surface, B-spl ine, Ship hu l l , CADCAM, Numerical Control.

CR CATEGORIES: 3.20, 3.21, 3.26, 5.13, 8.2

INTRODUCTION

Computer Aided Ship Hull Surface Design (CASHSD) has been investigated by a number of groups and several systems have been developed. Among these are Autokon, Viking, BRITSHIPS, FORAN, etc. In each case, a ships l ines approach has been used, i .e . the surface of the ship is described by a net of l ines. Usually the t rad i t iona l three mutually orthogonal planes containing water l ines, stat ion l ines, buttock l ines used by the naval archi tect have been chosen. Each set of l ines is " fa i red" and then cross-faired unt i l the surface is considered defined. The l ines fa i r i ng techni- ques have been many and varied. Typical ly para- bo l i ca l l y blended curves, cubic splines, c i rcu lar arc interpolated curves or rat ional polynomials have been used. Either in terac t ive or batch type processing has been used. Implementation has t yp i ca l l y been on a large scale computer.

Recently a CASHSD/CADCAM system cal led CAMILL (~omputer Aided Mi l l ing) for the design of ship hul ls and the production of towing tank models has been developed [5] , [6]. CAMILL is implemen- ted on an Evans and Sutherland Picture System graphics display driven by a PDP 11/45 minicom- puter. I t is highly in teract ive . Again, the ba- sic philosophy was to use a net of l ines to define the surface. The designer is free to choose the type of fa i r i ng algorithm for each l ine from among parabol ica l ly blended, cubic spl ine, Bezier or B- spline curves up to order s ix. The system has been used to develop the f a i r hull shapes and pro- duce towing tank models for several ships. The most recent e f for ts involved two icebreakers for the U. S. Coast Guard [5] . Although CAMILL has been successful, i t has become obvious that a fu l l surface descript ion technique is more desirable.

THE NEED FOR SCULPTURED SURFACES

True sculptured surfaces in engineering have t r a d i t i o n a l l y been used in the f i e l d of a i r c ra f t , ship, and automobile design. They have also been extensively used in consumer product design.

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Ai rc ra f t fusalages, wing-body f i l l e t s , cockpit can- opy fa i r ings , and engine in le ts require sculptured surface treatments. In ship design the bulbous bow, sonar domes, tunnel and conventional sterns as well as almost the ent i re underbody of a sa i l ing yacht are sculptured surfaces. Because of s ty l ing considerations, automobile design has been a heavy investor in sculptured surface techniques. Con- sumer products as wide ranging as the shower sham- poo bot t le and the case for the la test computer graphics device use sculptured surfaces.

SURFACE DESCRIPTION TECHNIQUES

The natural extension of the l ine or curve net surface descript ion technique is the Coons sur- face patch [3]. The Coons patch, although more general than recognized, is normally implemented as a Cartesian product surface. In a Cartesian pro- duct surface the edge curves of a four sided sur- face patch are defined by algorithms s imi lar to those used to define f a i r curves. Thus a Cartesian product Coons surface may be a parabo l ica l ly blended surface, a cubic spl ine surface, a Bezier surface or a B-spline surface. As such, the sur- faces reta in the advantages and disadvantages of the curves describing the edges of the patch. These advantages and disadvantages have been dis- cussed in [4] in the context of ship hul l design. Reference [4] indicated that no one curve descrip- t ion technique was a panacea. However, the most f l e x i b l e of the curves for in terac t ive ship design was found to be the B-spline basis curve. This is because the B-spline basis curve allows knuckles or sharp d iscont inu i t ies within a contiguous curve descript ion. I t also provides s ign i f i can t data compression and convenient in teract ive curve mani- pulat ion handles. Thus, an extension to B-spline surfaces appeared to be of in terest .

POLYGONAL NET MANIPULATION

The shape of a B-spline surface is contro l led by a polygonal net of points. In the implementa- t ion described here, the polygonal net which de- f ines the B-spline surface, seen as a perspective or orthographic project ion with in tens i ty (gray) scal ing, can be oriented in space by manipulating a set of control d ia ls . The control d ia ls provide ro tat ion about the three coordinate axes, t ransla- t ion along the three coordinate axes and overal l scaling. An addi t ional control dia} is used to adjust the sens i t i v i t y of the other d ia ls .

In in te rac t i ve ly manipulating indiv idual poly- gonal net points the basic problem is to acquire a defined polygon point in three space by manipula- t ing various controls, i . e . to execute a three d i - mentional h i t test or pick. Several d i f f e ren t techniques were implemented and tested. Three are described.

The f i r s t implementation uses a three dimen- sional cursor in the form of a small cube. Figure la shows the CRT display with the cube and a poly- gon net. The polygonal net can be manipulated in three space independently of the cursor cube. The or ien ta t ion of the polygonal net is shown by the 3-D axis system in the lower r ight hand corner. The cursor cube can be independently translated in the x,y,z direct ions by means of three control

B-~IP!.INE ~,Ji~IE DISPI.AY

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Figure l : Three dimensional pick scheme using cursor cube.

212

dia ls . The cursor cube is translated in the d i - rect ion given by the 3-D axis system. The posi- t ion of the cursor cube is continuously updated and displayed on the screen (Fig. la , CURX, CURY, CURZ). The object ive was to place the cursor cube "over" one of the polygon net points to select that point for modif icat ion. Once the polygon point is selected, the three control d ia ls are used to translate i t to the required posi t ion. Figures Ib and Ic generated from the display, i l - lus t ra te the fundamental problems. Figure Ib suggests that the cursor is over one of the points while Fig. Ic, in which the polygonal net has been reoriented, shows that this is not the case. While su f f i c ien t f a c i l i t i e s are provided to accom- pl ish the required match, the process is slow, f rus t ra t ing , and fat iguing. The basic d i f f i c u l t y is the necessity of manipulating two independent objects in three space with l imi ted or ienta t ion cues. The technique is not recommended.

The th i rd implementation is a resul t of the rea l i za t ion that i t , in fact , is not necessary to consider a three dimensional picking problem. To see th is , note that the surface is control led and shaped by the polygon net, which exists as an ordered set of points in the data base. What is seen on the display is e i ther a perspective or orthographic project ion of the polygonal net. The net is displayed by a simple drawing algorithm. Two dimensional h i t test ing can easi ly by per- formed by successively attempting to pass the project ion of each polygon point through the ' h i t ' window. When a match is obtained, a ' h i t ' has occurred and the par t i cu la r polygon point can be easi ly i den t i f i ed from the ordered l i s t .

The tab le t is used to posi t ion the h i t win- dow, which is made v i s ib le by means of a small tracking cross as shown in Fig. 3a. Once selec- ted, the posit ion of the polygon net point is con- t r o l l ed by three d ia ls . Figure 3b shows that se- lect ion in various or ientat ions is quite easy. The control d ia ls can be used to posi t ion the en- t i r e polygon net or surface to f a c i l i t a t e selec- t ion of a par t icu lar point of in terest (Fig.3c).

Figure 2: nodes.

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The second implementation uses the fact that the nodes of the polygon net are numbered l i ke the rows and columns of a matrix or the pixels of a raster scan display. A small keypad is generated in the lower l e f t corner of the display. The tab- l e t pen and a screen cursor is used to indicate the number of a node by select ing d ig i t s from the key pad. [Fig. 2]. Once selected, the nodes are moved by using control d ia ls to provide x,y,z t rans la t ion. The re la t i ve locat ion of the cursor is indicated as CURX, CURY, CURZ while the cursor is being moved. Again the ent i re net or surface can be manipulated in ro ta t ion and t rans la t ion by control d ia ls . The method is quite fast. How- ever, Fig. 2b shows a basic f law in the scheme. In many or ientat ions i t is d i f f i c u l t to keep track of the numbering of the nodes. Thus, continuous reor ientat ion of the polygonal net is necessary. Further, because the hardware character generator is used for speed, a phenomena s imi lar to wrap a- round occurs when displaying the numbers. Figure 2c shows that adding the surface fur ther excaber- ates the problem.

C

Three dimensional pick using numbered

213

A minimum amount of reor ientat ion between picks is required. Figure 3c also i l l u s t r a t e s that addi- t ion of the surface does not overly complicate the operation. The scheme is fast and easy to use. Further, the overhead of supporting a three dimen- sional cursor is el iminated. User acceptance is good.

B-SPLINE SURFACE ALGORITHM

A Cartesian product parametic B-spline sur- face is given by

m n

Q(u,w) : ~ i:O j=O

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where

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Mj,l(W) = I I i f y j Sw < Yj+I

0 otherwise

(w-yj)Mi,~_l(W)+ (Yj+~-w)Mj+l,~_l(W) Mj,c(w) = yj+~_l_y i Yj+~-Yj+I

and the x i ,Y i , are the elements of a knot vector

[3] . The shape controls for a B-spl ine surface are the order of the surface, the locat ion of the poly- gon net points, whether mul t ip le points occur on the polygon net, and whether mul t ip le knots occur in the knot vectors. References [ 3 ] - [ 6 ] implement B-spl ine curves such that mul t ip le vert ices imply mul t ip le knot values in the knot vector. This a l - lows sharp corners or knuckles . The implementa- t ion of the same concept for the surface is not as straightforward. For example, the ca lcu la t ion of the knot vector and the associated basis func- t ion for the schematic polygon net conf igurat ion shown in Fig. 4a is stra ight forward. However, i t is not as stra ight forward for the pplygon net con- f i gu ra t ion shown in Fig. 4b when mul t ip le net points imply mul t ip le or repeating knot values in the knot vector.

In Fig. 4b, the polygon net points ( i , j ) at (3,1) , (4,1) , and (5,1) and those at (3,2) , (4,2) , ~nd (5,2) are coincident or mul t ip le ver- t ices. I f mul t ip le vert ices implied mul t ip le in- ternal or repeating knot values in the knot vec- to r , a fourth order B-spl ine surface would y ie ld a sharp d iscon t inu i t y or knuckle along the l i ne formed by these two sets of mul t ip le ver t ices. However, the gr id in Fig. 4b for net points ( i , j ) i = I . . 7 j>2 does not have any coincident or mul t ip le ver t ices. This would imply no mul t ip le in ternal or repeating knot values in the knot vector. Ex- amination of the basis funct ion in the def in ing re la t ionsh ip for the B-spl ine surface implies that a s ingle knot vector in e i ther the u or w di rec- t ion is used to generate the basis funct ion for

8-,~PLZNE f~JRFRCE DISPLRY

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XTRN 0 YTRN 0 ZTRN 0 SCRL B8 J z ~ l l l l Ie t m

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Figure 3: Three dimensional picking in a two dimensional pro ject ion.

the ent i re surface. Without independent control of the knot vector th is would preclude the ex is- tence of sharp d i scon t inu i t i es or knuckles w i th in the surface patch i t s e l f . One possible scheme is to adjust the knot vector and thus the basis func- t ion from net l i ne to net l i ne based on the char- ac te r i s t i cs of the polygon net. For the polygon net in Fig. 4b, th is would imply that the knot vector in the u d i rec t ion would have repeating

214

knot values for the B i+ l , j+ 1 corresponding to i<l

and non-repeating values for i>l

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B-spline polygon nets.

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215

IMPLEMENTATION

The algorithm is implemented in an in te rac t i ve p i l o t program cal led BSSD (B-S_plineSurface De- sc r ip t ion ) . BSSD is modularly designed and imple- mented such that i t can be subst i tuted for the l ines descr ipt ion techniques presently used in CAMILL. I t is implemented in structured FORTRAN (RATFOR) on a PDP 11/45 under the RT-II Version I I IB operating system. The in te rac t i ve display system is an Evans & Sutherland Picture System I . The program operates in two overlay segments in ap- proximately 42K bytes of memory with data arrays declared in v i r t ua l memory. In terac t ive shaping and or ienta t ion of the surface is contro l led by menu picking using a tab le t , by funct ion switches and by control d ia ls .

The B-spline surface algorithm as implemented allows the net configuration shown in Figure 4b. However, multiple net points do not imply multiple knots• Thus, hard chines and knuckles within the patch are not possible. Figure 4c shows the planar surface generated by the polygon net in Fig. 4b. Figures 4d and 4e show the polygon net and assoc- iated B-spline surface generated by the implemen- tation when the polygon net has no multiple vert i - ces. In contrast, Figs. 4f and 4g show a polygon net with multiple vertices at (2,2) and (2,3), (2,5) and (2,6), (3,2) and (3,3), (3,5) and (3,6), and the associated surface. In Figs. 4f and 4g, multiple vertices do no__t imply multiple or re- peating knot values in the knot vectors• Figure 4g shows that even without this capability, a 'cor- ner' with a very small radius can be generated. As implemented, the algorithm is limited to a fourth order B-spline surface, i . e. a bicubic sur- face.

TEST CASE

BSSD is designed to test the feas ib i l i t y of interactively, i . e., by eye, matching a B-spline surface to a known ship hull. The system can, of course, also be used for ab in i t io hull surface de- sign. In contrast to the technique implied by Munchmeyer et al [5], the objective is to use as few surface patches as feasible• Using only a few surface patches signif icantly reduces the data storage requirements for the hull surface and provides a smoother (fairer) surface• I t also allows the designer to work on larger unit portions of the ship at one time, and thus, re- duces the time required to obtain a fa i r hull with the desired characteristics•

The forebody, i . e. the portion of the ship hull surface from the bow to midway to the stem, of an existing U. S. Navy ammunition ship (AE 23) was chosen as a test case. The ship has a sharp bow with a mild underwater bulb f laring back into a parallel mid-body and an essentially f l a t bottom• [Fig. 5a]. Since the ship is symmetrical about the longitudinal plane, only half of the forebody need be considered. The ship hull sur- face does not have any knuckles or sharp corners in the area under consideration and is thus within the limits of the present B-spline surface imple- mentation•

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Figure 5:

• -

d Test ship hull surface.

216

The body or stat ions l ines* for the forebody of the ship were d ig i t i zed from the ex is t ing l ines drawing. D ig i t ized points l i e on the hul l surface, and, thus const i tu te a three dimensional data base for the surface. These d ig i t i zed points were d is- played on the Picture System I using BSSD. [Fig.5a] An impression of the extent of the shape of the surface can be obtained from Fig. 5a which shows a quarter section of the f u l l ship. The f u l l ship is 564 feet long, has a beam or width of 81 feet and a depth of 48 feet.

A single, i n i t i a l l y planar 5x7 B-spline poly- gonal net was created. This net was then interac- t ive ly manipulated using the technique described above unt i l the resulting B-spline surface matched the ship hull surface described by the digit ized points. In manipulating the B-spline surface, an additional grid or net l ine was added in the area of the bow in order to more completely match the bulbous bow. Thus, the f inal B-spline polygonal net is 5x8. The digit ized points and the assoc- iated polygon net are shown in Fig. 5b. The poly- gon net and the resulting fourth order B-spline surface are shown in Fig. 5c. This i l lustrates that the shape of the surface corresponds to that of the ship. Display or removal of the surface, polygon net, digit ized points or any combination is controlled by function switches. Figure 5d shows a section through the ship. The short straight lines are the B-spline surface. The points in the v ic in i t y of the surface are the or- iginal digit ized points and those to the r ight and outside are the polygon net points. The match be- tween the surface and the original digit ized points is quite acceptable. These results were obtained in approximately l - l I/2 hours. More effort would yield an even better match.

CONCLUSIONS AND RECOMMENDATIONS

A B-spline surface patch generation and ma- nipulation scheme has been implemented and ap- plied to a test case using an existing ship. A single 5x8 surface patch has been acceptably f i t to the entire forebody of the test ship using interactive techniques. The results are encoura- ging.

Future work w i l l concentrate on extending the algorithm to include sharp discontinuit ies, knuckles or hard lines within the patch and to in- creasing the speed of the algorithm. This techni- que should then allow even relat ively complex sur- faces to be to ta l ly represented by from three to f ive surface patches.

REFERENCES

(1) Izumida, K. and Matida, Y. Ship hull defi- ni t ion by surface techniques for production use, Proceedings of ICCAS '79 Computer Appli- cations in the Automation of Shipyard Opera- tions and Ship Design, Nc.rth Holland, 95-I04.

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Munchmeyer, F.C., Schubert, C., Nowacki, H. In terac t ive design of f a i r hul l surfaces using B-spl ines, Proceedings of ICCAS '79 Computer Appl icat ibns in the Automation of Shipyard Operations and Ship Design, North Holland, 67-76.

Rogers, D.F. and Adams, J .A. , Mathematical Elements for Computer Graphics, McGraw-Hill, New York, 1976

Rogers, D.F. B-spl ine curves and surfaces for ship hul l design, Proceedings SNAME, SCAHD '77, F i r s t In ternat ional Symposium on Computer Aided Hull Surface De f in i t i on , (September 1977), Annapolis, Maryland. 26-27

Rogers, D.F., Rodriquez, F., Sa t te r f i e ld , S.G. Computer aided ship design and the numerical ly contro l led production of towing tank models, Proceedings, 16th Design Automation Confer- ence, (June 1979), 25-27. San Diego, Cal i - forn ia .

Sa t te r f i e ld , S.G., Rodriguez, F., Rogers, D.F. A simple approach to computer aided m i l l i ng with in te rac t i ve graphics, Proceedings of SIGGRAPH '77, Computer Graphics I I , 2 (1977), I 0 7 - I I I .

Stroobant, G. Soprindus, Bruxel les-Belgigue, pr ivate communication.

Yu i l l e , I.M. The forward design system for computer aided ship design using a minicom- puter, The Naval Architect, ]20, (Nov 1978) 6, 323-341.

*body or stat ion l ines are formed by cutt ing~planes through the ship hul l surface perpendicular to the longi tud inal axis.

217