Modelling of Sculpturesque Surfaces

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Modelling of Sculpturesque Surfaces V.A. Ossipov

VologodskiPolytechnicalInstitute,UI. Voroshilova3, 160600 Vologda,USSR

The paper presents two modules of a 15-module library for computer-aided geometric and graphic modelling of sculpturesque surfaces. A special contour method provides for the design and computation of surfaces enveloping transport machine-building objects, such as those for exteriors of ships, automobiles, etc. A "method of two relations" provides for the design and computation of any contoured surface.

Keywords:Mathematical

modelling, Geometric modelling, Computer-aided graphic modelling, Sculptured surfaces, Sculpturesque surfaces, CAD, CAM, Design, Aircraft design, Ship design, Automobile design, USSR.

V.A. Ossipov, born in 1936, graduated in 1958 from the Ufa Aviation Institute and went on to take a Cand. Sc. at the Moscow Aviation Institute in 1962. He obtained a D.Sc. degree in 1974. From the very beginning his research was concerned with the problems of automating design, engineerhag and product development process of sculpturesque surfaces of complex engineering shapes, as well as with geometrical and mathematical modelling of objects and processes in mechanical engineering, He headed the Chair of Descriptive Geometry and worked in close contact with industrial fellow researchers and engineers, helping them to deploy the resources offered by CADAM. V.A. Ossipov is the author of two textbooks and 4 monographs and the number of his scientific papers comes to 150. Currently, V.A. Ossipov is Rector of the Polytechnical Institute in the North oft the Soviet Union. His work now is concerned with CADAM, expert systems, and the use of computers in education. North-Holland Computers in Industry 11 (1989) 239-248

1. Introduction Strictly speaking, the m o d e l l i n g of sculpturesque surfaces is to b e d i v i d e d into the m o d e l l ing of objects p r o p e r ( s c u l p t u r e s q u e surfaces of various types) a n d m o d e l l i n g of processes (design, calculation, r e p r o d u c t i o n ) . H e r e we will speak a b o u t the m o d e l l i n g i n v a r i a n t as to " o b j e c t p r o p e r " o r " p r o c e s s " . Otherwise, it m a y b e called " o b j e c t of a t t e n t i o n " (Oa). T h e core of the t h e o r y of m o d e l l i n g in C A D is d i v i d e d into two intellectual levels (Fig. 1). The u p p e r level is a b s o l u t e l y subjective; its p r o d u c t is visualized o n l y in the m i n d of a designer as some k i n d of " i d e a " . A s a rule, with rare exceptions, the i d e a is b a s e d on a definite p r o t o t y p e . T h e designer c o m p a r e s the two a n d this enables h i m to isolate specific g e o m e t r i c c o n d i t i o n s . Fulfilling t h e m t h r o u g h certain " g e o m e t r i c t r a n s f o r m a t i o n s " the designer can realize the " i d e a " on the basis of its " p r o t o t y p e " , i.e. to synthesize the object of attention ( " o b j e c t " p r o p e r or " p r o c e s s " ) . H e r e then the o b j e c t of a t t e n t i o n is r e p r e s e n t e d b y its geometrical m o d e l ( G M O ) . All the a b o v e - l i s t e d p r o c e d u r e s of analysis a n d synthesis are a b s o l u t e l y subjective since they are b a s i c a l l y b r a i n - w o r k of the designer. A p a r a l l e l m a y be t r a c e d to h a r d w a r e representation, a n a l y sis a n d synthesis of an o b j e c t of attention. C o m p o n e n t s of the lower level: linguistic m o d els of O a ( d e s c r i p t i o n , users' instructions, designers' m a n u a l , general technical instructions, etc.);

i

ic~e,z ~ proto~d/~

Fig. 1.

0166-3615/89/$3.50 © 1989 Elsevier Science Publishers B.V. (North-Holland)

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qualified software designers as effective as possible.

3. Principal Definitions

i !

<)~,/ec/ o/ o['Z'e:z t:: ~,/:

Fig. 2.

mathematical models of Oa (equations, inequalities, domains of parameters, etc.); digital models of Oa (optimum digital data arrays); graphical models of Oa (bidimensional and three-dimensional drawings); and physical models of Oa, etc. are used to translate GMOa into some exterior form of information (Fig. 2).

2. Models of Objects of Attention (Oa) As to their structure models of Oa may differ as follows: "wired", "piecewise-framed", "allframed", "solid", etc., and each appears in more than 5 or 6 forms. Hence, we come to the conclusion that the core of the theory of modelling attends to every CAM problem of parts for mechanical engineering: CAD, robots, and flexible computerized manufacturing. The aforegoing statement testifies to the fact that the problem of modelling exists at all stages of production and is invariant of industry specifics. If we take into consideration the lack and high cost of software development for modelling, and the scarcity of powerful interactive hardware networks (processors and peripherals), and also the high level of invariance of modelling problems in the field of geometry and graphics in particular, it would be only safe to say it is high time we solved our problems of establishing special-purpose computer centres and laboratories of computer geometry and graphics serving a wide variety of enterprises or research institutes as well as groups of higher schools. This would give the opportunity to ultimately load computing facilities and use highly

Let us formulate now a number of principal definitions of fundamental components of the core of the modelling theory of Oa. By geometrical model of a geometrical object (GMO) or process (GMP) we mean a set of geometrical technicalities specifying object or process with required degree of detail and also geometrical transformations which enable the designer: - to mentally visualize the object or process; - formulate the representation of the object or process in the only variant possible way (linguistic, physical, mathematical, digital, or graphical) by the generation of the respective algorithm. Mathematical modelling of objects amounts to the mathematical description of its geometrical model. Let us call a mathematical model of a geometric object (MMO) a set of equations, inequalities, and restrictions and their consistency conditions (including domains of all shape and position parameters) which solely define the object in the given coordinate system, in other words, there is a unique value of, say, the applicate corresponding to every two particular coordinates of the point of the object. The mathematical model of process (MMP) is a set of equations, inequalities, and some other restrictions and their consistency conditions (including domains of all shape and position parameters) which solely define process in the given coordinate system. The graphical model of a geometrical object is its projective drawing. G r M O has accuracy depending on the mode of its implementation, e.g, accurate to drawing and instrumental errors (manual implementation); accurate to techniques of mathematical modelling and automatic plotting devices' errors (computerized hardware implementation). GrMO or GrMP are derived on the basis of their mathematical models and the algorithm of their representation. Graphical model interpreting (GMI) refers to projective drawing presenting (graphically in-

V.A. Ossipov / Modelling of sculpturesque surfaces

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terpreting) the results of hardware processing of the information. Graphical model constructive (GrMC) refers to projective drawing presenting (graphically interpreting) the results of hardware processing of the information and providing decision procedures of position and metric development problems with GrMI and computer graphics and geometric algorithms. By a digital model of a geometrical object (DMO) we mean three coordinate arrays Xi, ~, Z i of geometrical object points definitely arranged (e.g. on the basis of MMO) and optimized according to the criterion of minimum number of points and desired computing accuracy of object. Then, a digital model of a process (DMP) is an array of digital parameter values. Their interaction is claimed by a set of algorithms and functional being fundamental for MMP. DMO and DMP offer different services: information storage, substitution of one MMO or MMP by another, development of approximation or interpolation models, etc; and estimation of accuracy rating characteristics, etc. The process of the ordered substitution of continuous MM and DM is, in fact, what we call an optimized discretization of object or process. This is what the process of digital modelling is.

241

Consider some theoretical and practical questions as far as the development methods and corresponding software in CAGGr are concerned. The variety of existing intricate engineering surfaces calls for adequate modelling methods. In this line every surface type is served by the definite method of modelling. When choosing one must take into account the criteria of maximum accuracy rating and at least time consumption at all stages of modelling. Amalgamated classification gives rise to the following techniques: frame-wired modelling, modelling on the basis of basic elements, frame piecewise modelling, and framed kinematic modelling. During the last 25 years our geometrical school obtained certain results in the development of the latter two techniques of surface modelling of intricate engineering forms in the field of transport machine-building in particular and especially in cases when the surface being designed is streamlined by the surroundings (gas, fluid, conglomeration mixtures, and free-flowing bulk materials).

5. The Library of Modelling Methods 5.1. A Method for Special Contours

4. Computer-Aided (CAGGr)

Geometry

and Graphics

The theory of modelling, being developed within the scope of CAGGr, is instrumental in the optimum solution of object and process modelling problems in mechanical engineering. Extensive efforts have been and are under way at present to develop software in the field of computer-aided geometry and graphics. 30 former and 10 present postgraduates, and a number of researchers and engineers under the supervision of the author of this report are taking part in the work. Part of the software has been reduced to engineering practice, some software product is being developed, and other modules are passing evaluation tests in industry. The ultimate goal of these scientific efforts is the methodological, system, and structure component development of CAGGr and development as a whole.

A library of sculpturesque surface modelling methods functionally connected with the flow of the surroundings has been formed on the basis of the presently available results. Two methods are presented here. (1) A method for special contours provides for the design and computation of surfaces enveloping objects of transport machine-building. The surface in question is specified by continuous frame of transversely generating closed or open surfaces perpendicular to the flow of the surroundings. Generators of such type may be intersection curves of the planes perpendicular to the longitudinal axis of the product and surfaces of the exterior lines of the ships, automobiles, etc. The following main requirements are imposed on the method of special contour: maximum simplicity of analytical job, possibility of designing fairly intricate contours to meet dynamic and assembly demands, design of contours symmetrical as well as nonsymmetrical about axes Y and Z of

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+? A

I

.030?.~Y

I

I/..

PIN I

i

/I

t

.

I

I\

\i

k k

i

?>.- / \ I

I

/[

-

Fig. 3.

the coordinate system O X Y Z of the product, where X is the axis of distance. Our study has shown that the equation

from the corner of the epure coordinate system origin (ECS) with the corners of the square (Fig. 3). Thus, we think that

( 2 2 - 1)(Y 2 - 1) = 0

m = F(8),

(i)

may be considered the analytical expression of a unite (epure) square, two identity elements on the side (Fig. 3), and coordinate origin O in the intersection point of diagonals ~ 3 5 is referred to further in the text as orthographic epure coordinate system (ECS). On the basis of (1) we come to the conclusion that the domain of existence of epure square including bounderies is defined as follows: Y = _+1; Z = +_1. We can r e w r i t e ( l ) a n d it is good as contour computation algorithm: y=

+¢

1-Z2

-

1

-

Z

2= + l -¢y z -1 - y 2

2

'

(2)

8 = O D = ~yeTo+ Z 2 .

Parameter of contour shape m is introduced to one of the equations (2). After that it describes the epure contour with umbilical angles (Fig. 3). It is good to describe contours symmetrical about yand 5-axes by equations derived from formulae (2):

F= +¢ 1-22 -

1 -- m Z 2 '

1- F2 Z= + ¢ y_m~2 .

(4)

We evaluate m for the given by the theoretical drawing contour by expression (4) (Fig. 3): M= ~2_(1_2

Let us introduce parameter of shape contour control m. It is geometrically interpreted by value e of the distance of the contour diagonal points

(3)

2)

(5)

Now we study the domain of shape parameter m. Thus, m = + 1 corresponds to epure square (1),

V.A. Ossipov / Modelling of sculpturesque surfaces

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(2), and m = 0 corresponds to incircle of this epure square (4) (Fig. 3) at YD = Z-o = 0,707, and 8 = 1 accurate to engineering calculations. The research has shown that the range - 3 ~
Z0

243

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,~ >.

/

2

%

~n

z

\

Fig. 4.

If rn + 0, then length of segments tends to zero (Fig. 3). It is convenient to deduce points of practical contour separation from the sides of the square joining accuracy criterion A to the computation algorithm. The criterion is equal to the difference of the corresponding coordinates of epure contour and square functions. For instance, the conditional equation for coordinates of separation points of horizontal sides can be written as follows:

eters of longitudinal and cross-mapping transformations into the equation. The latter provide for Y and Z compression and extension of the whole contour, half-contour (the left and right contour sections abott.t Y-axis, the upper and lower sections about Z-axis, Fig. 4), any quarter-contour above these axes. MSC is good to quantify product line involving horizontal and vertical rectilinear sections, sine and cosine curve intercepts, and curve intercepts of any shape. If we substitute shape control function m = m( Z, A, B, C . . . . ), where shape control parameters A, B, C correspond to present contour points for shape parameter m in the above formulae, then contour shape may vary widely giving diverse convex and concaved sections.

IYRI ~ < I - A ,

5.2. A Method for Any Contour

--i --i

6oRj

--i --i

--i --i 60L = 6 0 P = 1

ZR = F(Y, )

(6)

(7)

(lock for the corresponding contour equation). In many cases we expect the value A = 0.001 to satisfy the accuracy of engineering calculations. Thus, interdependent parameters m and ~ of the contour shape characterize the degree of its growing round as to the initial epure square. The interaction of the support function of the epure square and of parameter of shape control m characterizing the degree of the contour "umbilicability" gives the shape of the resulting contour. We may proceed with the shape variation of the acquired contour introducing additional parameters into its geometry equation. Working algorithms leave the possibility to introduce param-

The method of two relations (MTR) provides for design and computation of any contoured surfaces. M T R is based on continuous transformations of Lie and Klein type of point sets. The development of M T R pursues the following objectives: setup of software tools for outline design satisfying as many as 6 or more conditions; and outline design of complicated shapes, i.e. closed nonsymmetrical, to put it shorter, not quite to the pattern. To accomplish the stated objectives a method of two relations is proposed using the techniques of nonstandard matrix of shape parameters for the construction of discrete range of curve points. The change in values of matrix elements results in the

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Fig. 5.

deformation of the curve, thus giving the opportunity to control the shape of outline over a wide range. Computer calculations of point coordinates of the outline in the form of a matrix meets the challenges of CAD. In the Lie and Klein theory of continuous transformation groups nonstandard matrices of shape parameters designate every plotting point of the outline as a result of continuous transformations of points into one another. Our method envisages the construction of the curve of any point density definitely designated on the plane. A pair of points (A, C) given graphically (Fig. 6(b)) or through coordinates and tangents in them (tA, t~) serve as initial data. Triangle A B C formed by tangents tA, tc, and a chord is assumed for a figure of constant elements. By

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/

,

P

a ~

tc

tA

/

c Fig. 6.

t~ ,\\

d

Lagrange theorem there always exists a point t.'~ AC of the continious curve on intercept AC in which the tangential curve is parallel to the chord contracting the arc ends. Let's introduce parameter X = B D / B A = B F / B C specifying the tangent, position parallel to the chord and parameter/~ = D E / D F , specifying point of tangency E. The construction algorithm reads as follows. In triangle ABC, knowing parameter X, we draw tangent parallel to chord A C: knowing parameter ~, we find point E being sought for. As a result, we get two triangles A D E and EFC. We treat both as basic units and using parameters values X and /~ plot the consequent tangents and their points of contact. We get the two successive points K and L of the curve. Now we have four triangles and find four tangents and four points of contact, and so on. The number of plotted tangents and their points of contact increases in geometric progression. The desired curve is constructed enveloping the obtained tangents or arbitrarily dense set of points. Thus, the curve construction algorithm enables one to determine position n of the tangents, each of them specifying a single point of the outline. In other words, the algorithm gives the possibility to obtain the sufficient number of the outline points in which tangents are determined. The shape of the curve bears a certain relationship to the variation of angles of the figure of constant elements in cases where parameter values X and /~ are constant. The curves constructed by the method of two relations are determined by the figure of constant elements (two points and tangents in them) and by shape parameters X and /~. What invariant shape parameter values X and ~ the geometry of the curve deforms depending on the variation of angles L~ and L 2 of the tangent inclination in the desired points of intersection? With the variation of the angles the figure of constant elements (triangle A B C ) converts into triangle AB~C, where B ~ is point of intersection of tangents t a and t~ (Fig. 6(b)). There is the perspective affine correspondence of the two triangles. Chord A C is in fact the axis of affinity as a geometric locus of fixed points. Points B and B i are the pair of corresponding points since they convert one to one into another. The perspective-affine transformation of the plane is determined by axis of affinity A C and the pair of corresponding points B and BL

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To sum it up, the curve is subjected to the perspective-affine transformation with the variation of angles L 1 and L 2 in the figure of constant elements. Let us qualify the dependence of shape parameter values and constant for every curve inside the invariant figure of constant elements. The variation of shape parameter values X and /~ sets up the vaguer limit of the curve arcs. In this way with the variation of parameter from 0 to 1 and constant value of parameter we get a set of curve arcs, each having its own constant value of parameter X (Fig. 6(b)). The maximum of arcs (points E . . . . ) pack on a straight line Bb which divides chord A C in the ration equal to the accepted value of parameter/~. At X = 0 the curve generates into chord AC. At small values X (0; 0,2; 0,3) the branches of the curves approach the given tangents but in the neighbourhood of point E the radius of curvature of the curve tends to infinity. The greater the value of parameter X, the more the radius of curvature. So with the change of X from 0 to 1 and at constant /~ the curve as if "rises" or "lowers" along straight line Bb inside the figure of constant elements. With the change of parameter ~t, on the other hand, from 0 to 1 and at constant X, we get a set of curve arcs, their points E packing along tangent t e (Fig. 6(d)). At ~ = 0 the curve generates into a broken line going a part along left tangent t A. At /~ = I the curve generates into a broken line going partly along right tangent t c. When/x ranges from 0 to 0,5 the curve as if "shifts" to left tangent tA, the left branch of the curve approaching this tangent. When ~t ranges from 0,5 to 1, the curve is "shifted" to the fight tangent, its right branch approaching then to tangent t~. Thus, with variation of parameter/~ from 0 to 1 the curve as if " m o v e s " from the left to the right on tangent t E, its position being determined by the accepted value of parameter )~. Furthermore, the field of the figure of constant elements is actually covered with curve arcs while ranging parameter values/~ and X. Parameters X and /~ are called here shape parameters of the curve since their variation permits to change the shape of any intercept of the curve. We increase or decrease the radius of the curvature in the region of, say, point K. In case we want to shift this point either to the left or to the right we have to change parameter value/~.

245

Let us consider the curves described with the matrixes of shape parameters X and /~. When constructing plane curves by the method of two relations shape parameters X and ~ are used, n --* ~ times depending on the required number of outline points. Value parameters X and /~ of every successive point constructed may be either equal to the Preceding )~ and /z or differ, and still otherwise they may be their functions. Shape parameters X and /z involved in the construction of points of the curve may be represented in the form of nonstandard matrixes (7) and (8). The curves obtained have several parameters. Elements of the first matrix line are involved in the construction of one curve point (corner E ) and elements of the second line are involved in the construction of the two successive points (vertices of arcs A C and EC). Every successive matrix line gives twice the number of the points as the preceding one. The shape of the curve obtained greatly depends upon the change in parameter value of the first matrix line (Fig. 6). The more the number of the line of the matrix, the less the dependence of curve distortion upon the change of parameter X and bt. Thus, in case of the curves of several parameters )~ and ~ parameter matrices can serve as means of adaptive control of the shape of the curve. Curves of several parameters are effective to approximate of the outlines of complex contoured engineering forms. Every successive point is said to be constructed on the basis of two preceding points according to curve point construction algorithm. Consistently, coordinates of every successive point are converted from coordinates of preceding points. Here is a formula for coordinates of intermediate points. To deduce this formula we place the figure of constant elements on Cartesian coordinates xOy. To get the solution, let us consider a particular problem. We are given points of the figure of constant elements

A ( XA, YA), C( Xc, Yc), B( XB, yh). Point E ( X E, YE) is to be found provided we know parameters and ~ and /~. We determine point D knowing shape parameters

BD BA

XB -- X D XB-X A '

x~= x ~ - xx~ + xxA, X D = ( 1 -- X) XB + XX,4.

(8)

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Then we find point F knowing BF

X B - X£

BC

Xe - Xc '

(9)

x(.= x ~ - x xB + x x,, = ( 1 - ) ~ ) X B + X X c.

Now we find point E knowing the shape parameter DE I~= D F -

XD - XF XD_XF"

It appears that

XE= X~-- .XD + ~,XF = ( 1 - t t ) X D + # X F.

(10)

We substitute values from expressions (8) and (9) into expression (10) and get a recursion formula: XE= (1 - ~)[(1 - X)XB + XX~] +~[(1 -X)X~+XXc]

(11)

We write expression (11) in the machine-processible form of a matrix;

(1 -x)x~+xG x~= I1-~, ~li(I 1 X ) X B + X X c II-X,

Xl X~

jl-X

xl x,, {12)

~ 2 ~ YB ~ ~ X ~

(1 - X ) Y B + X Y c

i

I ] l - X , Xl = I I - t ~ , ~1

rBil

to set up mathematical models for the range of the points straight lines, broken lines intercepts of the curves of second order (parabola, huperbola, ellipse, circumference), and intercepts of parabolic and transcendental curves (cordiods, astroids, Mungler ovals, cycloids strophoids, Cartesian folium, etc.). Decision procedures to quantify parameter X are listed here for the curves of second order provided curve arcs, position of the centre of the curves, and length of the segments on the axes are graphically given. Continuous affine transformations of arbitrary shaped curves determined by matrixes of constant shape parameters and /t have been investigated. The development of MTR [2,3] was preceded by the works by Riesenfeld, Satherland, J.H. Clark, Chajkina, and some other American and European authors [4-9]. We can say that all the techniques mentioned here are actually particular MTR cases of the method of two relations. We claim that the development of these techniques was accomplished quite independently. At present software consisting of 15 modules has been set up under the author's supervision. The software serves all imaginable types of sculpturesque surfaces. Each of the remaining 12 modules not included in the article can be published on request. Surface design and computation algorithms based on the above-mentioned 14 modules of computation complex plane generators can be divided into two types: design algorithms for all-framed surfaces; - design algorithms for piecewise framed surfaces. All algorithms afford smoothness of the second order of the designed surfaces.

I

IYA

References

[ll_X,X[I YBfI l

rc

6. Closing Remarks The article [3] covers in detail the results of the investigations related to mathematical modelling of the most common contours by the method of two relations. The author claims that it is possible

[1] V.A. Ossipov, Problems of forming of computer-aided system of geometry and graphic, in: T.M.R. Ellis and O.J. Semenkov, eds., Advances in CAD~CAM, North-Holland, Amsterdam, 1982, [2] V.A. Ossipov, Computer Techniques for Design of Allframed Surfaces, Mashinostroenie, Moscow, 1979. [3] S.M. Bajkalova, Design and computation of plane outlines of several parameters to the predetermined conditions, in: Kibernetica Graphiki & Priklanaay Geometriay Poverkhnostei, MAI, Moscow, 1975, pp. 37-39. [4] S.A. Coons, Surfaces for computer-aided aircraft design, Vd, Vol. (4), 1968, pp. 402-406.

Computers in Industry [5] G.H. Clark, Designing surfaces in 3-D, Comm. ACM, Vol. 19 (8), 1976, pp. 454-460. [6] A.P. Apmit, A multipaten design system for Coons patchs, 1968, pp. 152-161. [7] S.A. Coons, Surfaces patches and b-Spline curves, in: P.E. Barnhil, P.F. Riesenfeld, eds., Computer aided Geometric design, Academic Press, New York, 1974. [8] W. Gordon and R.B. Riesenfeld, Spline and Surface. Computer-aided Geometric Design, Academic Press, New York, 1974.

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[9] M.G. Cox, The numerical Evolnation of B-Splines, Inst. Mxms Aprelis, 10, 1972, pp. 134-149. [10] Yu. V. Davydov, Shape parameter of generalized equation of special contour, in: Mashinnoye Proektirovaniye, Uoyzka 7 Vosproizvedeniye Slozhnykh Detaley, Irkutsk, 1976, pp. 104-106. [11] V.A. Ossipov, O.I. Konovalov and V.A. Andreev, Computerized design of continious frames of canal surfaces, in: Izvestiya VUZOV, Seriya "Aviatsionnya Tekhnika", Kazan, 1975, N 4, pp. 17-24.