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Dallas, August 18-22 Volume 20, Number 4, 1986 I

The Synthesis of Cloth Objects

Jerry Weft AT&T Bell Laboratories

Murray Hill, New Jersey 07974

Abstract

In image synthesis, cloth objects such as clothes are most often modelled as textures mapped onto rigid surfaces. However , in order to represent such objects more realistically, their physi- cal properties must be examined. This paper describes a method for modelling cloth material hanging in three dimensions when supported by any number of constraint points. The cloth syn- thesized with this model contains folds and appears more realistic than ~imple texture mapping. This paper also describes a method for rendering the cloth once its free-hanging shape has been determined.

The computation of the surface of a free-hanging cloth is performed in two stages. The first stage approximates the shape of the surface which is interior to the constraint points, and the second stage performs a relaxation process on all points on the surface to arrive at a close approximation to its shape. The rendering of the surfaces is done using a ray-tracer which treats the surface as a mesh of line segments.

Introduction

In the field of computer graphics, objects made of cloth are usually modelled as rigid surfaces with textures mapped onto them [1,2,3,4]. These surfaces do not have the proper t ies of cloth, such as folds, and they therefore lack a degree of realism. Taking into account the physical propert ies of such objects would lead to more realistic looking scenes. Not only is the shape of the surface important in achieving realism, but the method for rendering the cloth objects in an image is also important.

The applications for modelling cloth accurately are varied. Aside from the industrial applications in the fashion and textile industries, realistic looking clothes and other cloth objects could enhance computer generated animation or computer synthesized advertisements. The research described in this paper relates to methods for modelling arbitrary cloth objects, but the focus of the paper deals with one specific problem.

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The Problem

This paper examines one solution to a very specific problem. A piece of cloth exists in three dimensions, and it is fixed in loca- tion at chosen constraint points. The problem is to de termine a possible solution to the way in which the cloth will hang from these constraint points. Determining a smooth surface for the cloth will not suffice, since, in reality, folds may occur in the cloth.

The Method

First, it is necessary to find a way to represent the cloth to be modelled. For the purposes of this paper , the cloth will be assumed rectangular, and will be represented as a grid, or two- dimensional array, of three-dimensional coordinates. The grid is t reated as a two-dimensional coordinate system, the axes of which consist of the row and column axes. The grid coordinate system should not be confused with the object 's coordinate system, which is a standard three-dimensional system with x, y and z axes. By increasing the density of the grid, greater resolution of the sur- face model may be obtained.

There are two stages to the method described here. [n the first stage, an approximation is made to the surface within the convex hull of the constraint points in the grid coordinate system. Some natural constraints of the cloth are ignored during this stage of processing, therefore the folds which would in reality appear over the surface may not appear after the completion of this stage. The constraints of the cloth are applied during the second stage of processing, which involves an iterative relaxation process [7]. The relaxation of points is i terated over the surface until the maximum displacement of the points during one pass falls below a predetermined tolerance.

Surface Approximation

A differentiation must be made between the interior and exterior points on the grid. The interior points are those which lie within the convex hull formed by the constraint points in the grid. coordinate system, and the remaining points are considered as exterior points. The surface approximation stage is completed when all interior points have been positioned somewhere in three dimensions by the method described below. At the start of this process, only the constraint points have been positioned, there- fore we begin by positioning points lying between pairs of con- straint points.

For a piece of cloth made of woven threads, the positioning of points along any given thread can be determined by examining the physics of such a model. The curve which an ideal thread naturally follows when suspended by two points is called a catenary [8], and is of the form:

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m S I G G R A P H '86

a x v Y = ~(e 7+e--~)=aeOsh(~)

By tracing catenar ies between each pair of constraint points [13], the grid points which lie along the lines between constraint points can be positioned. A line be tween constraint points refers to the (row,column) coordinates through which a line scan-converted from one point to the other would pass in the grid coordinate sys- tem. Thus , if one constraint point was at grid coordinate (2,3) and another was at (5,3), the line between the two points would include grid coordinates (3,3) and (4,3). What should be done in the case of a grid coordinate through which more than one such line passes? Figure 1 i l lustrates such a case. In reality, the correct position for the point through which the lines pass may be somewhere between the two curves traced along those lines, thus adding a new constraint point to the model. To avoid the compu- tational complexity of adding such new points each t ime two curves cross, one of the two curves will simply be removed. Ignoring the forces which the two crossing curves may apply to each other, the points along each curve have been positioned as low as they will natural ly fall (by definition of the catenary curve). According to this model , no point on the upper curve can move any lower; therefore, .the lower of the two curves is chosen to be e l iminated, and the upper one remains.

Y

. , . . . . . . x

F igure 1. Two crossing catenary curves

The surface approximat ion stage consists of this tracing of catenary curves. In the first step, a curve is t raced from each constraint point to each other constraint point. The curve equa- tion can be found from the two endpoints and the length of the thread hanging between (see Appendix for der ivat ion of this equation). One result of the process of posi t ioning points between constraint points is a series of t r iangles of connected constraint points. As constraint points are connected (by tracing catenaries between them), these tr iangles are c rea ted and added to a database of such tr iangles. This list of t r iangles will be used in the next step of the surface approximat ion stage. Of course , if a curve is removed due to a crossing, any tr iangle utilizing the removed curve as an edge must also be removed.

A similar process to the one described above is now per- formed on each triangle in the current list. Each tr iangle, which represents a section of the cloth, is t reated as a separa te enti ty, and will be repeatedly subdivided until each grid point in its inte- rior has been positioned. To de te rmine how to subdivide the tri- angle, three catenaries are traced, one from each vertex to the midpoint of its opposite edge. The highest of the three is chosen to subdivide the triangle. Each triangle is subdivided by this pro- cess until all interior points have been posit ioned. Figure 2 illus- trates a sample of such processing. After every tr iangle has been processed this way, the interior surface will be closely approxi- mated.

Relaxation

The next step in the process of de termining the surface of the cloth involves an i terat ive relaxat ion stage. The relaxat ion of the surface is achieved by propagat ing the d isp lacement of grid points over the sur face until the m a x i m u m disp lacement in a sin- gle pass falls below a certain tolerance. The d isp lacement of the points during each pass is de t e rmined through approximat ions to physical constraints . This is by no m e a n s an exact solution, but it is merely meant to achieve a reasonable looking surface through s t ra ightforward means .

imm imm Imm imm Imm Film imm i l l l l imm Imm i l i l l i m l l i ra l l lm I l l i l l i l l i l l i l l i l l

-H- -HH- +H- -H- +H- +H- -H- -H+ -H+ -H- -H+ -H+ -H- +H- +-H- -H- -H+ -H+ -H- +H- -H+ - H - q - H - - H + -H- +H- -H+ -H- -H+ ~-H-

a b

(2

a) original constraint points

immmmlmmmmmmmmmmmmmmm mmmmmmmm m

,muuumminnnnnmumn l i | | i l m m -" i l l l l l l l l l l l l i N I I I l l l l l l I l l l I N i i l l l l l l i i i l l i l l l l l l l l i n • i l l i l l . . . . . . . . , . , . . .

iii-" i l l I I i l l

l l l l • J i l l l l l l l l • l l l l

• l l l i

iiiii----, . " n l , ,.... '.--. . . . . lmmmmmmmmmm mmmml mmmmmmmmmmmmm mmmmm immmmmmmmmmmmmmmmmmmmm

b) t r iangles fo rmed by connect ing constraint points e) subdivision of tr iangle d) subdivision of newly fo rmed triangle

Figure 2. Four stages of surface approximat ion

For the sake of simplicity, gravity is ignored dur ing the relaxat ion process. In order to achieve a hanging effect in the direction of gravity, the exter ior points are initially placed at 3 = -oo. This s imula tes a downward pull, and the relaxat ion pro- cess gradually reposit ions these points upward until the con- s t raints are met . The effects of gravity have already been accounted for by ca tenary model l ing of the interior region• If the cloth is not free-fal l ing, the exterior points may be initially placed at some other locations. For example , to model a cloth being lifted f rom a flat surface , the exterior points would be placed at their initial locations on that surface.

To de te rmine the d isp lacement of each point, the con- s t raints of the cloth mus t be examined. The p lacement of any given point on the surface is one unit dis tance from each of its four-connected neighbors (without stretching). In order to deter- mine the d i sp lacement of a point in a given pass, d isp lacement vectors are de t e rmined to posit ion the point at the correct dis- tance f rom each of its neighbors. By adding these vectors, the direct ion of d isp lacement is found. The optimal magni tude for the final d isp lacement vector is difficult to de te rmine , since it would be necessary to predict the d isplacement of the future points to be examined . To emphas ize the inf luence of larger dis- p lacement vectors, the squares of the magni tudes of the vectors are averaged, and the square root of this result is used as the final magni tude . It is possible that , following the d isp lacement of a grid point, the surface will intersect itself. For simplicity, sur- face intersect ions are nei ther tested for nor corrected in this implementa t ion , and this has not proven to be a problem in the cases tested.

Some mater ia ls are s t i f fer than others , and therefore will

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Dallas, August 18-22 Volume 20, Number 4, 1986 I

not bend as much. This property is easily incorporated into the technique already described. One way to accurately model the proper ty of s t i ffness is to measure the angle formed by three con- secut ive grid points; the st iffer the cloth, the less the angle may devia te f rom 180 ° - - a comple te ly stiff mate r ia l would not allow this angle to deviate at all. This calculat ion would be fairly costly, and even more diff icul t would be de te rmin ing how to rotate the points in order to increase the angle if necessary . A s impler solution which achieves a similar resul t is to examine the dis tance between each grid point and the points two rows or co lumns away. This dis tance mus t be g rea te r than a cer ta in m i n i m u m distance, de t e rmined by the s t i f fness of the cloth. An added advantage to this me thod is that it fits in perfect ly with the me thod described above for de te rmin ing d i sp lacement vectors. If a point is de te rmined to be too close to ano the r point two rows or co lumns away, ano ther vector is added which places it at the m i n i m u m distance f rom that point.

Several t imes, the to lerance for re laxat ion has been men- tioned. This value is more or less arbi t rary , depending on the accuracy desired. However , this to lerance is somewha t re la ted to the const ra ints placed on the cloth. For example , the st iffer the cloth is modelled to be, the more i terat ions may be necessary for convergence . We do not a t t empt to es tabl ish a re la t ionship be tween the degree of s t i f fness and the value chosen for the tolerance; the to lerance values for this research have been deter- mined by exper imenta t ion .

A l though the sur face approximat ion s tage can be com- pletely e l iminated, us ing only the re laxa t ion s tage, the initial approximat ion can greatly reduce the n u m b e r of i terat ions neces- sary dur ing the re laxat ion stage. There are s i tuat ions when it is more practical to use the re laxat ion s tage by itself. One such example is in animat ion, when the constra int points are gradual ly moved for each f rame. In this case, comput ing each f rame could consist of updat ing the posit ions of the const ra in t points fol lowed by relaxat ion processing on the remain ing points . The re laxat ion s tage would also be useful by itself if the cloth had already taken on some genera l shape - the shape of the h u m a n body if model led as clothing, or the shape of fu rn i tu re if model led as uphols tery .

Rendering

Once the sur face of the cloth has been re laxed, it can be easily converted to a polygonized sur face , ready for any of several render ing techniques . Rende r ing the sur face in this manne r may appear realistic for some mater ia ls , but, in genera l , the sur face will still not have a cloth-like quali ty. A cloth texture can be mapped onto the surface , but the t r ans lucen t effect of some types of cloth may still not be achieved. The re are o ther ways to render the cloth more realistically, such as using Kajiya 's recent work with anisotropic surfaces [9], but probably the most realistic way to render the cloth is to actually render each th read individually. This is certainly computa t iona l ly more expensive than the other me thods men t ioned , but this k ind of detail allows for realistic close-ups of the cloth.

The render ing technique used here is a ray- t racer [141 which treats the cloth as a collection of line segment s , the endpoints of which are four-connected neighboring points in the grid. These lines are not t reated as one-d imens iona l l ines, but as shapes with th ickness and depth. In reali ty, the threads of a piece of cloth are somewhat cylindrical in shape. As will be seen, this cylindri- cal shape is s imulated by the per turba t ion of normals [2],

In order to achieve a surface which does not appear like a fish net, a very fine mesh of lines must be fit to the surface . However , the surface approximat ion and relaxat ion s tages become computat ional ly intensive when run on a fine grid of points. Fu r the rmore . there is not much d i f fe rence between the overall s t ruc ture of a sur face ca lcula ted on a very fine grid f rom that ca lcula ted on a much coarser grid. T h e r e f o r e , the two-stage process descr ibed above is run on a relat ively coarse grid, and the remain ing points are filled in by fi t t ing splines to the calculated

grid coordina tes [12]. A f iner mesh is c rea ted by first f i t t ing splines to the grid points a long each co lumn of the grid. Cor responding points along each of these splines are then used as the knots for splines to be fit along the rows. The points along this last set of splines are used as the sur face points by the ray- t racer ( Figure 3 ).

6 O

Q

O

t

O

Q

O

Original Points Column Spline Fit

• e • • • • • • • • •

• : : . . • . . • e e e

• • • • e •

• • • • • • e

. . . i "" Ruby Spline Fit

Figure 3. Spline f i t t ing of sur face

Next, the necessary informat ion for each line is placed into a database. This in format ion includes the endpoints of the line as well as three shading values for each endpoint . Th ree shading values are used to model the line as a cylinder. The three values are computed based on three normal vectors, which are deter- mined in the following manner . The directions of the first two normal vectors are found by project ing the line s egmen t onto the z=0 plane and then f inding the two oppositely d i rected vectors which lie in that plane and are perpendicu la r to the project ion, i.e. (-dy,dx,O) and (dy , -dx ,0 ) . The third normal lies along the vector which is perpendicular to the line segment as well as the first two normal vectors. These three directional vectors are all t rea ted as normal vectors, and thus the three shading values can be computed ( Figure 4 ).

Assuming the scene has been normal ized to a rec tangular view volume, rays can be cast into the scene perpendicu la r to the viewing plane to be tested for in tersect ions with the th reads in the scene. To de te rmine if a ray intersects a thread , the m i n i m u m distance is found from the ray to the line segment represent ing the thread. If this d is tance is within a chosen toler- ance, the ray intersects a cylinder sur rounding the line segment and therefore intersects the thread. This dis tance tolerance represen ts the radius of the cyl inder , and it can be al tered to represent various th icknesses of cloth - the larger the tolerance, the thicker the threads , and, the re fore , the th icker the cloth. Since calculat ing the exact distance f rom the ray to the line is a bit t ime consuming , this dis tance is approximated . Only the hor- izontal or vertical dis tance from the ray to the line is found, depend ing on the slope of the line. This approximat ion will result in variations of the th icknesses of the cyl inders by a factor of "V/2-, but this variat ion is hardly noticeable because of the size and densi ty of the threads in the cloth. This approximat ion also results in the cyl inders appear ing two-dimensional ly as paral lelo- g rams ra ther than rectangles . Aga in , this factor is hardly notice-

51

/ / ~. S I G G R A P H '86 I I I I

able in the final image. To compute the distance from a ray at location (x,y) to a line from (xl,yt,zl) to (x2,yz,z2), two cases are possible:

dy =~z--Y 1)

dz =(z z - z 0

Case 1: Idyl > Idx[ Y--Yl

dy

distance=x- (xl +t dx)

z =z I ~t dz Case 2: Idyl ~ Idxl

X - - X 1

dx

distance=y- (yl+t dy)

z=zl+t dz

(Distance)

(z-intersection)

(Distance)

(z-intersection)

If the computed distance is within the chosen tolerance, the dis- tance is normalized to lie between - 1 and 1, and, by linearly interpolating between the three shade values, an appropriate shade can be determined. Nonlinear interpolation could be used to model other three-dimensional surfaces as well. Note that division by zero is a special case.

Conclusion

The techniques outlined in this paper lead to the creation of more realistic looking images of cloth objects. Many of the methods described involve approximations rather than exact solu- tions; however, the approximations achieve results which appear realistic. A very specific problem has been addressed here, which is to model the appearance of a piece of cloth which is suspended at certain constraint points. The algorithms described for solving this problem can be extended for other uses, such as for model- ling clothes or for use in animating cloth objects. A method for rendering cloth has also been described in which a ray-tracer is used to render a mesh of line segments. Improvements might be made toward the time and space efficiency of such an algorithm.

Several enhancements can be made to the algorithms described here for more general situations. Such enhancements could include the addition of propagating forces to create wave patterns in animation, or additional constraints which would allow the cloth to be draped over solid objects.

Appendix

Solution to finding the catenary equation between two points:

Assume the two endpoints are (xt ,yl) and (xz,y2) and the length is L. To solve for the equation of the catenary passing between

the two points, y =c +acosh ( : ~ - I :

.=c+ocosh (1)

L osi h/ } } By subtracting (2) from (1), squaring, and subtracting (3) squared,

L 2-b'2-Yl]2 2a 2 "c°sh {L-~} -1 (4)

By the half angle formula and some rearranging,

~v/ L 2-D'2-)' l]2 =2asinh ( ~ a X ~ I (5)

A numerically. Since a =~-, from (3) we find that

Letting:

it follows that:

If N>M,

If M >N,

(7)

(8)

-~- ~M cosh I~'}-N sinh I a b~ } (9)

~=tanh-1 { NM~ ) (10)

M N Q= sinh(~) cosh(p.) (11)

[ t, ~a Jl

M N Q= cosh(~) sinh(~) (14)

b--a ~-cosh -1 ~- (15)

Knowing a and b, the solution for c is straightforward.

Acknowledgements

I would like to thank Larry O'Gorman for his continued help, advice and criticism with this research. Special thanks to John Hughes of Brown University for supplying the derivation in the Appendix and thanks to David Laidlaw for organizing it. I would also like to thank Dave Hagelbarger for introducing me to the catenary.

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Dallas, August 18-22 Volume 20, Number 4, 1986 I

References

(1) Blinn, J., Computer Display of Curved Surfaces, University of Utah, Salt Lake City, December 1978.

(2) Blinn, J., "Simulation of Wrinkled Surfaces," Computer Graphics, Vol. 12, No. 3, August 1978, pp. 286-292.

(3) Blinn, J. and Newell, M., "Texture and Refl~ction on Com- puter Generated Images," Communications of the ACM, Vol. 19, No. I0, Oct. 1976, pp. 542-547.

(4) Catmull, E., A Subdivision Algorithm for Computer Display of Curved Surfaces, University of Utah, Salt Lake City, December 1974.

(5) Foley, James D. and van Dam, Andries, Fundamentals of Interactive Computer Graphics, Addison-We~sley, Reading, Massachusetts, 1982.

(6) Hathorne, Berkeley L., Wo~'en Stretched and Textured Fabrics, Interscience Publishers, New York, 1964.

(7) Hsu, M.B., "An Interactive Graphics Program For The Equilibrium Shape Determination For Tensioned Fabric Structures," Engineering Software for Microcomputers, Proceedings of the First International Conference, Venice, Italy, 1984, pp. 227-237.

(8) Irvine, H. M., Cable Structures, M.I.T., 1981.

(9) Kajiya, James T., "Anisotropic Reflection Models," Com- puter Graphics, Vol. 19, No. 3, July 1985, pp. 15-21.

(10) Miller, L., "Computer Graphics and the Woven Fabric Designer," Computers in the World of Textiles, Annual World Conference, Hong Kong, Sept. 1984, pp. 634-644.

(11) Physical Methods of Investigating Textiles, Edited by R. Meredith and J.W.S. Hearle, Textile Book Publishers, Inc., New York, 1959, pp. 211-278.

(12) Rogers, David F. and Satterfield, Steven G., "B-Spline Sur- faces for Ship Hull Design," Computer Graphics, Vol. 14, No. 3, July 1980, pp. 211-217.

(13) Tensile Structures, Edited by Frei Otto, M.I.T. Press, Cam- bridge, Massachusetts, 1967.

(14) Whitted, Turner, "An Improved Illumination Model for Shaded Display," C..C.M., Vol. 23, No. 6, June 1980, pp. 343-349.

Surface Approximation

6 Iterations of RelaxaHon

Spline Fit

Figure 4. Rendering of single thread showing three normals used to compute shading values

Ray Traced Image Figure $. Four stages in synthesis of cloth suspended at corners

53

S I G G R A P H '86

Figure 6. Cloth lifted by five points (corners and center) Figure 7. Three stages of cloth lifted by corners (only relaxation was used)

Figure 8. Image with varying coarseness and thickness of threads (bars are threads also)

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