Dallas, August Volume 20, Number 4, 1986 I

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1 Dallas, August Volume 20, Number 4, 1986 I The Synthesis of Cloth bjects Jerry Weft AT&T Bell Laboratories Murray H, New Jersey 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 physical 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 synthesized 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 properties of cloth, such as folds, and they therefore lack a degree of realism. Taking into account the physical properties 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. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission, 1986 ACM /86/008/0049 $00.75 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 location at chosen constraint points. The problem is to determine a possible solution to the way in which the cloth w hang from these constraint points. Determining a smooth surface for the cloth w 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 w be assumed rectangular, and w be represented as a grid, or twodimensional array, of three-dimensional coordinates. The grid is treated 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 surface 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 iterated 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, therefore we begin by positioning points lying between pairs of constraint 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: 49

2 m a x v Y = ~(e 7+e--~)=aesh(~) By tracing catenaries between each pair of constraint points [13], the grid points which lie along the lines between constraint points can be positioned. A line between 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 system. 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 ustrates 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 computational complexity of adding such new points each time two curves cross, one of the two curves w 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 w naturally 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 eliminated, and the upper one remains. Y Relaxation S I G G R A P H '86 The next step in the process of determining the surface of the cloth involves an iterative relaxation stage. The relaxation of the surface is achieved by propagating the displacement of grid points over the surface until the maximum displacement in a single pass falls below a certain tolerance. The displacement of the points during each pass is determined through approximations to physical constraints. This is by no means an exact solution, but it is merely meant to achieve a reasonable looking surface through straightforward means. Imm Imm Film ll Imm il iml lira lllm Ill -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 (2 mmlmmmmmmmmmmmmmmm mmmmmmmm,muuumminnnnnmumn li ilmm -"i lllllll lllll in II Illllll Illl IN ii llllll ii llllll in ,.,... II llll J llllll llll llli iii-" iiiii----,."nl,,...'.--. lmmmmmmmmmm mmmml mmmmmmmmmmmmm mmmmm mmmmmmmmmmmmmmmmmmm b m a) original constraint points b) triangles formed by connecting constraint points e) subdivision of triangle d) subdivision of newly formed triangle., x Figure 2. Four stages of surface approximation Figure 1. Two crossing catenary curves The surface approximation stage consists of this tracing of catenary curves. In the first step, a curve is traced from each constraint point to each other constraint point. The curve equation can be found from the two endpoints and the length of the thread hanging between (see Appendix for derivation of this equation). ne result of the process of positioning points between constraint points is a series of triangles of connected constraint points. As constraint points are connected (by tracing catenaries between them), these triangles are created and added to a database of such triangles. This list of triangles w be used in the next step of the surface approximation stage. f course, if a curve is removed due to a crossing, any triangle utilizing the removed curve as an edge must also be removed. A similar process to the one described above is now performed on each triangle in the current list. Each triangle, which represents a section of the cloth, is treated as a separate entity, and w be repeatedly subdivided until each grid point in its interior has been positioned. To determine how to subdivide the triangle, 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 process until all interior points have been positioned. Figure 2 ustrates a sample of such processing. After every triangle has been processed this way, the interior surface w be closely approximated. For the sake of simplicity, gravity is ignored during the relaxation process. In order to achieve a hanging effect in the direction of gravity, the exterior points are initially placed at 3 = -oo. This simulates a downward pull, and the relaxation process gradually repositions these points upward until the constraints are met. The effects of gravity have already been accounted for by catenary modelling of the interior region If the cloth is not free-falling, the exterior points may be initially placed at some other locations. For example, to model a cloth being lifted from a flat surface, the exterior points would be placed at their initial locations on that surface. To determine the displacement of each point, the constraints of the cloth must be examined. The placement of any given point on the surface is one unit distance from each of its four-connected neighbors (without stretching). In order to determine the displacement of a point in a given pass, displacement vectors are determined to position the point at the correct distance from each of its neighbors. By adding these vectors, the direction of displacement is found. The optimal magnitude for the final displacement vector is difficult to determine, since it would be necessary to predict the displacement of the future points to be examined. To emphasize the influence of larger displacement vectors, the squares of the magnitudes of the vectors are averaged, and the square root of this result is used as the final magnitude. It is possible that, following the displacement of a grid point, the surface w intersect itself. For simplicity, surface intersections are neither tested for nor corrected in this implementation, and this has not proven to be a problem in the cases tested. Some materials are stiffer than others, and therefore w 50

3 I - a Dallas, August Volume 20, Number 4, 1986 not bend as much. This property is easily incorporated into the technique already described. ne way to accurately model the property of stiffness is to measure the angle formed by three consecutive grid points; the stiffer the cloth, the less the angle may deviate from completely stiff material would not allow this angle to deviate at all. This calculation would be fairly costly, and even more difficult would be determining how to rotate the points in order to increase the angle if necessary. A simpler solution which achieves a similar result is to examine the distance between each grid point and the points two rows or columns away. This distance must be greater than a certain minimum distance, determined by the stiffness of the cloth. An added advantage to this method is that it fits in perfectly with the method described above for determining displacement vectors. If a point is determined to be too close to another point two rows or columns away, another vector is added which places it at the minimum distance from that point. Several times, the tolerance for relaxation has been mentioned. This value is more or less arbitrary, depending on the accuracy desired. However, this tolerance is somewhat related to the constraints placed on the cloth. For example, the stiffer the cloth is modelled to be, the more iterations may be necessary for convergence. We do not attempt to establish a relationship between the degree of stiffness and the value chosen for the tolerance; the tolerance values for this research have been determined by experimentation. Although the surface approximation stage can be completely eliminated, using only the relaxation stage, the initial approximation can greatly reduce the number of iterations necessary during the relaxation stage. There are situations when it is more practical to use the relaxation stage by itself. ne such example is in animation, when the constraint points are gradually moved for each frame. In this case, computing each frame could consist of updating the positions of the constraint points followed by relaxation processing on the remaining points. The relaxation stage would also be useful by itself if the cloth had already taken on some general shape - the shape of the human body if modelled as clothing, or the shape of furniture if modelled as upholstery. Rendering nce the surface of the cloth has been relaxed, it can be easily converted to a polygonized surface, ready for any of several rendering techniques. Rendering the surface in this manner may appear realistic for some materials, but, in general, the surface w st not have a cloth-like quality. A cloth texture can be mapped onto the surface, but the translucent effect of some types of cloth may st not be achieved. There are other 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 thread individually. This is certainly computationally more expensive than the other methods mentioned, but this kind of detail allows for realistic close-ups of the cloth. The rendering technique used here is a ray-tracer [141 which treats the cloth as a collection of line segments, the endpoints of which are four-connected neighboring points in the grid. These lines are not treated as one-dimensional lines, but as shapes with thickness and depth. In reality, the threads of a piece of cloth are somewhat cylindrical in shape. As w be seen, this cylindrical shape is simulated by the perturbation 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 approximation and relaxation stages become computationally intensive when run on a fine grid of points. Furthermore. there is not much difference between the overall structure of a surface calculated on a very fine grid from that calculated on a much coarser grid. Therefore, the two-stage process described above is run on a relatively coarse grid, and the remaining points are fed in by fitting splines to the calculated grid coordinates [12]. A finer mesh is created by first fitting splines to the grid points along each column of the grid. Corresponding 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 surface points by the raytracer ( Figure 3 ). riginal Points Q t 6 Column Spline Fit e : :.... e ee e...i"" e Ruby Spline Fit Figure 3. Spline fitting of surface Next, the necessary information for each line is placed into a database. This information includes the endpoints of the line as well as three shading values for each endpoint. Three shading values are used to model the line as a cylinder. The three values are computed based on three normal vectors, which are determined in the following manner. The directions of the first two normal vectors are found by projecting the line segment onto the z=0 plane and then finding the two oppositely directed vectors which lie in that plane and are perpendicular to the projection, i.e. (-dy,dx,) 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 treated as normal vectors, and thus the three shading values can be computed ( Figure 4 ). Assuming the scene has been normalized to a rectangular view volume, rays can be cast into the scene perpendicular to the viewing plane to be tested for intersections with the threads in the scene. To determine if a ray intersects a thread, the minimum distance is found from the ray to the line segment representing the thread. If this distance is within a chosen tolerance, the ray intersects a cylinder surrounding the line segment and therefore intersects the thread. This distance tolerance represents the radius of the cylinder, and it can be altered to represent various thicknesses of cloth - the larger the tolerance, the thicker the threads, and, therefore, the thicker the cloth. Since calculating the exact distance from the ray to the line is a bit time consuming, this distance is approximated. nly the horizontal or vertical distance from the ray to the line is found, depending on the slope of the line. This approximation w result in variations of the thicknesses of the cylinders by a factor of "V/2-, but this variation is hardly noticeable because of the size and density of the threads in the cloth. This approximation also results in the cylinders appearing two-dimensionally as parallelograms rather than rectangles. Again, this factor is hardly notice- Q 51

4 // ~. S I G G R A P H '86 I III 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: Case 1: Idyl > Idx[ dy =~z--y 1) dz =(z z -z 0 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 distance 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 solutions; 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 modelling 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 ( ~ax~ 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 b~ a } (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 '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. 52

5 Dallas, August Volume 20, Number 4, 1986 I References (1) Blinn, J., Computer Display of Curved Surfaces, University of Utah, Salt Lake City, December (2) Blinn, J., "Simulation of Wrinkled Surfaces," Computer Graphics, Vol. 12, No. 3, August 1978, pp (3) Blinn, J. and Newell, M., "Texture and Refl~ction on Computer Generated Images," Communications of the ACM, Vol. 19, No. I0, ct. 1976, pp (4) Catmull, E., A Subdivision Algorithm for Computer Display of Curved Surfaces, University of Utah, Salt Lake City, December (5) Foley, James D. and van Dam, Andries, Fundamentals of Interactive Computer Graphics, Addison-We~sley, Reading, Massachusetts, (6) Hathorne, Berkeley L., Wo~'en Stretched and Textured Fabrics, Interscience Publishers, New York, (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 (8) Irvine, H. M., Cable Structures, M.I.T., (9) Kajiya, James T., "Anisotropic Reflection Models," Computer Graphics, Vol. 19, No. 3, July 1985, pp (10) Mer, L., "Computer Graphics and the Woven Fabric Designer," Computers in the World of Textiles, Annual World Conference, Hong Kong, Sept. 1984, pp (11) Physical Methods of Investigating Textiles, Edited by R. Meredith and J.W.S. Hearle, Textile Book Publishers, Inc., New York, 1959, pp (12) Rogers, David F. and Satterfield, Steven G., "B-Spline Surfaces for Ship Hull Design," Computer Graphics, Vol. 14, No. 3, July 1980, pp (13) Tensile Structures, Edited by Frei tto, M.I.T. Press, Cambridge, Massachusetts, (14) Whitted, Turner, "An Improved Illumination Model for Shaded Display," C..C.M., Vol. 23, No. 6, June 1980, pp 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

6 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) 54