Liao and Adanur [8] put up a method to 3D modeling using a computer aided geometric design (CAGD). In this approach, the yarn is assumed to be a uniform cylinder which has a constant cross-section and the center line is an arbitrary spatial curve. The base cross-section is determined first, and it is divided into many arc by several

points. A unique plane through any point on the center line and perpendicular to it can be defined. By sweeping the base cross-section curve along the points on the center line, all the points that determine the quadrilateral facets are automatically generated by coordinate transformation. The principle is well explained in Figure 3. Thus, the two 2D curves (yarn cross-section and yarn central line) are converted into a 3D surface that determines the shape of the object. In the model, the path of the yarn can be the combination of sinusoidal curve and segment line while the cross-section of the yarn can be circular, elliptical and race-track shapes which depend on the construction of the fabric. The structure of 3D woven fabric and braided fabric have been established. However, the cross-section of the yarn is uniform, which is unable to describe the true fabric structure and limits the application.

Lomov [9] proposed a 3D yarn model similar to Liao’s model, and software Wise Tex is developed based on Lomov’s work. Figure 4 demonstrates Lomov’s research. Lin and Long [10] made use of the similar technique to model the yarn comprising of many individual sections of regular shape such as ellipse, lens that can be locally edited to avoid yarns penetrating when used in fabric structure.

In order to reflect the effects of the uneven yarn, Li [11] builds a yarn which is made up with circles which has different diameters. The diameters can be acquired from the yarn uniformity tester. Segmentation on the surface of the yarn will be carried out by the evenly division of each circle along the center line of yarn. Capitalized on the number of the segment peaks, the contour of the yarn can be obtained by the list of the surface, edge and peak. This method is suitable to straight yarn and it can well reflect the variation of the cross-section. Figure 5 shows the 3D yarn contour in this method. However, the model can only show the straight yarn and cannot be applied to the yarn in fabric structure.

Much work has been done to represent the different shapes of the yarn for the changing of shapes varies due to the flattening and strain. Gong and Ozgen [12] [13] acquired the fabric images through ESRF (European Synchrotron Radiation Facility) without disturbing the yarn and fabric structure. The contours of the yarn in the woven fabrics were accurately measured. They confirmed the non-circular nature of yarn cross-sections and also the flattening of yarns at the warp-weft intersections. Two polynomial functions were used to describe the shape of the varied cross-section. The cross-section shape which was modeled by elliptical and polynomial curves was variable along the yarn path. Figure 6 is the 3D representation in this method. Unfortunately, Gong didn’t give a real 3D picture to demonstrate his research.

Similar to Liao’s method, Zhang [14] proposed that the central line of the yarn can be constructed by Bezier curve. The cross-section of the yarn is circular. First, a serial of interpolating points on the central line of the yarn are calculated or given. The control points determining the Bezier curve (central line) are reversely calculated by the interpolating points. Then, applying the sweeping technique, the vertexes of the control grid that determine the Bezier surface are calculated. Different from quadrilateral facets technology, the surface composes of many triangles which ensure the 3 vertexes of any triangle are at the same plane. The triangulation of the Bezier surface will be conducted automatically by OpenGL. The images of 3D yarn can be rendered by Computer Graphics principle.

Jiang and Chen [15] proposed an approach of super ellipse which can design the cross-section in arbitrarily shape. At the same time, the central line of the yarn is represented as spline curve that can pass the given control points. The shape of the cross-section can be designed manually, which makes it convenient to change the yarn shape. The uneven yarn can be modeled and bent. The algorithm can be used in constructing the fabric structure. The software Tex Gen is based on Chen’s work.

Nowadays, using B-spline surface is the most popular method to model the configuration of 3D yarn. B-spline curve technology gives the designer more convenience to model the shape of the yarn accurately. Lin and Newton [16] used the cubic B-spline curve to generate the yarn paths, and circle as the form of cross-section. Recently, and Sherburn [17] are able to design irregular yarn using B-spline curve technology. Like Bezier

curve, the control points have to be calculated reversely by interpolating points. Unlike Bezier curve, the control points are dependent on the initial conditions, which also causes no uniqueness in control the shape of the yarn. If there is a tiny difference in initial conditions, for example, the positions of the end points change, there may be a big difference in the shape of the curve generated.

Zheng [18] [19] proposed an algorithm that the yarn can be modeled by NURBS (Non Uniform Rational B-Spline Curve) to keep a stable shape of the yarn. The shape preserve quasi uniform cubic B-spline curve is applied to fit the contour of the yarn path accurately, smoothly and stably. Two more adjacent points are inserted before and after a interpolating point respectively according to the lending direction of the all interpolating points, which ensures that the curve go through all the interpolating points. Therefore, the complicated reversely calculation is avoided in determining the control points. In Zheng’s research, the quasi uniform quadratic B-spline curve is applied to approximately design the cross-section of a single yarn. The shape of the cross-section can be controlled by a 16-polygon formed by 18 control points. The shape of circle, ellipse, racetrack, lens, bowl, round rectangle can be designed along the yarn path simultaneously (see Figure 7). As long as the base cross-sections at given position are designed, the other cross-sections will be calculated automatically and smoothly. Each cross-section will be put perpendicular to central path. OpenGL is used to design NURBS surface and automatically generate the triangle facetsand give its shading effect. Moreover, some fancy yarns can also be designed by using different cross-section along the path, which will be introduced in the latter part.

Arif Kurbak [20] [21] also built 3D yarn model with 3DS-MAX for knitted fabric. The yarn paths were calculated according to the knitted structure. The cross-section of the yarn is set as circle, therefore, the yarn is actually a cylinder. The loop and tuck heads were taken as ellipses in two dimensions while the rest of the loops and tucks were taken as parabolic helices wrapped on circular cylinders. Thus, the path of yarn in different kind of knitted fabric can be clearly revealed.

To sum up, Table 1 gives a brief summary of characteristics of all the models introduced above.

Zheng [25] added 1D and 2D texture on the surface of yarn to simulate the twist of yarn. For 1D texture as shown in Figure 10(A), parallel lines are drawn on the surface of the yarn with the help of OpenGL. The slop of the lines is dependent on the twist angle of the yarn. The lines are uniformly distributed and the twist effect will be worsen when magnify the yarn. For 2D texture mapping technology as shown in Figure 10(B), an image or a picture scanned from the real yarn will be mapped on the surface of the 3D model to reflect the twist and even hairiness. Ma [26] (Figure 10(C)) and NedGraphics Jacquard Software use the similar mapping technology. Obviously, it requires numerous pictures to imitate different twist effects for actual application.

In order to improve the twist effects, Zheng [19] [27] [28] added bumping texture to acquire more realistic effect or 3D twist effect. By adding troughs and ridges on the contour of the cross-section of the yarn, the surface of the 3D yarn will appear bright and shade effect. When transforming the troughs and ridges at different sections, the twist effect will appear. If a trough or a ridge is deliberately discarded at random, the migration effect of fibers in the structure of the yarn is imitated. The principle is explained in Figure 11 and the imitated image is shown in Figure 12.

Zhao [29] proposed another algorithm to model the yarn based on slice superposition method and C-Cardinal spline curve, which is similar to the shape preserve B-spline curve. The curve passes through the interpolating points except the end points. The drawback can be avoided by repeating the end points two times. Similar to the bumping texture in Zheng [19] , the cross-section of the yarn is an irregular polygon as shown in Figure 13(A). By rotating the polygons, a number of polygons are obtained and the series of polygon slices are combined to form a yarn with 3D twist effect as shown in Figure 13(B). The advantages of using the C-Cardinal spline curve are embodied in easy calculation, small memory required. However, the disadvantage is that the curve has not been supported by OpenGL or other software package. If the 3D image is enlarged enough, the slices are separated and the yarn appears discontinuously.

Sriprateep [30] also proposed an algorithm based on fibers’ migration in yarns to imitate the 3D twist effect of the ring spun yarn. Each yarn is considered as the assembly of fibers. Each fiber is regarded as a helix along yarn path. The cross-section of the yarn is thought as elliptical shape. Therefore, the yarn can be approximately

constructed by NURBS. The migration parameters include twisting tension, migration period and amplitude, initial phase and frequency of migration pattern. The geometry of yarn structures is eventually modeled by using the Solid Works CAD software package and the model is validated by imitating the process of spinning rayon yarn (see Figure 14). The difference between the algorithm and Zheng’s is that Scriprateep uses many NURBS surfaces while Zheng only uses single NUBRS surface to imitate 3D twist effect.

A realistic model for 3D yarn should reflect the hairiness effect. Zhong [31] achieved the aim by using projected triangles to simulate the hairiness, but the effect could be weakened if the 3D image was enlarged. As Figure 15(A) shown, Xu [32] realized a photorealistic effect yarn by rotating an image including many lighten points. The shade effect, twist effect and hairiness effect were demonstrated excellently. However, it is not a real 3D model for yarn as the image generated cannot be rotated or viewed in different direction. Effect in this approach is in Figure 15(B). Zheng [19] proposed that NURBS surface be used to imitate the projected fiber similar to the technique for yarn. The length, projecting angle and the distribution of the hairs can be designed.

Folded yarn comprises of two or more single yarn, so it can be designed if the central paths of all the component yarns are determined. In Zheng [33] , a two-ply or three-ply folded yarn is modeled by rotating two or three circular cross-section around so called folded yarn center, therefore, the central line of each component single yarn is actually a helix. The sizes of the twist are controlled by the screw pitch of the helix while the direction of the twist is determined by the rotating angle. In Zhao and Zheng [34] , the cross-section of the single yarn in the folded yarn was assumed to be circular or irregular polygon by slice superposition method which is shown in Figure 16(A). In order to demonstrate the twist effect of the folded yarn, at least 4 slices of cross-sections and their centers of each component yarn will be calculated for every other 90˚. Figure 16(B) shows a bent folded yarn based on the algorithm similar to Zheng [22] [28] . If more than 20 single yarns are folded together, the folded yarn appears as a single yarn as each yarn is so slim that looks like a single fiber. Figure 16(C) gives the effects which is similar to Sriprateep’s work [30] .

It is certain that the fancy yarn can be modeled by 3D technology although it is rare. For fancy yarn, there are varied sections along the yarn path. It can be folded yarn whose components are of various color, thickness or configuration in space. Based on Zheng’s [19] [35] [36] algorithm, the cross-sections and configuration of the yarn can be changed based on NURBS, the fancy yarn will be modeled. Figure 17(A) demonstrates a slub yarn

modeled on the principle. In fact, loop yarn, knop yarn, slub yarn, snarly yarn can also be designed when the frequency/size of the effect spot, path/color of the component single yarn are designed. Based on C-Cardinal spline curve, Zhao [34] demonstrates a slub yarn as shown as Figure 17(B). The slub yarn includes even parts and uneven parts which change from thinner to the thicker diameter. The twist effect can be improved by increasing the rotating a larger angle of the two neighboring slices at the thinner area.

Zhuge [36] designed the contour of fancy yarn was drawn by Bezier tools. Then template of illumination factor was applied to the model. Then, loop hairiness, end hairiness or floating hairiness can be generated by random noise point.

If the geometric structure in a woven, knitted or braid fabric is calculated, or in other words, the shape and the path of the each yarn in fabric is determined, the 3D geometric structure of the fabric can be imitated. Figure 18(A) shows a plain fabric with the diameter of the yarn changes along the central path; Figure 18(B) shows the same fabric structure with 3D twist effect; Figure 18(C) shows a plain fabric comprised of folded yarn; Figure 19(A) and Figure 19(B) [35] show two fancy woven fabrics respectively; Figure 20 shows a knitted fabric structure.