1. Surface Flattening in Garment Design Zhao Hongyan Sep. 13, 2006

2. Surface Flattening Application: aircraft industry ship industry shoe industry garment industry

3. 3D-Computer Aided Garment Design 1. Import several patterns from other 2D garment CAD systems. 2. Obtain 3D garment patterns after a sewing simulation process. 3. Modify the 3D garment patterns by FFD (free- form deformation) tools. 4. Flatten the modified garment patterns.5. 2D comparison

4. 3D-Computer Aided Garment Design It is important to flatten the modified garment patterns properly, as the modification is always done in the flattened surface in practice.

5. Problem Definition Given a 3D freefrom surface and the material properties, find its counterpart pattern in the plane and a mapping relationship between the two so that, when the 2D pattern is folded into the 3D surface, the amount of distortion — wrinkles and stretches — is minimized.

6. Measurement of accuracy Area accuracy. A : the actual area of one patch on the surface before development; A ’ : the area of its corresponding patch after development. A can be approximated by summing the area of each triangle in the facet model:

7. Measurement of accuracy Shape accuracy. L : the actual length of a curve segment on the original surface before development; L ’ : the corresponding edge length on the developed surface after development. L can be approximated by summing the length of each triangle edge in the facet model:

8. Planar parameterization

9. Floater 97 ’ Fixing the boundary of the mesh onto a unit circlea unit square

10. Planar parameterization For interior mesh points: Forming a sparse linear system

11. Surface flattening based on energy model Charlie C.L. Wang, Shana S-F. Smith, Matthew M.F. Yuen CAD 2002;34(11):823-833

12. Mass-spring systems ◆ A mass-spring system is established for the deforma- tion of Ф. ◆ Ф is a planar triangular mesh pair (K, P)

13. Mass-spring systems

14. Elastic deformation energy function Tensile force

15. Discrete Lagrange Equation Mass-spring system is governed by Discrete Lagrange Equation:

16. Discrete Lagrange Equation f i is the external force. u i is the mass value; r i is the damping coefficient; g i is the total internal force acting on vertex i, due to the spring connections to neighboring vertex j;

17. Surface flattening based on energy model  Initial triangle flattening  Planar mesh deformation

18. Surface flattening based on energy model  Initial triangle flattening  Planar mesh deformation

19. ◆ Initial triangle flattening ◆ Assume one edge ( Q 1 Q 2 ) has already been flattened.

20. ◆ Initial triangle flattening §2.1 Unconstrained triangle flattening ◆ The third node ( Q 3 ) is going to be located on the flattened plan.

21. ◆ Initial triangle flattening(2) # Developable surface # Non-developable surface §2.1 Unconstrained triangle flattening

22. ◆ Initial triangle flattening(2) §2.2 Constrained triangle flattening When two edges are both available to determine the planar point corresponding to Q3, the obtained two points, shown as P ’ 3 and P ’’ 3, may not be uniform. Original mesh trianglePlanar mesh triangle

23. ◆ Initial triangle flattening(2.2) §2.2 Constrained triangle flattening In this case, a mean position is used

24. Surface flattening based on energy model  Initial triangle flattening  Planar mesh deformation

25. ◆ Planar mesh deformation Discrete Lagrange Equation can also be written in the following form: Discrete Lagrange Equation M : spring mass; D : damping matrix; K : stiffness matrix. Ignore the damping item

26. ◆ Planar mesh deformation For each node P i, the equation can be changed to m i : the mass of P i ; ρ: the area density of the surface; q i ( t ): the position of P i at time t ; f i ( t ): the tensile force on node P i ;tensile force

27. ◆ Planar mesh deformation Penalty function Goal: t o prevent an overlap

28. ◆ Planar mesh deformation the deformation process is described by the algorithm in the following

29. ◆ Examples Example.1 a ruled surface and its 2D patternExample.2 a trimmed surface and its 2D Pattern Table. 1 Calculation statistics of Example. 1 and Example 2

30. ◆ Additional phase: Initial Energy Release Since energy was generated in the first phase: Constrained triangle flattening, overlapping error would happen.Constrained triangle flattening Original 3D mesh surfaceSurface development without energy release

31. ◆ Additional phase: Initial Energy Release Therefore, the energy release is added

32. ◆ Additional phase: Initial Energy Release Original 3D mesh surface Surface development without energy release Surface development with energy release

33. ◆ Additional phase: Surface Cutting Surface cutting Some complex surfaces difficult to develop

34. Surface cutting Firstly, compute the energy on the developed surface. ◆ Additional phase: Surface Cutting

35. Surface cutting Second, determine a reference cutting line using an elastic deformation energy distribution map. ◆ Additional phase: Surface Cutting

37. Freeform surface flattening based on fitting a woven mesh model Charlie C.L. Wang, Kai Tang, Benjamin M.L. Yeung CAD 2005;37(8):799-814

38. Woven mesh model Planar woven fabric Weft / warp springs: tensile-strain resistance Diagonal springs: shear-strain resistance Node V i,j : intersection between springs

39. Woven mesh model: assumption 1. The weft threads and the warp threads are not extendable. 2. No slippage occurs at the crossing of a weft and a warp thread. 3. A thread between two adjacent crossing is mapped to a geodesic curve segment on the 3D surface.

40. Woven mesh model: assumption The directions of weft and warp springs are orthogonal to each other. Users specify Initial length of springs: r weft, r warp.(r diag ) Center Node: V i C, j C Tendon node: V i,j (i=i c, or j=j c ) Region node: otherwise. Type- I/II/III/IV node

41. Strain energy

42. Basic idea Fit a woven-like mesh (woven mesh) model onto a 3D surface M; Map the surface point onto the plane.

43. Basic idea Fit a woven-like mesh (woven mesh) model onto a 3D surface M; Map the surface point onto the plane.

44. Fitting methodology TNM (tendon node mapping) DNM (diagonal node mapping) Diffusion process

45. Fitting methodology TNM (tendon node mapping) DNM (diagonal node mapping) Diffusion process

46. TNM Specify a center point p C and a warp direction vector t warp on M. Compute the weft direction vector t weft. Call Algorithm ComputeDiscreteGeodesicPath(V i C, j C, t warp, M,r warp ). Iteratively until the boundary of M is reached. Determine all the tendon nodes.

47. Fitting methodology TNM (tendon node mapping) DNM (diagonal node mapping) Diffusion process

48. DNM Four quadrants For a type-I node V i,j (1) Assume V i-1,j-1, V i-1,j and V i,j-1 all have been positioned; (2) Determine two unit vectors and ; (3) Set the diagonal direction as t diag =1/2(t 1 +t 2 ); (4) Staring at V i-1,j-1, search the point on the geodesic path along the t diag direction with distance r diag, by calling Compute Geodesic algorithm iteratively;

49. Strain energy release (5) Locally adjust the position of V i,j

50. Boundary propagation

51. Fitting methodology TNM (tendon node mapping) DNM (diagonal node mapping) Diffusion process

52. Energy minimization by diffusion Goal: minimize the strain energythe strain energy Solution: let every node V i satisfy where

53. Insertion of darts 1. A specified space curve 2. Delaunay triangulation 3. The fitted woven mesh

54. Basic idea Fit a woven-like mesh (woven mesh) model onto a 3D surface M; Map the surface point onto the plane.

55. Surface to plane mapping

56. Experiments

57. Comparison