Surface Flattening in Garment Design

Zhao Hongyan

Sep. 13, 2006

Surface Flattening

Application: aircraft industryship industryshoe industrygarment industry

3D-Computer Aided Garment Design

1. Import several patterns from other 2D garmentCAD 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

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.

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.

Measurement of accuracy Area accuracy.

(4),s

A AE

A

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:

1

0

. (5)n

ii

A A

Measurement of accuracy Shape accuracy.

(6),C

L LE

L

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:

1

0

. (7)m

i

i

L L

Planar parameterization

Planar parameterization Floater 97’ Fixing the boundary of the mesh onto

a unit circle a unit square

Planar parameterization

For interior mesh points:

Forming a sparse linear system

Surface flattening based on energy model

Charlie C.L. Wang, Shana S-F. Smith, Matthew M.F. Yuen

CAD 2002;34(11):823-833

Mass-spring systems

◆ A mass-spring system is established for the deforma-tion of Ф.

◆ Ф is a planar triangular mesh pair (K, P)

Mass-spring systems

Mass-spring systems

Elastic deformation energy function

2

1

1( )

2

n

i i j jj

E P C PP d

Tensile force

1

( )i j

n

i i j j P Pj

f P C PP d n

Discrete Lagrange Equation

Mass-spring system is governed by

2

2, ,i i

i i i i i ijj

d x dxu r g f g s

dt dt

Discrete Lagrange Equation:Discrete Lagrange Equation:

Discrete Lagrange Equation

2

2, ,i i

i i i i i ijj

d x dxu r g f g s

dt dt

fi is the external force.

ui is the mass value; ri is the damping coefficient; gi is the total internal force acting on vertex

i, due to the spring connections to neighboring vertex j;

Surface flattening based on energy model

Initial triangle flattening

Planar mesh deformation

Surface flattening based on energy model

Initial triangle flattening

Planar mesh deformation

◆ Initial triangle flattening

◆ Assume one edge (Q1Q2) has already been flattened.

◆ Initial triangle flattening§2.1 Unconstrained triangle flattening

◆ The third node (Q3) is going to be located on the flattened plan.

◆ Initial triangle flattening(2)

# Developable surface

# Non-developable surface

§2.1 Unconstrained triangle flattening

◆ 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 twopoints, shown as P’3 and P’’3, may not be uniform.

Original mesh triangle Planar mesh triangle

◆ Initial triangle flattening(2.2)

§2.2 Constrained triangle flattening

In this case, a mean position is used

Surface flattening based on energy model

Initial triangle flattening

Planar mesh deformation

◆ Planar mesh deformation Discrete Lagrange Equation can also

be written in the following form:

0Mq Dq Kq M: spring mass; D: damping matrix;K: stiffness matrix.

0Mq Kq Ignore the damping item

◆ Planar mesh deformationFor each node Pi, the equation can be

changed to

2

, (10)3

, (11)

, (12)

. (13)2

i k

ii

i

i i i

i i i i

m A

f tq t

m

q t t q t tq t

tq t t q t tq t q t

mi: the mass of Pi ;ρ: the area density of

the surface;qi(t): the position of Pi

at time t;fi (t): the tensile force

on node Pi;

◆ Planar mesh deformation Penalty function Goal: to prevent an overlap

**

*1

1 ( )

0 ( )

(14)

mpenalty j j

penalty penalty j jj penalty j j

C h hF C h h n

C h h

(15)t ti i penaltyq q F

◆ Planar mesh deformation the deformation process is described by

the algorithm in the following

◆ ExamplesExample.1 a ruled surface and its 2D pattern Example.2 a trimmed surface and its 2D Pattern

Table. 1 Calculation statistics of Example. 1 and Example 2

◆ Additional phase: Initial Energy Release Since energy was generated in the first

phase: Constrained triangle flattening, overlapping error would happen.

Original 3D mesh surface Surface development without energy release

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

◆ Additional phase: Initial Energy Release

Original 3D mesh surface

Surface development without energy release

Surface development with energy release

◆ Additional phase: Surface Cutting

Surface cutting Some complex surfaces difficult to

develop

Surface cutting Firstly, compute the energy on the

developed surface.

◆ Additional phase: Surface Cutting

Surface cutting Second, determine a reference

cutting line using an elastic deformation energy distribution map.

◆ Additional phase: Surface Cutting

◆ Additional phase: Surface Cutting

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

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

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.

Woven mesh model: assumption

The directions of weft and warp springs are orthogonal to each other.

Users specify Initial length of springs: rweft, rwarp.(rdi

ag) Center Node: ViC,jC

Tendon node: Vi,j (i=ic, or j=jc) Region node: otherwise. Type-I/II/III/IV node

Strain energy

Basic idea

Fit a woven-like mesh (woven mesh) model onto a 3D surface M;

Map the surface point onto the plane.

Basic idea

Fit a woven-like mesh (woven mesh) model onto a 3D surface M;

Map the surface point onto the plane.

Fitting methodology

TNM (tendon node mapping)

DNM (diagonal node mapping)

Diffusion process

Fitting methodology

TNM (tendon node mapping)

DNM (diagonal node mapping)

Diffusion process

TNM Specify a center point pC and a warp dire

ction vector twarp on M. Compute the weft direction vector tweft.

Call Algorithm

ComputeDiscreteGeodesicPath(ViC,jC, twarp, M,rwarp).Iteratively until the boundary of M is reached. Determine all the tendon nodes.

Fitting methodology

TNM (tendon node mapping)

DNM (diagonal node mapping)

Diffusion process

DNM Four quadrants

For a type-I node Vi,j

(1) Assume Vi-1,j-1, Vi-1,j and Vi,j-1 all have been positioned;

(2) Determine two unit vectors and ;

(3) Set the diagonal direction as tdiag=1/2(t1+t2);

(4) Staring at Vi-1,j-1, search the point on the geodesic path along the tdiag direction with distance rdiag, by calling Compute Geodesic algorithm iteratively;

1 1, 1, 1 1, 1, 1/ || ||

i j i j i j i jt V V V V

2 , 1 1, 1 , 1 1, 1/ || ||

i j i j i j i jt V V V V

Strain energy release

(5) Locally adjust the position of Vi,j

Boundary propagation

Fitting methodology

TNM (tendon node mapping)

DNM (diagonal node mapping)

Diffusion process

Energy minimization by diffusion

Goal: minimize the strain energy Solution: let every node Vi satisfy

where

Insertion of darts

1. A specified space curve

2. Delaunay triangulation

3. The fitted woven mesh

Basic idea

Fit a woven-like mesh (woven mesh) model onto a 3D surface M;

Map the surface point onto the plane.

Surface to plane mapping

Experiments

Comparison

Comparison