Character Modeling using Physically based Deformable Curves

L. H. You

, E. Chaudhry

, X. Jin

, X. S. Yang

and Jian J. Zhang

National Centre for Computer Animation, Bournemouth University, Poole, Dorset, U.K.

State Key Lab of CAD & CG, Zhejiang University, Zhejiang, China

Keywords: Character Modelling, Physically based Deformable Curves, Sculpting Forces, Ordinary Differential

Equations, Analytical Solution.

Abstract: Curve and physically based surface modelling techniques are becoming more and more active in geometric

modelling of three-dimension (3D) objects since the former can create 3D models easily and quickly and the

latter can produce more realistic appearances. In this paper, we present a curve and physically based surface

modelling technique to create 3D models of virtual characters. This technique is based on physically based

curve deformation. With such a technique, a character model is created from a number of surface patches.

Each surface patch is obtained from physically based deformable curves. We introduce sculpting forces into

a vector-valued ordinary differential equation to control the physically based deformations of curves. By

describing concentrated sculpting forces with a sine series, we present an efficient analytical solution of the

ordinary differential equation which can determine physically based curve deformations quickly. An

example of character modelling is presented to demonstrate the application of our proposed technique.

1 INTRODUCTION

Virtual characters are widely applied in many

industries. Currently, the most popular modelling

techniques of virtual characters are polygon,

NURBS and subdivision.

A lot of literature addresses polygon modeling

such as Autodesk Maya Press (2006), Russo (2005),

Apostol (2012), Patnode (2008) and Flaxman

(2008).

NURBS tools can be found in various 3D

modelling and animation software packages. There

are many publications talking about NURBS such as

Piegl and Tiller (1997), Reese (2000), Rogers

(2000), and Farin (2001).

Subdivision modelling (Cashman 2012) depends

on subdivision schemes. Approximating schemes

include Catmull and Clark (1978), Doo and Sabin

(1978), Loop (1978), Peters and Reif (1997), Habib

and Warren (1999), and Kobbelt (2000).

Interpolating schemes include Dyn and Levin (1990)

and Zorin et al., (1996) etc.

Curve-based surface modelling creates free-form

surfaces from some characteristic curves. It was

investigated by Singh and Fiume (1998), Igarashi et

al., (1999), Karpenko and Hughes (2006), Nealen et

al., (2007), Liu et al., (2008), and Gal et al., (2009).

In addition to purely geometric modeling,

physics-based modeling methods have been

investigated by Terzopoulos and Fleischer (1988),

Terzopoulos and Qin (1994), Xie and Qin (2004),

Choi et al. (2004), Müller et al., (2005), Nealen et

al., (2006), McDonnell and Qin (2007), and

Swanson et al. (2009).

The work presented in this paper aims to

combine the high efficiency of curve-based surface

modelling with physically based shape deformation

together and develop a new technique of character

modelling.

2 MATHEMATICAL MODEL AND

ANALYTICAL SOLUTION

A character model can be decomposed into a

number of parts and some surface patches. Each of

the parts and surface patches can be created

separately. The parts can be produced by connecting

its surface patches together, and the whole character

model can be built by assembling these parts. In this

section, we discuss the mathematical model of

curve-based surface modelling and the solution of

the mathematical model.

119

H. You L., Chaudhry E., Jin X., S. Yang X. and J. Zhang J..

Character Modeling using Physically based Deformable Curves.

DOI: 10.5220/0004297601190122

In Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information

Visualization Theory and Applications (GRAPP-2013), pages 119-122

ISBN: 978-989-8565-46-4

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

2.1 Mathematical Model

In order to maintain a smooth transition at the

interface between two adjacent character parts, the

position and first derivative of the two adjacent

surface parts should keep the same at their joint

interface. A character part has two ends which are at

0u  and 1u  , respectively. Both ends are the two

interfaces of the part. If we assume that the vector-

valued position function and first partial derivative

to be met by the character part are

()vC

and

()vD

at one end

0u  and

()vC

and

()vD

at the other

end

1u  , the surface

(,)uvS

of the part should

satisfy the following boundary constraints

(,)

0 ( , ) ( ) ( )

(,)

1 ( , ) ( ) ( )

uuvv v



 



 



SC D

(1)

where () () () ()

iixiyiz

u CuCuCu







)1 ,0(



() () () () ( 0,1)

iixiyiz

uDuDuDu i







D , and

(,) (,) (,) (,)

xyz

uv S uv S uv S uv







S .

The surface

(,)uvS

of the character part can be

created by a set of curves between two interface

curves

()vC

and

()vC

and satisfying the

constraints of the first partial derivatives

()vD

and

()vD

, and the shape of the character part can be

achieved by deforming the curves.

According to the concept of physics-based

geometric modelling, a geometric object possesses

material properties and geometric attributes such as

length, width and thickness. For the physics-based

geometric modelling of a curve with an elastic and

isotropic behaviour, the material properties are

Young’s modulus

and Poisson’s ratio



, and the

geometric attribute is the thickness

h .

The lateral deformation of a curve is similar to

the bending of an elastic beam. Therefore, the

governing equation of the lateral deformation of a

curve can be derived with the same methodology as

that of the bending of an elastic beam which can be

described by

()



(2)

where

12(1 )







(3)

and () () () ()

xyz

u CuCuCu











is a vector-valued

deformation function of the curve, and

() () () ()

xyz

u FuFuFu











is a vector-valued

function of sculpting forces.

The solution to equation (2) represents an

arbitrary 3D curve. At the two ends of the curve, the

curve should satisfy boundary constraints (1) at a

certain position



. When v

changes from 0 to 1,

a set of curves are generated which define a 3D

surface

(,)uvS

. Therefore, equations (2) and (1)

form the mathematical model of a 3D surface

describing a part of character models. In the

following section, we discuss how to solve the

mathematical model.

2.2 Analytical Solution of

Mathematical Model

Equation (2) is a vector-valued nonhomogeneous

fourth order ordinary differential equation.

According to the knowledge of ordinary differential

equations, the general solution of Eq. (2) consists of

two parts: the complementary solution and the

particular solution which can be written as

12 3

()

[ ( , )sin ]sin

ij i

uuuu

uv nu nu









   



Cddd d

(4)

The following work is to generate a 3D surface

(,)uvS

from the curve (4) and boundary constraints

(1). At the position

v , the surface becomes a curve

defined by Eq. (4). Therefore, boundary constraints

(1) can be changed into

0 ( , ) ( ) ( )

(, )

()

( )

1 ( , ) ( ) ( )

(, )

()

( )

uuvuv

udu

uuvuv

udu

















SCC

(5)

Inserting Eq. (4) into the above boundary

constraints, we can determine all the four vector-

valued unknown constants

and

, and

obtain the mathematical expression of the curve.

Since

v can be any values within the range [0,

1], we change

v into

, and obtain the

mathematical representation of a 3D surface which

will be used to create various surface patches

GRAPP2013-InternationalConferenceonComputerGraphicsTheoryandApplications

120

between two boundary curves in the following

section.

3 APPLICATION EXAMPLE

In this section, we use our proposed technique to

build a horse model. A house is first decomposed

into parts. These parts are body, ears, legs and tail.

Taking one leg of the horse whose profile curve is

shown in Figure 1a as an example, we further

decompose the leg into 4 surface patches. Then we

draw the five boundary curves shown in Figure 1b.

a b

Figure 1: Creation of boundary curves of a horse leg.

From the first and second curves from the top in

Figure 1b, we create the first surface patch shown in

Figure 2a. The second, third and fourth patches are

from the second and third boundary curves, the third

and fourth boundary curves, and the fourth and fifth

boundary curves, respectively. These three created

surface patches are given in Figures 2b 2c, and 2d,

respectively.

a b c d

Figure 2: Surface patches of a horse leg.

Then we put the four surface patches together

and build the horse leg model shown in Figure 3a.

Similarly, we obtained the other three legs which are

shown in Figures 3b, 3c and 3d, respectively.

a b c d

Figure 3: Creation of horse legs.

With the same method, the other parts of body,

ears, and tail were produced and depicted in Figures

4a, 4b, 4c and 4d.

a b c d

Figure 4: Creation of horse body, ears and tail.

The last step of the horse modelling is to

assemble all parts together. For the surface patches

which share common boundary curves and the first

partial derivatives, the positional and tangential

continuities are maintained automatically. However,

for the surface patches which intersect each other,

we first determine the intersecting curves between

them. Then, two curves which are on the intersecting

surface patches and close to the intersecting curves

are determined. And a blending surface between

these two curves is created to smoothly connect the

two intersecting surface patches together. With the

above treatment, we built the horse model and

depicted it in Figure 5.

Figure 5: Created horse model.

4 CONCLUSIONS

A new modelling technique of character models has

been proposed in this paper. This technique aims to

combine curve-based surface modelling and

physically based deformations together, and creates

free-form surfaces using physically based

deformable curves. A vector-valued fourth order

ordinary differential equation involving concentrated

sculpting forces is introduced to achieve physically

based deformable curves, and sharing of boundary

curves and the first derivatives by two adjacent

patches is used to obtain the positional and

tangential continuities of two adjacent patches at the

shared boundary curves.

In order to make the proposed technique build

character models efficiently, we transformed

concentrated sculpting forces into an analytical

mathematical expression and derived the analytical

solution of the vector-valued fourth order ordinary

differential equation by means of the analytical

mathematical expression of concentrated sculpting

CharacterModelingusingPhysicallybasedDeformableCurves

121

forces.

We have applied our proposed technique in the

modelling of a horse model which demonstrates the

effectiveness of our proposed technique in character

modelling.

ACKNOWLEDGEMENTS

This research is supported by the grant of UK Royal

Society International Joint Projects / NSFC 2010.

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