ADAPTIVE CONSTRAINT AND 3D SKETCH-BASED

DEFORMATION FOR INTERACTIVE FREE FORM SURFACE

STYLING

Li Han

1,2,3

, Raffaele De Amicis

, Giuseppe Conti

Graphitech, Via Dei Molini, 2, 38100, Trento (TN), Italy

Information and Communication Technology Faculty, Trento University, Italy

College of Computer and Information Technology Of Liaoning Normal University, Dalian, China

Keywords: Physical-based modelling, adaptive constraint, free form sketching, local deformation.

Abstract: This paper tries to answer to the increasing demand in the domain of conceptual design for more intuitive

methods for creating and modifying free-form curves and surfaces. This is done by addressing the issue of

physical-based shape control by free hand spline sketching instead of the tedious mathematical parameters

adjustment. We present a novel approach capable of matching the designer’s requirements in terms of qual-

ity and accuracy of the produced model. The algorithm adopts a simple 3D sketching technique and a finite

element deformation method to create free-form models. In the method proposed the user applies interactive

sculpting to modify a surface in a predicable way. Our algorithm automatically extracts the key points from

sketched target curve and adaptively distributes the external-force constraints which impose the force energy

on the corresponding control vertexes along their normal. We have limited the influence of these constraints

to a localized area by attaching an influence factor to each control vertex of the parent surface. The smooth-

ing function introduced later further solves the transition interval and it provides for symmetry features. This

proposed method is finally implemented in a 3D scene environment and the results show how the designers

intuitively and exactly control the shape of the surface.

1 INTRODUCTION

Efficient and intuitive shape manipulation techniques

are vital to the success of geometric modeling, com-

puter animation, physical simulation and other com-

puting areas. Recently considerable achievements

have been reached through the adoption of Free-

Form Deformation (FFD) and Extended Free-form

Deformation (EFFD). These embed the whole object

into a tensor product volume, and the volume can be

deformed by means of spline control points while the

embedded object is deformed accordingly. Unfortu-

nately, manipulation of splines is not intuitive. Al-

though other physical-based manipulation ap-

proaches improve the natural operation and a new

medial Axial Deformation method (AxDf) is being

currently proposed to achieve better deformation

results, the degree of freedom available to control

the shape is still limited.

The technique presented here supports fully in-

teractive and intuitive shape control, ranging from

free-form surface creation to predictable shape de-

formation. We proposed three surface generation

modes based on easy freehand sketching. During the

implementation of the deformation, our method

adaptively extracts force constraints and it automati-

cally adjusts the corresponding control vertices in

response to the external force distribution. Further-

more, a series of linear influence functions are intro-

duced to improve the continuity and the symmetry.

The rest of the paper is organized as follows: in

section 2 we present the previous works. Section

describes the parent surface creation and relevant

concepts. In section 4 we detail the implementation

of the algorithm showing how to control the defor-

mation process. Section 5 describes some experi-

mental results. Finally in section 6 we conclude with

a summary and describe the directions of future work.

2 PREVIOUS WORKS

The first modeling deformation technique introduced

into the CAD/CAM field was the method relying on

392

Han L., De Amicis R. and Conti G. (2006).

ADAPTIVE CONSTRAINT AND 3D SKETCH-BASED DEFORMATION FOR INTERACTIVE FREE FORM SURFACE STYLING.

In Proceedings of the First International Conference on Computer Graphics Theory and Applications, pages 392-399

DOI: 10.5220/0001352803920399

 SciTePress

global and local deformations (Barr, 1984). This

method and its improvements (

Güdükbay, 1990) can

provide support for regular deformations such as

twisting, tapping, bending, rotating and scaling.

However the method does not easily yield arbitrary

shapes and, further, it involves tedious and unintui-

tive geometric operations. Free-Form Deformation

(FFD) (

Sederberg, 1986) tackles the issue of generat-

ing more complex objects through geometric model-

ing. Specifically a geometrical element such as a

point, a line, a plane etc. is chosen together with

special weight factors and by taking their weighted

average to express a complex shape. Then, to induce

deformations within such complex shape, those con-

trol points are moved, while the local object points’

coordinates are deemed to remain unchanged: i.e. the

topological structure of the deformed object remains

unchanged (

Bechmann, 1994). In fact, FFD is one of

the most versatile and powerful deformation tools,

however it is not easy for the user to exactly predict

the deformation to exactly reach a desired effect.

Further exact placement of object points is hard to

achieve. In literature (

Kalra, 1992; Lamousin, 1994;

Griessmair, 1989; Coquillart, 1991; Chadwick, 1989) it

is possible to find FFD improved in terms of shape

control functions, whose results have been exploited

in several domains including human body animations

and dynamic flexible deformations. However, while

these approaches increase control flexibility, on the

other hand they require the solution of complex

nonlinear equations with numerical methods.

Another approach which promotes an easier and

more intuitive interface is the so-called Extended

Free-Form Deformation (EFFD) technique proposed

in (

Coquillart, 1990). With EFFD, instead of starting

with the FFD’s set of parallelepipeds control points,

the user configures the initial lattice of control points

taking into account the approximate shape of the

intended deformed shape. However, the user must

know the general shape before starting to model, and

the interface is still a direct representation of the

underlying mathematics.

Thanks to the introduction of constraint points

and radius the authors in (

Borrel, 1994) present a

good technique for local deformation which was

further improved by (

Xiaogang, 2000) to the extent

which, not only it conducts to deformation of point

constraints, but also lines and surfaces constraints.

Léon et al. (

Léon, 1997; Léon, 1995; Léon, 1991) have

linked the control polyhedron of a surface to the

mechanical equilibrium of a bar network by using

the force density concept. Although its effect on the

deformation is very satisfactory in terms of aesthetic

feeling, it often needs the solution of high-order

systems of linear equations which generally need

demanding complex computations.

3 SURFACE CREATION AND

RELEVANT CONCEPTS

In the approach implemented we adopt a physical-

based method that incorporates both external force

energies and internal deformation distributions into

the parent NURBS geometric surface.

During the implementation both the “parent sur-

face” creation by user interactive sculpting and the

“target surface” construction through spline-driven

approach underline the physical meaning and pre-

dictable motion features.

3.1 The Creation of Parent Surface

The so-called “parent surface” is the surface which

will be affected by the deformation. As mentioned

earlier, it can be obtained in three different ways:

1. Geom-filling mode: the designer is allowed

to sketch two or more 3D curves which

serve as the constrained boundary of the

surface.

2. Skinning mode: the designer sketches a sur-

face by using the well-known concept of ex-

trusion. He/she first draws a free-form 3D

curve, then the curve is attached to the

pointer and when the pointer is moving, the

process of surface generation starts and the

shape is immediately shown.

3. Revolving mode: the designer can generate

a surface by revolving a curve around one

axis.

In this process, the technique developed supports

fully freehand curve sketching, and as a result, the

NURBS parent surface is constructed by a “multi-

patch” which is composed of a compatible network

of iso-parametric curves as it is shown in formula 1,

where numRow and numCol represent the number of

iso-parametric curve C(u,v) in U and V directions.

S (

)v,u

numRow

(

∑

; )v,u

numCol

(

∑

); (1)

In the following section, we will further describe

how the constraint-based resultant surface is finally

reconstructed by combining multi-patch with so-

called “physical force” distribution technique.

ADAPTIVE CONSTRAINT AND 3D SKETCH-BASED DEFORMATION FOR INTERACTIVE FREE FORM

SURFACE STYLING

393

3.2 Relevant Concepts

Before illustrating further details of the approach we

first introduce the mathematical representation be-

hind the process presented in the following sections.

Bound curve and Target curve

Let C: φ (u, v) = 0 be a sketched closed-curve in 3D

space. It will be used for deciding the region to be

deformed. The influence factor

E (Q

) (0 ≤ j≤ Nt) is

attached to each control vertex

within the parent

surface

S (u, v), where Nt is the number of the control

vertex in parent surface. If the control vertex Q

lies

inside the bound curve, E is equal to 1 and it can be

influenced by force constraints, otherwise E is set to

0 and it will thus keep a “static” status;

⎩

⎨

⎧

)Q(E

≤

)v,u(

(2)

Figure 1: Target curve (in purple) influences only the area

which is inside of the bound curve (in dark)

Likewise the sketched target curve is used to de-

fine the resultant shape by Ψ (u, v) = 0. A series of

force energies are adaptively produced through the

key points on this curve (see Figure 1).

The next section will illustrate how we effectively

obtain these force energies and how they influence

the whole parent surface.

Linear force energy fi (Ki ,P).

We define K

as the key points on target curve and D

) as the projection distance from K

to the parent

surface S (u, v) along the normal N

(see Figure 2 -

left). Q

is the closest control vertex to the projected

point P which is used for determining the corre-

sponding curve on the parent surface. In this way,

the energy of each force f

,P) will be distributed

among the vertices on the corresponding curve.

Therefore, the parent surface will be gradually ap-

proximated to the leading target curve (see Figure2 -

right).

⎩

⎨

⎧

)(

)()(),(

ijii

KDQEPKf

(3)

Figure 2: (Left) the multi-patch structure of a surface. K

the key point which imposes the force energy f

to the

patch, and Q

is the closest vertex to the projected point P.

(Right) the resultant surface under the influence of the

force f

The force intensity “α”

Within our model, “α

” represents the contribution of

the i

force energy to the parent surface S (u, v). If

the projected point P lies in one patch, the force will

be distributed among the neighboring four vertices

(see Figure 3);

, v

)

(4)

)

K(D)

Q(E)

)P,

)

Q(F

αα

== (5)

))v,u(Q(F))v,u(C(F

∑

(6)

Figure 3: Force Intensity distribution in one patch. P is

the projected points from key point K

; then as the inten-

sity of the force to the closest vertex Q

“α

” is varied

according to the extent of area x

Here x

, x

, y

and y

are defined as the distance

from P to the four neighbor vertices, and the unit of

the intensity “α

”

is set to “1”. Then the force exer-

cises its effect inversely to the extent of the area.

Parent Surface Key Point

Bound Curve

Target Curve

GRAPP 2006 - COMPUTER GRAPHICS THEORY AND APPLICATIONS

394

Therefore the force’s influence on the control vertex

Q can be described as )Q(F

(see formula 4, 5). In

formula 6,

))v,u(C(F

represents the force’s influ-

ence on the whole curve, where the Np is the number

of control vertex on this curve.

Resultant Surface

Finally we call D

the replacement function, which

represents the extent to which the parent surface is

influenced by the force f. We assume Nc as the

number of curves and Nt as the number of the con-

trol vertices on the parent surface, m is the number

of the force constraints. Then the resultant surface S’

(u, v) is obtained as:

∑∑∑∑

••=

)

K(D)

Q(E)

))v,u(

F)v,u(

1111

αδ

(7)

)v,u(S)v,u(D)v,u(S

′

4 SKETCH-BASED LOCAL

DEFORMATION

4.1 Local deformation algorithm

The algorithm followed briefly depicts the imple-

mentation of local deformation where applicable

through the help of pseudo code description.

Step1: The user creates a free form surface S (u, v)

in the preferred mode (e.g. by geom-

filling, skinning or revolving);

Step2: The user draws a bound curve φ (u, v) and

consequently the system calculates the in-

fluence factor

E (Q

) for each vertex.

Step3: The user sketches the target curve

(u, v);

IF (Over-constrained) then Goto Step 5;

IF (Under-constrained) or (Well-constrained)

1. The system predicts the motion tendency of

the target curve “_DiR” and it determines the

number of force constrains “m”

2. Switch (_DiR)

Case (U direction):

The curves in V direction evolve by

repositioning control vertexes accord-

ing to each force constraint;

Case (V direction):

Likewise, the curves in U direction

evolve by repositioning control ver-

texes.

3. The system resolves the transition intervals

and it improves the symmetry.

Step4: The system renders the resultant surface

Step5: End

In the following sections we will detail how to ef-

fectively obtain the key points on the sketched target

curve and how to classify three different constraints

configurations to further improve boundary features

of resultant surface.

4.2 The number of force constraints

“m”

Since the designer’s sketching activity produces only

an approximation of the desired shape, it is impor-

tant that the resultant surface captures the “shape”

features of the leading target curve. However, in the

free-form domain, the number of the constraints is

usually unknown. Most current approaches provide

only a solution that is a result of a predetermined

criterion.

We instead propose a method which adaptively

provides such criteria through the prediction of the

motion of target curve (see Figure 4).

We then adopt the partial derivatives θ

and θ

(see equation 8). As shown in Figure 4 we can easily

get the points P

) and P

) by projecting

and K

onto the parent surface S (u ,v). Then the

span of patches can be obtained, where Cs and Ce

are respectively the curve position in the V direction

while Rs and Re are the curve position in the U di-

rection.

Figure 4: The Parent surface with bound curve (in green)

and target curve (in orange); the yellow circles represent

the key points which are adaptively produced by consider-

ing the value of “m”.

ADAPTIVE CONSTRAINT AND 3D SKETCH-BASED DEFORMATION FOR INTERACTIVE FREE FORM

SURFACE STYLING

395

When θ

≥ θ

the target curve is leading towards

the V direction. Therefore the number of key points

(constraints number) on the target curve “m” is de-

termined by the difference between Ce and Cs. Vice

versa, when θ

< θ

, “m” is valued by the difference

between Re and Rs. In this way the key points on the

target curve will be adaptively produced and they

will impose the force’s spring to the surface.

4.3 Improvement of the boundary

feature of the resultant surface

During the deformation process, we have excluded

the option of fixing all the control vertices outside

the bound curve and operating only on those inside.

However this choice could still result in an inaccu-

rate and insufficient deformed shape around the

bound curve. Furthermore, the leading target curve

may result over-constrained or just show unaccept-

able undulations.

To avoid these issues, we propose two means to

improve the quality of the result of the deformation.

First, we classify the constraints into three cases:

• Over-constrained: if the target curve com-

pletely lies outside the bound curve.

• Under-constrained: if the target curve partly

lies inside the bound curve.

• Well-constrained: if the target curve lies well

inside the bound curve.

When the configuration is over-constrained, the

parent surface will not be affected. Conversely when

the configuration is under-constrained or well-

constrained, we will use the aforementioned Formula

8 and four extremes (see Figure 5) to calculate the

adaptive constraints.

Secondly, we introduce two factors to resolve

the undulations near the bound region.

1) Approximation Scale

We provide the scale factor “λ” (Li, 2005),

through which the users can interactively adjust the

degree of approximation to the target curve as de-

tailed in Formula 9.

⎪

⎩

⎪

⎨

⎧

)(

)()(),(

ijii

KDQEPKf

λλ

10 ≤≤

(9)

2) Relaxation Interval.

The so-called Relaxation Interval is used to pro-

vide the transition parts, from two ending points on

the target curve to the parent surface.

We define them through computing the mini-

mum bounding box of bound curve, and then four

extremes can be obtained: MinRow, MaxRow, Min-

Col and MaxCol. As shown in Figure 5-top, the

relaxation intervals T

P1,

are valued by the span of

the patches

MaxRowRe− and MinRowRs − . The force

),(

PKf

and

),( PKf

will gradually decrease to

reach zero within these two parts as it is shown in

Formula 10 and Formula 11.

{}

MinRowRs

)Minrowr)(P,

K(f

)

Minrow

Rsr

Minrow

Rsr

))v,u(

C(F

−

∑∑

αδ

(10)

{}

∑∑

−

Maxrow

Rer

MaxRowRe

)rMaxrow)(P,

)vj,

Maxrow

Rer

))v,u(

αδ

(11)

Figure 5: (Top) the Relaxation Interval T

and T

green line. (Bottom) an example of dealing with relaxation

interval.

;

u∂

∂

;

v∂

∂

⎪

⎩

⎪

⎨

⎧

−

≥−

θθ

≺;RsRe

;CsCe

Integer

Re,

∈

(8)

MinRow

MinCol

MaxCol

)P,K(f

MaxRow

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396

4.4 The smoothing function for

symmetry and discontinuity

features

Since the target curve used to drive the deformation

process of the surface might be characterized by a

sharp line behavior (see Figure7-left), we propose a

smoothing function to improve the symmetry feature

of the deformed surface. This provides strong visual

impact of the quality of the surface within such areas.

Without the need for any new patches insertion,

we maintain the same topology by symmetrically

distributing the external force influence to the corre-

sponding curve (see Figure.6).

)v,u(C

Tolerance

= (12)

Tolerance

)

K(D)

Q(E

Tolerance

)

K(D)

Q(E

))

u(Q(F))v,u(C(

αλ

∑

∑∑

),(

ttt

vuCQ ∈

1≤ t, r≤ Np (13)

Figure 6: The force f is symmetrically distributed along

the curve; (left) the “tolerance” serves as a step; Q repre-

sents different vertex in this curve. (Right) the deformed

curve is produced by symmetrical force distribution.

Figure 7: before smoothing (left) and after smoothing

(right)

The details are shown in formula 12 where the

value

),( vuC is the length of curve C, while Np is the

number of control vertices on each curve. The toler-

ance factor is used to determine the step of the dis-

tribution, along the corresponding curve.

From formula 13, we can achieve symmetric

space deformation by symmetrically and gradually

distributing

to different control vertex

the parent surface. The results can be compared in

Figure 7.

5 EXPERIMENT AND RESULTS

We have implemented our method in C++ with

OpenGL and OpenInventor 4.0 on a Pentium 4

1.6GMhz with 512MB of RAM. This implementa-

tion provides real-time feedback (approx 20 frames

per second for average 30,000 vertices) with a se-

quence of deformations.

In order to improve the interaction for the re-

quired shape, we have developed a 3D dragger

metaphor, which can be freely controlled in 3D

space. This is used for handling the plane in which

the object lies. So that the user is capable of dynami-

cally controlling the target curve and parent models

to reproduce a series of results. Meanwhile in our

application, all the objects can be adjusted by freely

“oversketching”.

Furthermore, we apply two methods to limit the

influence of the force to the surface. The first is that

the local area is directly obtained by projecting the

target curve onto the parent model. In this way a

series of springs are produced in the parent model

which are going to respond to the energy strains

from the target curve. In the second method, we

directly define a local region by sketching a bound

curve as aforementioned. The comparison is shown

in the following figures.

Our experiments indicate that our method is in-

tuitive and effective for creating and editing a large

variety of free form shapes (see Figure 8, 9, 10).

f ( k

, p)

Deformed Curve

Original Curve

j-2

j-1

j+1

j+2

ADAPTIVE CONSTRAINT AND 3D SKETCH-BASED DEFORMATION FOR INTERACTIVE FREE FORM

SURFACE STYLING

397

(a) (b) (c)

Figure 8: Dynamical shape control by spline-driven deformation method where the parent surfaces are respectively

generated by geom_fill, skin, and revolving. (a) The red sail is selected as the sensitive surface which is going to be

deformed. (b) The local area is automatically obtained by projecting target curve (blue spline) onto the surface.

When the target curve is close to the parent model the target curve imposes high force intensity to the local area. (c)

When the target curve is moving away from the parent model, the influenced region becomes smaller but the force

energy becomes stronger and consequently the surface will be dynamically adjusted in response to the energy dis-

tribution.

(a) (b) (c)

Figure 9: Three stages of the operation: (a) Before the deformation. (b) Without a predefined local region: the key

points in the target curve (blue balls) produce force springs in the surface and only impose strain to these sensitive

springs. However the influence can not be strictly localized. (c) With the predefined local region, the force’s in-

fluence can be well localized.

(a) (b) (c)

(d) (e)

Figure 10: (a) (b) (c) with the predefined local region we can properly control the result of the deformation. (d)

We provide interactive shape control which allows to re-deform any selected parts. (e) The texture is applied and

its mapping is dynamically changed with the user interaction.

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398

6 CONCLUSION AND FUTURE

WORK

In this paper, we present a physical-based deforma-

tion method. When working with our deformation

method, the designer does not need to manipulate

some non-intuitive mathematical shape parameters,

such as control points and control vectors. Instead,

he/she can work with the point constraints and

spline-based constraints, therefore designers can

easily and intuitively control the resultant shape.

The potential function is centered at the con-

straint and it symmetrically distributes the so-called

force energy. Compared with other deformation

methods, this approach has the following advantages:

intuition, locality and simplification since this

method combines shape creation and deformation.

Finally, it is possible to use various intensities and

smoothing functions to enrich the quality and accu-

racy of the deformation.

Although our method is intuitive and less compu-

tationally challenging for free form surface modeling

and styling, it still needs some time to create sophis-

ticated models; and there is a limitation in our de-

formation technique when we want to change the

topology of the model, such as creating a hole.

In the future work, we will further investigate in-

telligent operations for shape editing and multi-

surfaces modeling based on 3D sketching; such as

surface splitting and stitching. We also plan to im-

prove the connectivity and continuity between dif-

ferent surfaces based on declarative constraints.

ACKNOWLEDGEMENT

This research presented in this paper is a part of EU

project “IMPROVE” and of the Part of the project

InSIDe.

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