Context dependent digital shape editing in product design

Digital Shape Editing

in Product Design

Digital Shape Editing

in Product Design

Proefschrift

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. J.T. Fokkema,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen

op maandag 12 maart 2007 om 15:00 uur

door

Raluca DUMITRESCU

Diplom˘

a de Studii Aprofundate

Universitatea Politehnica din Bucure¸sti

Dr. J.S.M. Vergeest

Samenstelling promotiecommissie:

Rector Magnificus voorzitter

Prof. dr. H. de Ridder Technische Universiteit Delft, NL, promotor Dr. J.S.M. Vergeest Technische Universiteit Delft, NL, toegevoegd

promotor

Prof. dr. P.J. Stappers Technische Universiteit Delft, NL Prof. dr. F.J.A.M. van Houten Universiteit Twente, NL

Prof. dr. J.-C. L´eon INPG–ENSHMG, Grenoble, FR

Dr. W.F. Bronsvoort Technische Universiteit Delft, NL

Dr. F. Giannini CNR–IMATI, Genova, IT

Prof. dr. K.H.J. Robers Technische Universiteit Delft, NL, reservelid

Raluca Dumitrescu

Context Dependent Digital Shape Editing in Product Design PhD Thesis, Delft University of Technology

Copyright c 2007 by Raluca Dumitrescu

Printed by W¨ohrmann Print Service, Zutphen

1 Introduction

1.1 The product development process . . . 1

1.1.1 The design brief . . . 2

1.1.2 The product design process . . . 3

1.1.3 On the product shapes . . . 4

1.1.4 On the variations of product shapes . . . 7

1.2 Designers’ computer support . . . 8

1.3 Designers’ contextual shape editing . . . 12

1.4 Motivation of the research . . . 14

1.5 Aim of the research . . . 14

1.6 Overview of the thesis . . . 15

2 Graphical objects editing

2.2 Evaluation criteria . . . 19

2.3 Graphical objects editing categories . . . 20

2.3.1 Structure–Shape editing . . . 21

2.3.2 Structure editing . . . 31

2.3.3 Structure–preserving editing . . . 39

2.4 Conclusions . . . 41

3 Analysis of shape modelling activities

3.2 Goal of the experiments . . . 45

3.3 Experimental method . . . 46

3.4 The WebCam Experiment . . . 49

3.4.1 Procedure and Model . . . 49

3.4.2 Participants . . . 50

3.4.3 Results . . . 51

3.4.4 Summary . . . 62

3.5 The Soapbox Experiment . . . 62

3.5.1 Procedure and Models . . . 63

3.5.2 Participants . . . 64

3.5.3 Results . . . 65

3.5.4 Summary . . . 71

3.6 Discussions and conclusions . . . 71

4 Shape context: formalisation & experimental tool implementations

4.1.1 Shape context definition . . . 75

4.1.2 Shape context observable . . . 76

4.1.3 Shape control in freeform product design . . . 78

4.2 Experimental shape context editing tools . . . 79

4.2.1 Single–parameter controlled morphing–like editing tool . . 79

4.2.2 Multiple–parameter controlled morphing–like editing tool 83 4.3 Conclusions . . . 88

5 User evaluation

5.2 Evaluation of the single–parameter controlled morphing–like edit-ing tool . . . 92

5.2.1 Experimental set–up . . . 92

5.2.2 Results . . . 93

5.2.3 Summary . . . 98

5.3 Evaluation of the multiple–parameter controlled morphing–like editing tool . . . 98

5.3.1 Experimental set–up . . . 99

5.3.2 Results . . . 99

5.3.3 Summary . . . 104

5.4 Conclusions . . . 104

6 Curve and skeleton based shape editing

6.2 Curve and skeleton based shape editing . . . 110

6.2.1 The source surface . . . 110

6.2.3 The target surface . . . 112

6.2.4 The morphing–like editing method . . . 114

Chapter

1

Introduction

The high competition on the global marketplace of consumer durables is forcing companies to shorten their product development time and increase their effi-ciency by enhancing the interaction between the different actors involved in this process. This high competition is due to several factors. Two of the important factors refer firstly to the current state–of–the–art in injection molding technolo-gies, which compared to classical manufacturing processes allow easy and fast manufacturability of freeform shapes. And secondly, a high customer demand for personalised products forces companies and industrial design engineers to design and produce fashionable and diverse shapes for their products. There-fore, the notion of freeform shape has become relevant in competitive design and hence the development of better computer tools to support the design of freeform shapes.

1.1

The product development process

Figure 1.1: The traditional product development process.

solution emerges, many iterations are performed between the product design and the other departments to ensure the changes are appropriate.

Automotive industry was among the first industries to realise the drawbacks of SE and to develop and implement a new approach to improve the development process. This approach is known as concurrent engineering (CE), and employs a multidisciplinary team that concurrently designs and applies all necessary pro-cesses to produce an effective and efficient product satisfying the customer’s needs. A significant advantage of CE is that the interaction between the prod-uct designers and the manufacturers reduces the costs of iterations performed in SE and also provides the product a higher chance of market success. Real-ising the benefits of CE, other industries followed the automotive example and adopted CE. However, in many small and even medium-sized enterprises (SME) the multidisciplinary teams are truncated and very often one actor plays multi-ple roles. This may be a handicap when competing with large companies with aggressive marketing, and better supporting tools may help SME’s in this regard.

No doubt, all stages of the product development have their own influence on the product success and the company competitiveness. Product design has, however, a significant role in both the determination of product’s costs and its market success. It represents up to 60% of customer’s decision to buy a particular product [66], and often this decision is based on the product semantics, price or its functionalities. SMEs can, therefore, take some or even great advantage of this customer behaviour. Carefully analysing the target customer group, understanding the product semantics these customers are most interested in, and correspondingly designing the product shape may increase the chances of developing a successful product.

1.1.1

The design brief

product, quality issues, target group, company identity [31]. Fankhauser [41] says that people propose requirements both with expectations and an image in mind, meaning that these may be either very precise or very vague. The design brief might be a statement like the one made by J.F.Kenedy in 1961: before the end of the decade, to land a man on the moon and bring him back safely, where only the goal was fixed and nothing about the means to achieve this goal was mentioned. Or, in other cases, it can give a very detailed description of the prod-uct and how it should be produced, stored, etc. The prodprod-uct designers start the design process by materialising a product that fulfils these requirements. During development stages, the design brief is often reviewed by the different actors to ensure the validity of the end–product against these requirements.

1.1.2

The product design process

The product design process itself consists of a number of stages. The most important refer to: (a) conceptual design, (b) embodiment design, (c) detail de-sign [31]. In the conceptual dede-sign stage, ideas are generated mostly by means of pencil–and–paper. Radical design changes often occur in these concept–sketches, some concepts are even abandoned, while few concepts are carried over to the embodiment stage. In this stage, the concepts are analysed in greater detail and one sketch is chosen as the one to be further developed. Then, in the detail design, one is concerned with the accurate specification of the product charac-teristics.

Sch¨afer and Roller [95] report on four types of product design, namely: (a) design from scratch, (b) re-design, (c) adaptation design, (d) variational design. Design from scratch refers to new design, and all the other three can be grouped into the class of variational design. Based on the results of a company survey, the authors conclude that new design occurs in ≈ 40% of cases, whereas variational design accounts for ≈ 60% of the cases. Therefore, rather often designs are generated based on the adaptation and the variation of existing designs.

Two main actors are involved in the product design process [31]. Traditionally, they are the product designer and the industrial designer. Product designer has a broader responsibility, being concerned basically with all aspects of the design of the product (including also the engineering design of the mechanical components). Product design is therefore a mix of engineering and design. The industrial designer, however, is concerned with the aesthetics and usability of products. He1defines the overall shape of the object, the location of details with respect to one another, colours, texture, sounds, and ergonomics. Additionally he may specify the way the product is presented to the customer at the point of

It is therefore the industrial designer2 who is concerned with the product shape and its semantics. At the embodiment design stage, the stylist’s sketched shape is evaluated in the three–dimensional space by means of prototypes. Very of-ten adjustments on the prototype shape are performed together with the clay modeler to produce better quality shapes. Further on, the prototype is reverse engineered (scanned) to generate its 3D digital model necessary for the down-stream stages. It is at this step where the stylist’s involvement in the shape definition practically stops. The digital model is further tested by means of vir-tual simulations (like structural simulations). Materials are chosen considering these simulations, and then the production process is developed. However, the product character as designed by the stylist sometimes may get lost in case small changes in the shape are performed, for example, due to manufacturing issues. The stylist cannot perform changes on this digital model because of the difficulty to handle the mechanical computer–aided design (MCAD or just CAD) systems that are mainly designed to support the testing and production stages of the development process.

1.1.3

On the product shapes

As earlier mentioned, product semantics are very important in influencing the customer’s decision to buy a product. They determine the customer’s initial impressions about a product. Small variations of the product shape may de-velop into different semantics, which may send wrong signals to the customers. Product semantics study the relationships between the form produced by the de-signer and its meaning perceived by the customers. It refers to the appearance (aesthetics), use and context of the product and is a qualitative appreciation influenced by the individual’s cultural background and experience. As Vihma [126] pointed out, the aesthetic feeling is immediate, without comparison with other feelings, and cannot be described or analysed. Several studies in computer support for designers have been done to understand how product shape charac-teristics can be linked to aesthetics [112, 125]. No systematic answer was given to this question, and it is still the designer himself to find this answer. Aesthet-ics has therefore no representation and it is still unclear how it can be assessed. But the fact that aesthetics is inherent to any product is clear, making aesthet-ics an essential element to be considered when designing or analysing products. Altogether, product semantics can:

• contribute to make the use of products self-evident and culturally mean-ingful;

concavity

a) b) c)

Figure 1.2: Object decomposition: a) arrows indicate strong segment-ing regions of an object; b) interpenetratsegment-ing two shapes produces a pair of cusps; c) cusps are used for decomposing the object into components (approximated by geons). Adapted from [15].

• supply products with a distinct character;

• influence the customer’s decision to buy the products.

Although designers appear to perform well at designing products using tradi-tional tools, their activity is not fully understood from a theoretical point of view. Several attempts have been made in this regard and once the design pro-cess is understood, it will also be possible to better support it by computer tools. Mortensen [81] proposes a model of a design language (design grammar) based on design characteristics. These refer to: the class characteristics, the composition characteristics, the ”beschaffenheit” characteristics (influencing the functionality) and the organisational characteristics. The author distinguishes all these characteristics for each component of a product, after which a synthesis operation follows resulting in the final design.

Warell [130] proposes design syntactics as a modelling approach aiming at cap-turing the contents (form entities and form elements) and structure (the compo-sitional principles) of the visual product form. The author claims that products are made of parts that have external surfaces, arranged according to a structural skeleton that spatially defines the surfaces in relation to each other. A form element is defined by one or more constituent parts, and the interaction of the form elements creating a system of visual relations defines the end–product.

Figure 1.3: Example of a teapot class: a) The teapot archetype. b) Instances of the teapot class derivable from the archetype.

shapes, called geons (geometrical ions) (Fig.1.2c). Object abstractions are then possible in terms of simple geometrical representations as these are a natural way to interpret complex freeform shapes. Cylinders, blocks, wedges, and cones rep-resent a sufficient set of simple geometrical shapes whose combinations generate many more shapes than humans can actually learn.

Muller and Pasman [82] define a product class as a set of products that share a corresponding basic structure. The structure of the class linked with the spatial relations of the components form the mental frame of the class which humans employ when recognising instances of specific classes [66]. Slight variations in the spatial relations of the components, in their shapes or insertion of new com-ponents can still result in proper recognition of the class, as the mental frames of products classes are flexible and context dependent. The instances of a specific class can vary as much from their basic structure as their recognition still leads to the class to which they belong. Therefore, in this work, the set of all shapes within a class are said to be context dependent on the basic structure.

The archetype is the most representative instance of a class and it is often employed when representing the basic structure and the spatial relations of the components. We refer to Fig.1.3 for an example of an archetype and instances of a teapot class. In practice, instantiation and development of various real–life products usually starts from the archetype.

Although products belonging to the same class may be very similar, their se-mantics may be different, sometimes even opposite. Kreuzbauer and Malter [66] present such an example with the case of the redesigned Volkswagen Beetle automobile:

Figure 1.4: Example of how the product shape evolves during the design process, while fulfilling the design requirements.

Perceptual psychology offers knowledge about human perception, but seems not to be attuned to understanding product semantics [126]. Different meanings can be designed in a new product by simply modifying key characteristics of an existing product’s shape attribute, while preserving the product structure to guarantee the market acceptance of the designed product. The shape of a prod-uct is therefore of significant importance for the prodprod-uct success. It is of great concern for the designer to produce the right shape for each component so that its spatial relations produce an appealing product. Product shape should allow easy recognition of the product class and also of any newly added functionality.

1.1.4

On the variations of product shapes

The increasing importance of the product form for market success or gaining competitive advantage has led to a need to be able to efficiently and effectively perform and evaluate changes in the product’s aesthetic form [130]. Often design starts at improving older generation products or is heavily influenced by com-petitors’ existing products [66]. A proper workflow could require that an existing product is scanned or retrieved from a digital catalogue and that further adjust-ments of its shape are performed employing the variational design approach. In this way the class’ structure is preserved and shape changes are performed through intuitive shape handles3. Designers do so until the desired impressions in the shape are reached (Fig.1.4), and the need to explain the sketched design to the clay modeler is eliminated. The digital models can then be accurately passed on to downstream testing and production departments and any change in the shape of the digital model can be performed by any of the actors involved in the development process.

An important swift is currently happening in industry regarding the adoption of the digital environment for all stages of the product development process [18]. This is mainly due to the high competition that generated geographically distributed development teams in the need to cut the development costs. All elements of product design and production are linked together through a central hub, sometimes called the Virtual Product Space (VPS). Shape modifications necessary to result in an acceptable-cost production process while preserving the initially designed appearance are currently decided mainly via telephone and fax [97], but the geographically distributed development team can be in a better position to take such decisions through a VPS if appropriate shape editing tools exist.

Among the many types of computer support, shape modelling tools are therefore crucial for the design and modification of products’ shapes. Most of the newly developed CAD systems allow increasing control over the modelling of freeform shapes4 _{and such systems are referred to as computer–aided industrial design}

(CAID) systems. The de facto industry standard for the representation, design, and data exchange of geometric information in CAID is NURBS5_{. They can}

represent both analytic shapes and freeform shapes, are invariant under com-mon geometric transformations (translation, rotation, parallel and perspective projections), and NURBS algorithms are fast and numerically stable [87]. A NURBS curve of degree p with n + 1 control points {p0, p1, . . . , pn}, with

pi associated with weight wi ≥ 0, and a knot vector of m + 1 knots 0 ≤ u0 ≤

u1≤ . . . ≤ um= 1, where the fundamental equality m = n + p + 1 must hold,

is defined by:

C(u) = Pn

i=0Ni,p(u)wipi

Pn

i=0Ni,p(u)wi

(1.1)

where Ni,p(u) is the ith B-spline basis function of degree p.

A NURBS surface of degree (p, q) with a grid of (m + 1) × (n + 1) control points p_i,j, where 0 ≤ i ≤ m and 0 ≤ j ≤ n, and knot vectors 0 ≤ u0 ≤ u1 ≤ . . . ≤

ur= 1, and 0 ≤ v0≤ v1 ≤ . . . ≤ vs= 1. Each control point pi,j is associated

with weight wi,j≥ 0. The fundamental equalities: r = m+p+1 and s = n+q +1

must hold, is defined by:

S(u, v) = Pm

i=0

Pn

j=0Ni,p(u)Nj,q(v)wi,jpi,j

Pm

i=0

Pn

j=0Ni,p(u)Nj,q(v)wi,j

(1.2)

−→ u ↑ v

a) b)

Figure 1.5: Example of a NURBS surface: a) A trimmed surface as visualised on the computer’s screen; b) its corresponding parametric do-main as stored in the computer.

where Ni,p(u) and Nj,q(v) are the B-spline basis functions of degree p and q,

respectively. Fig.1.5a shows an example of a trimmed NURBS surface and its rectangular parametric domain is shown in Fig.1.5b. Trimmed surfaces are very difficult to handle during modelling. A mismatch often occur between the per-ceived shape (the dark region) and its digital representation (as stored by the computer). Designers treat trims as parts of the surface that do not exist, whereas in CAID systems the trimmed areas still exist but are hidden to the user. Work–arounds do exist to cope with trims. They mainly refer to the surface parametrisation but are improper solutions for designers.

NURBS object creation has many similarities with other CAD representation schemes, meaning that sphere, cube, plane, cylinder, and cone primitives are supported along with Boolean, trimming, and blending operations. Other spe-cific creation approaches exist which provide intuitive shape handles. Most no-tably are the skinning approaches where a surface is generated from a set of curves. When the minimum number of curves is two, surfaces of revolution, ruled, swung and swept surfaces can be created. Details about their algorithms are provided by Piegl and Tiller [87].

There are several ways to change the shape of a NURBS:

• interactively altering the position of the control points. Moving a control point causes a portion of the shape to move in the same direction. • directly altering the value of the weights. Increasing or decreasing the

value of weight wi pulls or pushes a portion of the shape toward or away

from control point p_i.

Figure 1.6: Various shapes belonging to the motor hood class whose differences can be controlled through intuitive shape handles; [courtesy of T. Wiegers].

There are a number of fundamental algorithms in NURBS design. The most important one is the knot insertion algorithm. Knot insertion means adding a new knot into an existing knot vector without changing the shape of the curve or surface. Degree elevation or reduction algorithms increase or decrease the degree of a NURBS while changing also the shape of the curve or surface. Both these algorithms are used when control over a larger or smaller area to be deformed is desired. Being able to locally deform a NURBS is a significant advantage of modelling with NURBS as this can help generate large sets of shapes. However, user desired changes like those shown in Fig.1.6 typically require manipulation and editing of many shape handles and are time–consuming and far from being intuitive. While local changes are possible, but tedious and time–consuming, a global change can be a very difficult task when the object’s creation history is not available. The NURBS’ shape handles provide more of an approximate rather than an exact control over the shape and their local influence make them unsuitable for global modifications.

de-not too many. . . just an INTUITIVE SHAPE HANDLE!

the intuitive shape handle – Constraint #1, – Parameter #1, – Parameter #2, – Constraint #2, – Parameter #3.

Figure 1.7: Symbolic representation of the mismatch between desired shape handles and currently available tools for shape editing.

formation to each patch. These are incompatible with the straight–forward WYSIWYG6 tools believed to stimulate designers to explore alternatives [82] and thus more suitable for them. In this respect, requirements for CAID tools adapted to the designer refer to:

• no shape parametrisation has to be known when modelling with NURBS; • possibility to perform meaningful editing with intuitive shape handles, like

higher–level or user–defined shape handles;

• easy specification of user–defined shape handles for controlling the shape editing;

• exploration and evaluation of the design space should be facilitated during shape editing.

CAID can only be successful for designers if developers are able and willing to go beyond the boundaries of traditional tools. The low–level shape handles

changes. Designers do prefer freedom while modelling, but this freedom seems not to be the freedom given by the control point editing of NURBS. Instead, designers would prefer to be able to specify the limits of a deformation, the behaviour of the deformation, and the system should be capable to implicitly set the necessary constraints (Fig. 1.7). A possible answer to this problem is given by the contextual shape editing proposed in this thesis.

1.3

Designers’ contextual shape editing

In general, context is credited to offer the minimal number of information pieces and essential functions that are necessary to the task at hand [46]. Its recognition helps to identify the task a designer is working on and offers the appropriate tool for further manipulation [51]. In this research, this translates to the right shape handle at the right time for the right task to help the designer explore the design space of the digital design and to evaluate different aesthetic impressions of the product shape.

Contextual deformations are somehow lacking in the specialised literature. The reasons for this refer to the difficulty to capture the right meaning of context being a rich term with too many interpretations. Then, how can one capture or detect the context and provide the desired shape handle to produce the de-sired change in the shape? Or how can one know which is the limit of a dede-sired deformation and how the surface should behave? Semantic transformations of digital designs are appropriate candidates for supporting contextual shape edit-ing. In this respect, Hsiao and Huang [55] use a neural network technique to generate new product forms from the existing forms of two similar products. The product is decomposed into its components (design elements). For each component, relations are established between the form parameters and its user perceived adjectives of the image. Shape generation rules have also been consid-ered for each product and the alteration of semantic differential sliders7 of the image adjectives generates new product forms. The system, therefore, allows users to modify product shapes according to new adjectives. A similar method is presented by Hsiao and Liu [56] for the generation of in–between products. Two products of the same class are similarly decomposed into their components to which image words are attributed. The user selects one product as the initial shape and another one as the final. The morphing-ratio between the initial and the target product controls the shape of the new product. Hsiao and Chuang [54]

a) b) c) d) e)

Figure 1.8: Shape averaging applied on a bottle: a) & c) Initial and target bottle shapes, b) the interpolated shape, d) & e) the extrapolated shapes to evaluate product trends, [24].

continue this approach of product morphing, this time using reverse engineered products which may even have different structures. The morphing is performed through characteristic curves extracted from both scanned products.

A method that helps designers explore product trends is proposed by Chen and Parent [24]. Their Shape Averaging produces new shapes by linearly interpo-lating and extrapointerpo-lating a set of corresponding piecewise linear 2D contours of two 3D objects. The 3D shape blending is therefore reduced to the 2D level through these contours (Fig.1.8). It is a syntactical blending whose transforma-tion is controlled by the user, but the authors also mentransforma-tion the necessity to have semantic blending in design. They define semantic blending as the blending of the meaningful components of a product (e.g., a component for a cup is repre-sented by its handle). Semantic blending can also be performed with the Shape Averaging method but user interaction is necessary to specify the boundaries of each component.

Smyth and Wallace [106] focus on generating variants of brand identity elements that need to fit into new designs. The elements are reduced to skeletons, lattices or bounding boxes and the system uses some genetic operations (mutation rate) to generate new shapes from these skeletons, lattices or bounding boxes. The designer can then select a new generated shape for further evaluation.

Affordable digital systems are becoming common tools for both SME’s and free-lance designers. CAID offers the possibility to directly design the product and eliminates the need to explain the design idea to the clay modeler. Experience has demonstrated that creating a CAID design sometimes requires a fraction of the labour needed to generate the corresponding clay model [132]. Hence, far less cost is involved in working with CAID compared to a physical model. The digital model can also directly serve as the starting point for the down-stream departments, eliminating the need to scan the clay model and generate surfaces from the point clouds. Once in CAID, designers have direct control over their designs and are also provided with exceptional visualisation and rendering capabilities for better presentation of the design to the client.

Designers need freedom in shape modelling to explore alternatives or to person-alise existing products. The freedom in deforming NURBS shapes is appreci-ated, but designers are also forced to think about the mathematical limitations of NURBS parametrisation and not just about their designs. Available CAID tools are still lacking in supporting designers with intuitive shape handles to perform fast and easy shape exploration [39]. Therefore, the central question of this research is:

Given the digital shape of a product that has no creation history and new design requirements for a new similar product, how can the designer efficiently edit the shape of the existing product to generate a set of similar but new products?

Fig.1.3b presents examples of new products that can be generated from an ex-isting object (Fig.1.3a).

1.5

Aim of the research

Semantics

Figure 1.9: Semantics, structure and shape as levels of abstraction in product perception, and as implementation levels in CAID.

structure, which is built on its shape. This approach to shape perception and modelling is widely accepted, [1]. CAID typically operates at the shape level, while structure is rarely explicitly supported. Modelling at the semantics level is currently extremely difficult to be handled in CAID by the designer. User– specifications of shape handles seem appropriate for designers, but a tedious or time–consuming specification of a shape handle would still not make the defor-mation more efficient compared to control point editing. A possible way to have meaningful shape handles for controlled contextual deformation is manipulating at the product structure level. Our aim is to investigate CAID activities, to understand how designers can be better supported in terms of intuitive shape handles for fast and easy editing, and to propose a methodology for the devel-opment and implementation of contextual shape editing tools.

1.6

Overview of the thesis

Chapter

2

Graphical objects editing

Shape modelling has two main branches: shape creation and shape editing. In Section 1.2, we have briefly mentioned about the skinning creation approaches that are most common in shape creation. These are versatile and powerful, allowing the creation of complex shapes by interpolating a network of curves. Depending on the creation level, they may require the trimming of an existing surface to accommodate a new element and blending afterwards to smoothly connect the new element to the existing surface. Blending with geometric con-straints is presented in [118, 44]. For a review of currently implemented creation approaches, the reader is referred to [114, 87, 88, 90, 91, 42] or the standard user manuals of commercial surface modelling software (e.g., [5]). In this chap-ter, we focus on the other branch of shape modelling, namely shape editing, by presenting a review of graphical objects editing approaches from a designer’s perspective.

2.1

Graphical objects

Gomes et al. [49] give a broad definition of a graphical object O as consisting of a finite collection U = {U1, . . . , Um} of subsets Ui ⊂ Rn, of some Euclidean

space Rn_{, and a function f : U}

1∪ . . . ∪ Um → Rp. The family U is called the

geometric data set of the object. The union U = U1∪ . . . ∪ Umdefines the shape

of the object, and f is the attribute function of the object. The dimension of the shape U is called the dimension of the graphical object. The shape U defines the geometry and the topology of the object, and function f defines the different attributes of the object. Depending on the dimension of the graphical object, ranging from 0 to 3, their geometric representation can be classified into, [127]:

(0) Discrete models: they refer to point clouds, particles or mesh represen-tations of 3D objects. They have high storage requirements, are relatively easy to render, but difficult to handle during interactive manipulation. The shape’s degrees–of–freedom (DOF) equal its vertices, implying that their interactive editing is difficult and tedious. Point clouds and particle rep-resentations are just emerging in the field of shape editing, whereas mesh representations have been already successfully used in computer graphics.

(1) Wireframe models: they represent objects as a topological network of lines. They produce an ambiguous visual representation of a 3D model and are, therefore, rarely used.

(2) Surface models: they refer to continuous functional representations, and are mainly of implicit or parametric form. In most cases, the object is decomposed and represented as a set of patches resulting in a piecewise description. Topological and geometrical constraints, like the relations imposed on two or more neighboring components or patches, or the G0– position, G1–tangency and G2–curvature continuity are imposed preserve the perceived structure or guarantee the object’s validity.

(3) Solid models: they represent the boundary of the objects together with their interior. They can employ space partitioning trees, using decomposi-tion or constructive techniques, or boundary informadecomposi-tion (B-rep). B-reps eventually describe objects in terms of their lower dimensional entities, like surfaces, curves, or points. Due to their capacity of storing information about the inside of the objects, they are very suitable for the production departments for their numerically controlled process–planning operations.

Figure 2.1: Computational geometry representation; adapted from [49].

2.2

Evaluation criteria

Shapes have a variety of handles that can be employed to modify their appear-ance, spatial position, or behaviour. For example, control points can be used to modify the shape of a surface, or the radius of a blending surface can be varied while constraining its boundary edges. Shape handles are therefore important to define and control a deformation. They define the amount, behaviour and limits of the deformation, and constrain some of the parameters of the shapes, differ-ential properties or degrees–of–freedom. Parameter values can be changed to reshape the object while constraints define relations between the geometric enti-ties [34], forcing the latter either to preserve a constant value or to vary within a tight interval. Examples of parameters refer basically to all defining variables of the composing primitives of a model (like the radius of a cylinder, the height of a parallelepiped or the centre point of a sphere). In addition, for freeform shape representation they also refer to the position of control points, the direction of normal/tangent vectors, the shape of the defining curves of skinned surfaces, or the continuity between patches. Among several types of shape handles evalu-ated by Dijk et al. [33], it appeared that users prefer higher–level handles (like tangent plane manipulation or curve–on–surface manipulation) instead of us-ing low–level or far too abstract handles (like control point or embeddus-ing space manipulation).

Besides analysing the types of shape handles supported by the various methods presented in the literature, we will also consider the following issues:

• How can the user define his desired deformation?

• What kind of input is necessary for this and how tedious is its specification? • Is there any interactivity supported?

interpolated. These are all issues to be supported by any deformation and for some of them an intuitive correspondence and interpolation must also hold be-tween the surfaces involved.

2.3

Graphical objects editing categories

Shape editing allows the user to treat an object as if it were built from clay and could be repeatedly transformed into a final desired shape. In practice, the designer’s effort is concentrated on the extremely tedious set–up of the editing to generate sets of similar shapes with different expressions. The existing edit-ing techniques are commonly split into two categories: geometric–oriented or physics–oriented; which can be both applied locally as well as globally. While the geometric–oriented deformations involve some kind of geometric constraints that the deformed surface should fulfil (like point or curve interpolation), the physics–oriented approaches consider physical properties of materials or theories of elasticity and plasticity to produce fair surfaces (class A surfaces). These clas-sification criteria fail, however, to properly support designers with deformations focussing on semantics specific to product classes (Fig. 1.9). With respect to a product, a deformation should support changes of its shape to generate different meanings of the product, or should support product structure alteration to gen-erate products with new added functionalities. Therefore, we review the shape editing literature considering the following capabilities:

Structure–Shape editing A graphical object can be transformed into a com-pletely different object when both its underlying structure (i.e. the skele-ton) and its form (i.e. the shape) are changed. Typically, the resulting object belongs to a different product class than the initial one.

Structure editing Alterations of the object structure can be performed by modifying the underlying structure itself, or by inserting components into its underlying structure. The shape corresponding to the deformed skele-ton preserves the initial characteristics with respect to (w.r.t.) the skeleskele-ton and, as such, is said to be unchanged. The resulting object may belong to the same product class, a similar one, or even a different one.

Figure 2.2: Object twisted with Barr operator, [9].

For each of the three categories, we focus on shape editing techniques for graphi-cal objects with either parametric or mesh representations. Designers prefer the smoothness of the parametric representations; however, due to their relatively easy computational manipulation, mesh models present many intuitive solutions regarding shape editing. The review covers topics like feature modelling [113], shape reuse [127], shape morphing [2, 71], skeleton–driven modifications and space deformations [10].

2.3.1

Structure–Shape editing

Free Form Deformations and variants

In shape modelling, tangent vectors are used for delineating and constructing the local geometry, and normal vectors are used for obtaining surface orientation and lighting information. The early work of Barr [9] introduced the idea of procedural global and local deformation by using transformation functions for the surface tangent and normal vectors. A geometric mapping of three–dimensional space F : R3_{−→ R}3_{, for example, given by:}

F (p) =   cos θ − sin θ 0 sin θ cos θ 0 0 0 1     px py pz  

maps the vectors of an initial surface into those of a final surface causing the object to twist about the z–axis (Fig. 2.2). Other related mappings can cre-ate rigid motion, tapering, stretching, or bending around a central axis. More complex deformations can be constructed by composing mappings under hierar-chical arrangements. Barr operators are extremely efficient but of limited utility since they offer only specific stylised deformations and the user’s control of de-formation is not very intuitive. When freeform shapes are involved, unexpected resulting shape sometimes can appear.

Figure 2.3: Telephone handset resulting from successive local and global FFD of an initial simple bar; [100].

through interactive repositioning of control points, generates a corresponding deformation of the embedded object. These deformations are commonly known as Free Form Deformations (FFD) and are formulated as a mapping F : R3_−→

R3 −→ R3 that takes a point within the lattice, P = (x, y, z), from the world space, through the local parameter space of the lattice, to the deformed world space, P0 = (x0, y0, z0). The local coordinates of the points are formulated as a sum of control points weighted by polynomial basis functions, as follows:

P0 = F (u, v, w) = a+l−1 X i=0 b+m−1 X j=0 c+n−1 X k=0

Bi,l(u) · Bj,m(v) · Bk,n(w) · Si,j,k

where F (u, v, w) specifies the local parameter space coordinates of P0, B are the basis functions (the first subscript denotes the index and the second the order), and (a, b, c) are the lattice S control points in its three directions. The use of these trivariate Bernstein polynomials generates smooth deformations and pre-serves object continuity. The method requires, however, that the control lattice is of standard geometry which makes it difficult to approximate objects with arbitrary shapes. Coquillart [29] proposed a general Extended Free Form De-formation (EFFD) by joining multiple regular shaped lattice structures to form arbitrary shaped spaces. The EFFD can be positioned over the object for se-lective arbitrary control of sub–regions of the object’s surface. G1 _continuity

during deformation is guaranteed by merging the control points of neighbour-ing lattices. This constraint makes the EFFD computationally expensive and reduces the user control of the surface corresponding to these constrained areas.

of tetrahedral B´ezier lattices. Points P (r, s, t, u) lying within the lattice can be formulated as:

P (r, s, t, u) = X

i+j+k+l=n

Pi,j,k,lBi,j,k,ln (r, s, t, u)

where Bn_i,j,k,l(r, s, t, u) = _i!j!k!l!n! risjtkul, with r + s + t + u = 1, and Pi,j,k,l

are control points of the tetrahedron. The combination of such tetrahedrons allows creation of arbitrary lattices, but preserving G1 _{continuity during the}

deformation requires defining constraints on the displacement of five control points of a pair of opposite tetrahedrons. Three of these five control points belong to each tetrahedron.

Other developments have concentrated on generating FFD lattices as similar to the objects as possible. In this respect, MacCracken and Joy [74] used a special Catmull–Clark subdivision algorithm to refine the lattice and provide the desired shape. Critical lattice points along sharp edges and corners have to be handled separately. Kobayashi and Ootsubo [65] and Shao et al. [102] developed similar methods using triangular and Doo–Sabin subdivided 2D manifold control meshes respectively. In both cases, object points are parameterised by the local coordinates of each point of the control mesh, and new object point positions are blended with respect to the deformed control mesh point positions, generating in this way relatively smooth results. Comparably, Singh and Kokkevis [105] developed a surface–oriented FFD where a surface S is modified by a deformer O which is defined as a triple hD, R, locali. D and R are the driver and reference surfaces of matching topology with bijective correspondence, while local is a scalar that controls the locality of the deformation. Initially, points on the surface are bound to the reference surface using a distance metric represented as a scalar function f (d, local) = 1/(1 + dlocal_{), which decays in value as the}

distance d increases. Each deformed point Pdef of the surface is computed with

Pdef =P n k=1u P kP def k , where u P

k is the normalised weight of control element k

for point P defined as uP_k = w_kP/Pn

1w P k.

a) b)

Figure 2.4: a) Initial model and the embedding axial curve. b) De-formed model based on the deDe-formed axial curve; [69].

squares energy to produce the desired deformation. As a result, the constraint point does not entirely resemble the desired one and the final result is actually an approximation.

None of these methods associates any physical constraint to the deformation of the objects and some researchers claim that deformations of geometric models should preserve the physical properties of the object to provide users with intu-itive deformations. The most common property in shape deformations is volume preservation, and this is believed to provide means of deformations similar to physical clay modelling. Aubert and Bechmann [7] proposed therefore a method using a generalised direct manipulation FFD based on a least-squares energy function [57] to guarantee total volume preservation of the embedded object. Because the method uses approximated volume computation, it is not ideal for very large deformations, and is limited to polyhedra. The volume change pro-duced by each face of the polyhedra is computed based on the area of each polygonal face and its normal as : V (P ) = 1₆P

f FAf· nf. Then, the volume

difference between the deformed and un–deformed polyhedrons is minimised.

Feng et al. [43] introduced a different global deformation by manipulating the object to be deformed according to the editing of two parametric surfaces: a shape surface S(u, v) and a height surface H(u, v). A control point or vertex, P , of the object is projected onto the shape surface along its normal and correspon-dence is established between these two points: P = −→S (up, vp) + hp

− →

Ns(up, vp).

The object is embedded into the parametric space defined by this shape sur-face and any further editing of the shape sursur-face is computed with reference to the height surface and produces displacement of the object’s points: Pnew =

− →

Snew(up, vp) + Hnewhp−→nnew(up, vp), where −→nnew(up, vp) is the unit normal of

− →

Snew(u, v) at (up, vp), and Hnew is the new height w.r.t. to H(u, v). The

a) b) c)

Figure 2.5: a) The initial object defined coarsely by the wires. b) The wires. c) Deformed object corresponding to deformed wires; [104].

Compared to the more general and interactive FFD techniques and its variants, Barr’s [9] original work on axial procedural deformations is considered relatively specific and therefore limited. Lazarus et al. [69] have generalised Barr’s 3D axial deformation to allow user–defined curvilinear paths and interactive defor-mations. The degrees–of–freedom of the surface to be deformed are attached to points of the 3D axial B–spline curve (Fig. 2.4a). A local coordinate frame for each point is linked to the parametrisation induced by the axis (by finding the closest point on the axis). Reshaping the axis generates the modification of the cylindrical space surrounding it as well as the object immersed in this space (Fig. 2.4b). Although powerful in providing the user with higher–level shape handling compared to the lattice control points of the FFD, this method fails to provide a one–to–one correspondence between the curvilinear path and the object and can generate unexpected results.

To overcome these limitations, Singh and Fiume [104] present another inter-esting contribution in giving the modeler direct control over and specification of the deformation. Their wires geometric deformation primitive incorporates a familiar interface to sculptors: armatures. Wires, represented by parametric curves, are used to coarsely define an object (Fig. 2.5a–b) and their further local editing controls the deformation of the object around the wire (Fig. 2.5c). Wires are defined as tuples hW, R, s, r, f i, where W and R are the freeform parametric curves, s is the radial scalar around the curve, r is the radius of influence around the curve, and f is a density function. The deformation is computed based on vector displacement fields defined through a smooth falloff function around the wire and its reference curve dragging the surface’s control vertices along. The region of influence is a function of distance in R3_{. As a result, the object’s}

Figure 2.6: A 5–step process to deform a sphere into a spork. One or two handles are employed to deform the shape similarly to clay mod-elling; [73].

Barr’s approach [9] has been further developed by Llamas et al. [73]. Gluing one or two handles to a surface and transforming these handles will drag–and– rotate the corresponding surface points and their associated surface normals and tangents. In their Twister shape editor, the transformations are interpreted as time–dependent screw motions. A decay function is used to distinguish between the surface points attached to the handle undergoing a full screw motion and the neighbouring points which are only partially transformed (Fig. 2.6). The proto-type uses two Polhemus trackers through which the user can attach the handles to the model and then simultaneously stretch, twist, and bend the surface of the virtual 3D object.

The FFD techniques have found applications in modelling [65, 102, 73] as well as animation [30]. They gained widespread applicability due to their local and global effect capabilities and their property to deal only with surface information. No information regarding the object structure (e.g. skeleton) is employed, yet the object’s structure may change after the deformer is applied on the object. While theoretically the deformation acts on every point of the object, in practice objects are sampled and only representative points are used. They can handle a variety of object representations as long as sample points can be extracted from the embedded object. Objects described by parametric representations will still appear smooth as the deformation is conducted on its control points. However, when few points are sampled from polyhedral objects, the actual deformation will be far from the expected result and in the worst case the objects may appear unchanged. Thus the sampling problem is important for the practical use of FFD techniques.

are relatively intuitive and efficient, those methods can only offer limited defor-mations and applying them several times conflicts with the designer’s creative work.

Shape morphing and blending

Metamorphosis is of Greek etymology and is composed of two words: meta and morphosis. Meta means between or after, and morphosis refers to the way a form or a structure changes. In the specialised literature, the term morphing is used as a short–hand form of metamorphosis. Shape morphings are a mix of structure and shape transformations and represent the continuous transformation of a source shape into a target shape. In Computer Graphics, morphing requires solutions to two sub–problems: correspondence and interpolation. As there is no agreement about which point on the source corresponds best to which point on the target, and which travelling path from one to the other (interpolation function) generates an intuitive transformation, the quality of the generated morphing sequence is thought to be in the eyes of the beholder. There exist several approaches to shape morphing.

Sederberg and Greenwood [99] defined a physically based quality measure for the correspondence of two–dimensional simple polygonal contours, which min-imises the energy of the morphing. The quality measure intuitively assumes that similar shapes need to undergo much smaller transformations to resemble each other’s shape compared to dissimilar ones. The authors also considered cases where source and target polygons contain different number of vertices. Here, they propose vertex multiplication to yield a globally optimal solution to the energy minimisation problem. Further, they used linear interpolation to gen-erate in–between shapes. However, this seems to be inadequate for cases like the interpolation of two parallel segments oriented in opposite directions which collapses for some interpolation value. Therefore, Sederberg et al. [98] proposed interpolating both the edge lengths and the vertex angles, instead of just the vertices. The intrinsic qualities of the source and target shapes are preserved during the generated morphing sequence, but special attention has to be paid to guarantee the result is a closed polygon.

C2(v)

C1(u)

T2(u)

T1(v)

Figure 2.7: Matched tangent fields of two freeform curves; [27].

the skeletons of the source and target shapes and then unfolding the separate star polygons. Although the technique is versatile for objects presenting simi-lar skeletons (isomorphic), self–intersections do occur when the dissimisimi-larity of the two shapes is too large. Therefore, Mortara and Spagnuolo [80] proposed blending two polygonal shapes if they are similar. The authors use approximate skeletons (a variant of the medial axis) to morphologically define the two shapes and assume that the two shapes are similar if their approximated skeletons are isomorphic as graphs. The similarity measure considers the size and the rel-ative orientation of the arcs of the graphs and is defined as: SIMa(e1, e2) =

αSIMangle(e1, e2) + βSIMlength(e1, e2) + δSIMarea(e1, e2) + γSIMtype(e1, e2),

where e1and e2are two arcs of the two skeletons, and α, β, δ, γ are weights that

can be changed according to the requirements of the specific input. The global correspondence is performed firstly by a partial matching of the sub–graphs and then by finding correspondence between the components. Linear interpolation is further used as a proof for the correspondence issue.

In the context of twist–free parametric surface blending, Cohen et al. [27] intro-duce an inter–fairness matching algorithm that matches the relative parameteri-sations of two or more freeform parametric curves using the first order differential properties. The unit tangent vectors sampled uniformly in the parametric space of the two curves guarantee that features with similar tangent fields of the para-metric curves are preserved (Fig. 2.7). The arc length or curvature–based sam-pling can result in a better matching but at the drawback of a time–consuming and expensive shape interrogation process. After matching is established, in– between curves C(t) can be generated by employing an affine combination be-tween the two curves C1(t) and C2(t) as follows: C(t) = (1 − s)C1(t) + sC2(t),

intersection is eliminated by comparing the points where the intersection occurs and evaluating instead their left and right mirror points values when creating the intersection free in–betweens.

Three-dimensional morphings are much more difficult to compute compared to their two-dimensional counterparts, and to make things easier, Hui and Li [61] proposed a feature–based shape blending technique. Initially, an automatic anal-ysis of 2D polygonal contours is introduced. Given two contours, the source and the target, the algorithm first detects their meaningful features. Features are found by detecting their boundary vertices represented as sharp turns. These turns are detected when the ratio of the radius of the curvature over the total curve length is less than a predefined constant value. Then the algorithm estab-lishes global correspondence relationships by linking the similar features. The generated correspondence fails, however, to provide an acceptable result when the two contours have different relative orientations. Nevertheless, the proposed 2D curve blending method has been successfully applied in 3D shape design for the blending of homeomorphic products by combining their 2D profile curves. Instead of automatically determining the shape features, Cohen–Or et al. [28] introduce user–defined anchor points which are used to align the two objects and provide significant user control over the overall morphing process. A rigid (rotation and translation) transformation and an elastic warping then form a two–step warp which provides reduced distortion in the intermediate models. The rigid transformation coarsely matches the source to the target, while the elastic warp manages the finer features of the models through a distance (or volume) field interpolation. Interactive surface decomposition is also proposed by Gregory et al. [50]. In their approach the user specifies the feature–patches, while some patches are automatically re–meshed until the corresponding morphing– patches have similar topologies. The morphing trajectories are represented here as cubic B´ezier curves whose shapes can be interactively changed by the user so as to produce an accelerated or slower transformation of some vertices. The drawback of the method is that it can be applied only on homeomorphic models and they have to be aligned as to prevent self–intersections.

In order to provide an intuitive morphing, Lazarus and Verroust [70] argue that the issues of correspondence and interpolation should be dealt simultaneously with. They therefore proposed a common parameterised polyhedral mesh built from the two defined skeletal structures of the source and target meshes respec-tively. The two objects have to be star–shaped around an axis and they can have a cylindrical part and one or two hemispherical parts so that cylindrical and spherical parameterisations can be employed to yield natural correspon-dence. During morphing, the skeletal curves are blended and their embedded meshes are interpolated to generate the in–betweens.

Figure 2.8: A five step morph history generated between a cube and a six-pointed star. The textures of the objects have also been morphed; [16].

anymore as the correspondence is established through the trimmed skeleton of their symmetric difference (Fig. 2.8). Given two shapes A and B, their sym-metric difference is A4B ≡ (A − B)S(B − A), and their trimmed skeleton is trim(Skel(A4B)) ≡ Skel(A4B) − CSkel(A) − CSkel(B), where CSkel is ob-ject’s complete skeleton. The contact points of the maximal spheres centred on the skeleton are used to establish points correspondence and linear interpolation is used for the generation of the morphing sequence.

Piegl and Tiller [89] present a set of symbolic operators for the geometric com-puting of spline curves and surfaces. The most interesting for us is the linear combination of curves or surfaces. Given the rational surfaces represented as Si(u, v) =

Ni(u,v)

di(u,v) =

[xwi,ywi,zwi]

wi , with xwi = x(u, v)wi(u, v) and a scalar α,

their sum can be computed with:

S(u, v) = (1 − α)S1(u, v) + αS2(u, v)

and the α scalar controls a morphing sequence as in Fig. 2.9. This operator acts on corresponding control points while knot refinement and degree eleva-tion are necessary to make the initial entities compatible. Therefore, piecewise representations or trimmed patches cannot be handled by this operator.

Following the idea presented in [27], Surazhky and Elber [108] propose a match-ing algorithm for parametric surfaces. They use the normal field, which is the counter–part of the tangent field from the curves’ case, and a resemblance metric between the discrete sampled surfaces’ unit normal fields as matching criterion. The two surface parametric domains are then separated into corresponding sub– parts, and the matching process is performed at this level and mapped back to generate the 3D in–between surfaces. This relative parameterisations matching preserves object features and guarantees a more intuitive morphing compared to the convex combination commonly used.

z

y x

S1(u, v)

S2(u, v)

Figure 2.9: Linear interpolation between two NURBS surfaces; [89].

manual correspondence (anchor points, patch border) and automatic correspon-dence (skeleton, feature detection) are supported. The few approaches we found for parametric representations employ shape interrogation to find matching tan-gent or normal fields of the two surfaces. After correspondence is established, most of the techniques use linear interpolation of vertices or sampled points to generate the in-between shapes, while Gregory et al. [50] propose using B´ezier curves for describing the travelling path of the vertices. While morphing can be a powerful tool for designers to help them continuously transform a shape into another, the presented approaches produce automatic transformations and none of them allow designers to control the amount of the morphing.

2.3.2

Structure editing

There are two types of deformations which involve structure changes. The first one we deal with refers to the changes occurring at the spatial relation level between existing components of the object. The other structure deformations refer to alterations of the product structure by inserting new components or features into the product.

Skeleton–like driven editing

Figure 2.10: Local and Global skeleton–based shape editing; [136].

for our research in product modelling. A skeleton–like driven deformation would generate in most cases products belonging to the same class, sometimes to similar classes, and very rarely different products can be reached. A skeletal structure is an abstracted object representation, that encodes the hierarchical decompo-sition of the object into its meaningful components. These can refer to purely geometric or application–dependent components, but it very often conforms to human perception.

Yoshizawa et al. [136] introduced the concept of mesh deformations based on skeleton editing. A one–to–one mapping exists between the initial and the skele-tal mesh, as the latter is a Voronoi–based skeleskele-tal mesh. The deformation of the initial mesh M is then represented as a set of vertices displacements d from the skeletal mesh S as M = S + dN , where N is the unit normals field N defined at the vertices of the mesh S. A point–constraint [57] is used to edit the skeletal mesh, and the initial mesh is reconstructed from the new skeleton. However, self–intersections are avoided by employing a homotopy method to de-compose the deformation into a sequence of L deformations connecting the non– deformed skeleton S0= S and the deformed one SLgiven by: Sj= S0+ jSL−S_L 0,

j = 1, 2, . . . , L. In contrast, Katz and Tal [64] firstly segment the mesh into its meaningful subparts analysing their geodesic paths, and then proceed with ex-tracting a skeletal line for each part. From these a skeletal structure of the object is constructed and then the object can be deformed through skeletal editing. As Fig. 2.10 shows, skeletal structure editing can produce both local and global deformations, which can represent a powerful tool for industrial designers.

require-Figure 2.11: A skeleton–based structural deformation; [60].

ment by allowing extrusion of open surfaces. The two open curves sketched by the user determine the shape of a new element to be inserted into the model. However, their implementation cannot avoid intersection between the extruded surface and the existing surface.

Most of the proposed algorithms are suitable for mesh representations, but re-cently other representations are being considered. Hui [60] developed a spe-cial type of axial deformation that interactively supports bending, twisting and stretching of objects. The method is object representation free and is an exten-sion of previous works reported in [9, 69] being interactive and allowing twisting. Axial curve–pairs (primary and orientation curves) are defined as 3D B–spline curves. The primary curve is a skeleton–like curve, while the orientation curve is an approximate offset of the primary. The object is located with respect to the Frenet frames of these two curves, oppositely to only one in [69]. A typical ses-sion is shown in Fig. 2.11. Any modification of the orientation curve is reflected on the primary curve and then the object is deformed according to the two sets of Frenet frames. This method, however, allows only deformations of the object structure, and new elements cannot be inserted into the object structure while deforming the curve–pairs.

Features have initially emerged in the mechanical design field. They are defined as sets of faces or regions of one component (or assembly) with distinct topo-logical, geometrical and engineering information [101]. Features are therefore building blocks of simple shapes used for product definition or geometry reason-ing and allow designers to operate with higher–level more intuitive parameters and constraints, rather than with geometric constituents such as points, curves and individual patches. Different feature classes can be found in the literature depending on their application domain, like design features (having only topolog-ical and geometrtopolog-ical information) or manufacturing features (having topologtopolog-ical, geometrical and manufacturing information). They are sometimes used to char-acterise the mechanical product using numerical or knowledge-based systems.

In the industrial design field, features are defined as shape details that enhance the aesthetical appearance. Examples include character lines of a product [47], or styling lines [68]. They are often linked to perceived rather than functional qualities, and are rather difficult to define or store in a feature library. Any shape editing inserting a feature into the object produces a structurally different object. This is usually done by constraining an area on the surface that is then deformed, or by trimming an area and inserting the feature either through repositioning the trimmed part and blending it with the main object or copy–and–paste it from another object.

Constraint–based deformation on limited areas Celniker and Welch [21] developed techniques for interactively sculpting deformable B–spline curves and surfaces through slide–bar variations of some individual sculpturing forces. The sculp-turing forces presented include pressure, springs, and gravity for the effects of enlarging, attracting, and flattening. Springs attract a surface point to a point in space, gravity pulls on the surface only in one direction, and pressure acts in the direction of the surface normal. The user specifies one or a set of constraints that the deformed surface has to fulfil. Constraints refer to a specific point or curve interpolation, or the orientation change of a surface through the specifica-tion of point, curve and normal direcspecifica-tion constraints. Fair deformed surfaces are achieved by constructing an optimisation system which minimises certain objec-tive functions, such as thin plate energy (stretching and bending) subjected to these constraints: R

σ(α stretch + β bend)dσ −→ min, where α and β are weights

on stretching and bending. This optimisation problem is discretized to arrive at a finite dimensional linear system of weighted first and second derivative squared norms which generate an approximate solution.

point and curve handles to a surface, and then moving or deforming these han-dles. Surface points can be selected and moved to new locations, while curves may be inscribed onto the surface and manipulated. To guarantee minimum errors of the optimisation system, surface refinement is automatically performed on selected regions.

A direct surface manipulation (DSM) is proposed by Marsan et al. [76]. As in [21], the user sketches a boundary curve on the surface to be deformed and a target constraint to be interpolated by the local deformation. In addition, radial and Dirichlet parameterisations are used, which allow generation of various shapes. A point target constraint produces a bump–like feature, an open curve target constraint generates a ridge–like features, while a closed curve creates a protrusion–like feature.

Another approach to surface editing through user–defined target curves is pre-sented by Michalik and Bruderlin [78]. A user–defined curve C(t) is projected on the surface S domain, and its pre–image from the surface domain extracted, S(u(t), v(t)). The deformed surface is produced by minimising the square dis-tance, J , between C(t) and S(u(t), v(t)): J =R_C1₂[S(u(t), v(t)) − C(t)]2dt. The result suffers, however, from 3D aliasing effect due to possible mismatches be-tween the iso–curves of the surface parametric domain and the curve pre–image. Therefore, a solution is proposed by reparameterising the surface such that the curve pre–image matches an isoparametric direction. This solution gives good results with rather simple curves, but it fails in supporting highly freeform target curves due to the surface’s rectangular domain.

A touch–and–replace method to allow designers reshape a curve on a paramet-ric surface has been developed by Michalik et al. [79]. The method can be applied locally to generate new components or eliminate existing ones, or can be applied globally to change objects expressions. The user–sketched stroke is approximated with a B–spline curve C(u). Using the end conditions from C(u), a selected curve–on–surface is reshaped to approximate C(u). Initially, a para-metric interval of the surface domain corresponding to C(u) is sought. Then the positions of the two neighbouring knots of this interval are found and a transition geometry is generated by interpolating between these two knots and the differ-ential properties at the ends of C(u). Finally, the new curve is reparametrised to fit the surface knot sequence. To limit the change introduced in an existing surface, curve constraints can be placed on the surface in a similar manner.

Feature Generation and Manipulation Cavendish and Marin [20] were the first to allow the design of user–defined pockets or protrusions. Starting from a given surface, S0, and specifying a secondary surface, S1, as the floor of the desired

pocket, a feature can smoothly join these two surfaces between two closed curves ¯

C0and ¯C1of the two surfaces (Fig. 2.12). An assumption is made that the

Figure 2.12: Primary S0 and secondary S1 surfaces before feature

cre-ation; [20]

z = f0(x, y) and z = f1(x, y). A smooth transition is constructed in the ΩT

re-gion generated by the difference between the plan–view projections of ¯C0and ¯C1.

For the construction of the feature z = g1(x, y), a transition function Φ(x, y) that

increases monotonically and guarantees tangency continuity between the two surfaces is used, where g1(x, y) = (1 − Φ(x, y))f0(x, y) + Φ(x, y)f1(x, y). The

main drawback of the approach is that the feature is automatically generated and any further modification is difficult to perform.

An interactive freeform displacement feature method, similar to the previous one, has been developed by van Elsas and Vergeest [117]. A sketched curve C(u) on a B–spline surface S(u, v) bounds a surface area to be displaced at a user– specified height (Fig. 2.13). A transition geometry is then constructed between the base and the displaced surfaces as a ruled surface with G1 continuity and without self–intersections. The method is highly interactive and allows users to control the shape of the feature much as is the case with mechanical feature. Thus, a set of adjusting slide–bars control various defining parameters of the feature, like the height and scaling of the displaced top, and the two rounding parameters at the feature boundaries.