Steven Melemis
email: ssmelemis@gmail.com

Damien Masson
email: damien.masson@grenoble.archi.fr

Representing Spatial Hybridities. A Pragmatic Approach to the Representation of Emergent Figures of Public Space Employing Conventional Computer-Based Tools

As this paper's title suggests, it will associate a number of terms among which the relations may not immediately be obvious. One of these is the "figure" -roughly, the figure in the sense of figure-ground relations as they are present in architectural discourse as well as in the domain of the psychology of form. A second term, no less important here, is that of hybridity; it refers to emergent spatial environmental conditions that can be defined in terms of easily identifiable existing elements which recombine to produce new territorial figures that seem to resist classification within pre-existing categories.

The objective of this presentation is to examine each of these terms separately in light of certain difficulties we are constantly faced with as we work in the contemporary territory, and then to propose a mode of association between them carrying some consequences for how we think about and practice design in the contemporary territory.

This work has been developing in our research and, at the same time, in the context of the design studio. In both of these contexts, we have worked in situ on studies of spatial conditions around recently constructed tramway lines in Cologne and Grenoble conurbations , and on microclimate-related situations around Grenoble (2007). Both of these subject have served as the basis for devising and testing specific methods for identifying and intervening in what we have called hybrid figures. The second theme allowed us to extend ideas about spatial hybridity which had previously been formulated. Among these was the question central to this paper: that of the invention of specific modes of computer-generated graphic representation related to the notion of hybridity.

"Figures", "Hybrids"

First it will be necessary for us to consider the notion of the figure in its relation to the territory and this will mean re-examining a few founding references on the question in the domain of architectural theory. Derived in the first instance from gestalt form-psychology, it refers to that instant in which a set of relations -be they temporal as in the case of music, visual as in pictorial representation, or of another nature -appears to constitute a whole that is something more than the sum of its parts. The perception of form thus occurs through this "coming forward" of the figure or, to put it differently, in the act of "pointing-out", of designation. Since the form-experiments and theorizations of Klee, Albers, Moholy-Nagy and others at the Bauhaus, the determination of viable, strong forms within the rapidly changing visual context of the modern world has been very frequently explored in terms of relations between "figures"-which one generally wishes to be a sharply defined as possible -and "grounds" often of great complexity.

These ideas make it to the United States in the post-war period, brought there by the key people who had first explored them -Rudolph Arnheim at the New School, Moholy-Nagy at the New Bauhaus and Gyorgy Kepes at MIT and at Harvard. It was perhaps Kepes that provided the most forceful and public expression of what he called "the language of vision" in art, design and architecture; it was then that the young Kevin Lynch picked up from him ideas of gestalt form-psychology, devising methods for architects and urban designers working within the already problematic framework of the American city. (In a moment we'll come to the still more problematic current state of the territory.) The work of his that we all of course know sets out a paradigm of reflection that has proven extremely durable; it involved resituating the architect's gaze at least momentarily at the level of the users of urban space in an attempt to capture the ways in which strong form-consciousness emerged -or didn't -as one moved around in city space.

In opening up this path to the user's apperception of the city's plan, if rather crudely at first, through the notion of the mental map, Lynch lent credibility to what the senses -for him mainly that of sight -could convey, independently of words or concepts. Some distance was thus obtained from the functionalistic terms dear to many architects and planners, technicians and certain sociologists. In a sense, his approach contributed to the opening of a sort of common middle ground between empirical observation in the "scientific" realm and that of the everyday experience of space.

Since then of course, the question of the figure has held a central place in urban design theory. Its remarkable longevity is certainly linked to the fact that it informs a situation of rapid and profound change in the urban environment that causes us to reconsider over and over the principles of intelligibility that best apply to the project situations we are faced with. Today, as the very idea of the city appears fatally compromised -and with it of course that of the country (and also that of "wilderness") -the question of how one might understand spatial forms in the territory is profoundly modified. Rather than clear, structural oppositions between overriding figures -city and country in the first instance -we are confronted with mixtures of small or large fragments of each. The discourse on the "fragmentation of experience" is perhaps as old as modernity itself but, if we believe Henri Lefebvre, Françoise Choay and many others, that fragmentation would appear to have definitively reached the city. The evolution of techniques has of course played a role in this. As many authors (including Latour) in the domain of the sociology of science and technique have insisted, the greater the promise of control or of power a scientific discovery or a technical invention offers, the more unexpected and potentially uncontrollable the effects it can unwittingly unleash. As any architect or urban designer or planner knows by now, power and singlemindedness in the technical realm often still go together. In the territory of course, the sideeffects of a project often emerge literally and physically alongside or around the site on which something has been built; a piece of transport infrastructure, a shopping mall, a residential subdivision or just around an ordinary apartment tower.

In such contexts, if the urban or post-urban territory continues to contain intelligible spatial figures, these would appear to belong only in part to familiar systems of architectural and urban form, or to the specific pictorially-derived frame that we call landscape… It would appear that a good deal of the "things" we see and live with cannot be said to belong to one or the other of these two frames and that, as a result, we would have to content ourselves more and more with the term "environment", as abstract as it can seem at times to relate to our lived experience of space. This brings us to the question of hybridity, in the first instance not in the realm of representation but in the territory itself. We have come to adopt the term -at least temporarily -in a sense close to that which Bruno Latour has given it. Latour's idea concerns the way in which "networks" of relations form not only within the social realm in the strict sense, but between it and the world of things (one might say of inanimate matter) and of living, non-human beings. The idea here is that these elements tend more and more to combine and produce unexpected phenomena that demand our attention. It should be stressed that these phenomena are implicit aspects of our environment. They only assume the status of "things", tentatively at first, in the instant in which we designate them.

The act of designating an "element" or a "thing" in the territory is the one in which one points and says, "there is something there, though I don't yet know what it could be called". We are interested in how we might position ourselves as designers with respect to this moment, be it we or others who are doing the pointing. We are most interested in those situations in which a variety of individuals simultaneously designate more or less the same thing, perhaps based on different forms of expert scientific or professional knowledge or simply on their experience as users of space, and how such situations can initiate more or less formal processes of collective designation.

In this sense, designating an "element" or a "thing" or a "figure" in the territory amounts to pointing at something and thinking, "there is something there, though I don't yet know what it could be called". We are interested in how we might position ourselves as designers with respect to this moment, be it ourselves or others who are doing the pointing. We are most interested in those situations in which a variety of individuals simultaneously designate more or less the same thing, perhaps based on different forms of expert scientific or professional knowledge or simply on their experience as a users of space, and how such situations can initiate more or less formal processes of collective designation.

Constructing an Approach

In the construction our approach, we have insisted on the importance of the framework provided by architectural theory while introducing a number of different and partly new premises that associate the notion of the spatial figure with that of. In doing so another of our intentions has been to construct more explicit ties between research done on architectural and urban atmospheres in our research called the CRESSON since it was founded in 1976 following the first publications of Jean-François Augoyard. Looking at space from the perspective of pedestrian practices (Augoyard, Thibaud, Thomas), or understanding urban space according to different sensory components, focusing on such themes as sound phenomena in the urban setting, nighttime lighting, sensory accessibility and others, a field of research was gradually built by the CRESSON on the theme of architectural and urban ambiances. The group approaches space as a combination of physical, social and sensory components. In addition, such approaches to space have led inevitably to new forms of representation. How does one qualify a sensory route through a given space? How does one illustrate the reach of sound, thermal or olfactory phenomena? How, in general, does one express a practical and involved rapport with a space? These approaches, by nature complex in that they most often position themselves in the dialectical relationship between body and space, considering them not as separate but rather as permanently in relation to each other, require modes of representation which go beyond simple cartography and text. Intermediary forms of representation had, therefore, to be invented, combining text and image, frontal and overhead views, analytical elements and projections etc. (see Grosjean and Thibaud, 2001).

Informed by these past studies, our position in this new one might be described according to the following principles:

1) That of an engaged, pragmatic (in the sense of "practice-oriented") perspective with respect to the space of the territory, which gives primacy to empirical knowledge gained through practice and experiment, sensory experience and apperception.

Clearly, this approach bears some relation to Lynch's explorations in the early sixties. At the same time, opening this perspective up to include both the knowledge of users and a diversity of forms of expert knowledge can bring a new, properly environmental aspect to it. 2) An emphasis on spatial narratives that reveal both how space is apprehended and how it comes to make sense for those who live in it. Approaches that explore spatial narratives and theatricalities traverse the whole of the last fifty years, from the Townscape of Cullen to interesting if obscure recent approaches to cinematic experience and to "event cities". 3) An openness to sensory experiences of all kinds, thus not only to visuality. Following Gyorgy Kepes' lead, Lynch develops the idea of a "language of vision" somewhat distinct from discourse. His approach reflects the Kepes' criticism of the "hegemony" of abstractions and concepts like that of the function in favour of a more concrete rapport to city space. This interest in what could be called the figural dimension remains an interesting response to the question of form-defining in a context marked by rapid and profound change; on the other hand, an environmental approach might benefit from an openness to figuration that includes all of the senses. 4) A priority given to redefining the orientations of the design-disciplines "from the outside", that is as a position within a constellation of disciplines. However brilliantly argued, the idea of the city as dependant on an array of archetypal architectural forms and the dictum "Architecture is architecture" of Aldo Rossi epitomize the idea of architecture and the city as one entirely self-referential and closed "system". The gestalt themes of O.M. Ungers or the manipulations of figure-ground relations of Colin Rowe and Frank Koetter also define city-form almost entirely in visual / architectural terms. We would argue that the very terms city and also landscape have definitively lost much of their power and we are now condemned to look beyond them. 5) The assertion that the weakening of the very ideas of city and landscape require that we put an emphasis not always on transcendent, apparently permanent qualities of urban space but rather adopt an attitude that is more concerned with imminent, with what might be emerging. Approaching the "hybrid" spatial figures we are concerned with requires very specific methods of identification and description and we suspect that these methods could very well change the way we understand and practice the act of designing itself. 6) Another assertion, that approaching hybridities in the territory implies a mixing, or perhaps even a hybridization of modes of representation. Interesting as they are, strategies such as that of Xavier de Geyter which attempt to embrace the complexity of form at the territorial scale using layered maps and groups of photographs can appear very unconvincing.

Observing and Rendering Visible

Thus with this last point we return to the central question concerning modes of architectural representation. Representing something implies that one has already at least observed or "seen" it in some preliminary way. At the same time, to create a representation is to participate in its definition. There is usually some degree of concurrent development of the mode of representation and of what it is to render visible.

The ability of architects and other designers to manipulate what has be called "visual language" and spatial syntaxes clearly gives them a major role in what we would define as the collective process of spatial configuration. The importance of their role is linked to the primacy of the sense of vision itself in modern society and the fact that images, unlike sounds or smells for example, to show up on paper next to the printed word. It is true that the printed image tends to play a secondary, illustrative function with respect to the text, but as we all know image and text can combine into figurative forms that render visible thought processes in spatialized terms, and in the figurations of spaces themselves. More rarely, images can momentarily adopt the role of "carrier" of discourse in their own right, for example in the texts of excellent architectural historians or theoreticians (S. Gideon and Colin Rowe).

All of these uses are of course indispensable but there is another point to be made which is of particular importance with respect to the notion of hybridity which interests us here. The first concerns the capacity of the graphic image to "unhinge" or to "pry apart" the rapport between what we perceive and our capacity for discourse. In his book Discours, Figure,

François Lyotard uses the metaphor of "figural space" to describe how in the "moment" of apperception, our mental apparatus ranges freely over what we see and, as a function of desire or will, explores the figural possibilities of the stimuli surrounding us. Lyotard sees the "figural energy" deployed in these moments as the source of a liberty to reconsider the world in terms of future life-possibilities. He sees certain artists' explorations of pictorial space in the twentieth century as an mode of exploration of figural space which in some sense contradicts a definition of good form as the highly determined visual figure to which a precise denomination can be given and attributes to pictorial space figural possibilities with respect to sound, to time, to tactility and to the kinesthetic.

A liminal definition of the figure -one stressing psychic activity at the threshold of consciousness -is important for the idea of hybridity in that it implies the possibility of stepping out of ordering principle in order to form another in its place. In the present context, this implies the possibility of sorting-through the experience of complex and nearunintelligible urban situations. Such a possibility could suggest ways of recombining and transforming existing elements of the environment into new modes of intelligibility, notably concerning the status of public spaces, "urban" and "rural", "manmade" and "natural" etc.

Within this process, graphic representations which architects and planners produce might play a double role: 1) one of integrating existing representations and forms of knowledge about the territory into heterogeneous or hybrid graphic representations that suggest relations between different ways of knowing the territory; 2) one of combining the elements, which might allow the identification of common figures that can be developed as projects.

Latour and Callon, and also Michel Serres, have spoken of "translation" in referring to the ways in which the thoughts and actions of actors can in some sense be absorbed and reacted to by others, thus producing new forms of association. Other specialists in the field of science studies have spoken of the constant need of contemporary technicians and scientist to invent "border languages" or intermediate vocabularies in order to be able to interact with each other.

This is precisely what we are working on: how architectural and planning drawing might play such a role between different professionals, researchers and users and the idea that all this implies about what a design-approach to the territory might be.

Two Themes, Two Experiments With Methods

During the last two years, in research at the CRESSON and in the design studio, we have worked on two themes. One concerned the possibilities for future development of a variety of local conditions following the creation of a new tramway line that extends across the entire Grenoble conurbation. This tramway line traverses a number of "natural" spaces -and rivers and their banks -that divide up the territory. It crosses the borders between a number of different boroughs (communes) that have long existed in a fairly autonomous way. Most importantly perhaps, the tramway and the renovation that occurred around its stops affirm the presence of the pedestrian in places that were previously only hospitable to cars. These give a conventional impression of urbanity to spaces that were formerly crossed at a higher speed and that had a more clearly fragmented character resulting from a number of uncoordinated, juxtaposed, layers of development that mixed rurality and even something approaching wilderness with suburban and urban fragments.

The fieldwork which the research-team carried out along the tramway demanded a study and survey process that could itself be considered "hybrid". It unfolded in two stages:

The first reflected the diversity of disciplines represented by the members of the researchgroup: architecture, geography, sociology, photography, urbanism. The advantages and difficulties that this diversity implied were treated as an essential part of the work itself. Each member first studied the terrain independently, deploying his or her own methods and grid of observation. Each freely described what was seen and experienced according to his or her own way of seeing. In this sense, the approach encouraged a form of engagement with the terrain rather than a distanced, « overhanging » attitude. It is important to note that at no time was an over-riding, specifically architectural point of view expected or encouraged except or course when it was an architect who was speaking in his own name.

At the end of this first phase, each member of the research-group had produced an individual monograph most frequently combining both text and image. Each was then read by all the other authors and commented upon. After this point, a single co-written monograph -which we have qualified as « equivocal » -was constructed. A number of means were employed to draw out the different forms of relation that existed implicitly or explicitly between the various authors' points of view. The final result of the research was not only the monograph itself but an enumeration of the different, eminently practical tactics and strategies deployed to write such a co-disciplinary document.

The second theme -the one we will concentrate on here -involves the problem of what it is like to live in Grenoble's increasingly hot, humid and polluted summer climate. Working with a variety of professionals and scientists, the students were asked to portray and to combine forms of scientific and empirical knowledge within the same drawing. The assimilation of the different discourses had to be carried out in such a way as to render each form of knowledge as communicable to others as possible. Care was to be taken to consider the particular graphic conventions employed by different actors in inventing those of the actual drawing. At the same time as different forms of knowledge were being integrated, the shared, empirical realm of experiences of the hottest days of the year in Grenoble was to be brought forward.

In order to capture all of this, students were to start by collecting data that fell into two broad categories which we called "clinical", that is related to methodical forms of observation, and "metonymic", understood in the sense of "creating an opening towards narrative". These two registers were in the end to be combined as seamlessly as possible.

The type of projection to be used was also given; the students were to draw sections at 1/100 scale covering a kilometer of the conurbation, thus producing drawings ten meters long and in general a little more than a meter high. The graphic means employed could be as heterogeneous as one chose and there were no restrictions on the means to be used. The only limit to the heterogeneity of the drawing was posed by the clarity with which it could communicate.

The transects across the territory were roughly determined by the professors; they were somewhat arbitrary though care was given to choose a number of quite different climatic conditions and places (or milieux) of a more or less iconic status.

The choice of the section reflects a choice that is also a refusal. In the face of such insidiously global phenomena as climate and global warming we chose to at least temporarily avoid all use of plans and especially all recourse to panoptic, totalizing forms of imagery such as aerial views or Google-Earth inspired zooms; the transect considered to portray global phenomena in terms of what Latour has called "the local in movement".

The sections in question were to be drawn using computers. The point was not by any means to innovative in a deep sense in this domain (as many are today in domains such as that of geo-localization), but to be inventive with the tools architects have at hand. Computers mattered in this process because they allow for a relatively effortless manipulation of found imagery, new graphics and photographs. All these could be combined into a seamless whole, something entirely different from the technique of collage.

At the root of all this is an objective that relates the activity of modifying the urban or posturban milieu through design. It has to do with the fact that we expected the sections to contain areas of a particular graphic density or intensity. These would be areas in which a diversity of persons, in different capacities and possessing different forms of knowledge, all designate more or less the same thing and then, around the thing in question or rather the controversy the thing provokes, can start to interact in ways that give it a more explicit and more common definition. In fact, when the students moved on to design their projects (which they are now in the process of finishing) we asked them not to choose a site or a program in the first instance, but rather to choose the controversy they wanted their project to address and to work on the territorial figure that it corresponds to.

One last word about these sections: the decision to proceed through large sections of this kind reflected the fact that the drawings would later be presented within the framework of an exhibition and seminar during Grenoble's Biennale of Sustainable Habitat. The drawings are thus intended for a particular form of public existence. It is our hope that those who see them will want to take a marker and add to or modify what they see.

Conclusion

Based on a variety of contemporary approaches by architects and urban planners and from the social sciences (notably those of the CRESSON research group on urban atmospheres), we have been attempting to associate environmental issues with questions of perception of spatial form. In order to do so we were led to criticize and to look for ways to extend approaches concerning figure/ground relationships in the domain of architectural theory.

We have insisted here on the idea of identifying not much pre-existing figures but rather emergent ones, and of getting as far as possible beyond the domain of the visual to embrace other forms of sensory experience in order to enter more fully into what could be defined as an environmental approach.

The student work that has been produced suggests that it may be possible to invent hybrid forms of representations that allow us to identify and engage with figures of hybridity in the territory ; at the same time, it has made us aware of many methodological flaws that remain to be addressed. We intend to pursue the experiments in the design studio and through practice in coming years while continuing to develop the theoretical perspective we have presented today.

We would like to conclude by coming back to the term "paradigm" that appears in the title of this conference. The definition Thomas Kuhn gave to the term several decades ago has been of vital importance for the sociology and history of science and techniques and seems relevant here. Kuhn defines the paradigm as a framework that orients the observation of the world and thus what we choose to understand about it. In a sense, what we have been attempting to do through the work we have just presented is develop a means of opening up design-practice to a paradigm -that of emergence -which has appeared in the so-called hard sciences and in the social sciences over the last twenty years or so. The paradigm reflects the fact that in contemporary society virtually no phenomenon that can be observed is free of human influence, thus of its potential observer; it would be hard to deny that this condition holds true for the territory in which we as designers intervene.

Figure 1 :Figure 2 :

 12 Figure 1: "Climatic section", Grenoble-Echirolles, detail 1 (F. Ivol, A. Bordean, masters students, 5 th yr, ENSAG, 2008)


Cédric Zanni

Evelyne Hubert

Marie-Paule Cani

M.-P Cani

Warp-based Helical Implicit Primitives

Keywords: Descriptors (according to ACM CCS): Computer Graphics [I, 3, 5]: Computational Geometry and Object Modeling-

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Implicit surfaces, defined as f (x, y, z) = c, where f is a smooth function and c a given iso-value, provide an easy way to design organic-looking shapes. They can be used in a constructive modeling framework where smooth shapes are progressively blended into more complex ones by simply summing their field functions. Among them, convolution surfaces define f as the integral of a kernel function K along a skeleton S of lower dimension (a set of points, curves and surfaces). This skeleton, along which we can assign a fluctuating weight, provides an intuitive control of a shape with varying thickness. Thanks to the additivity of the integral operator, different partition of the skeleton leave the convolution function unchanged. Convolution surfaces thus naturally lend themselves to skeleton subdivision techniques for refinement. However, convolution surfaces require a closed form expressions of their field functions to achieve reasonable computing time. So far, this has been found for planar skeletons such as line segments, arcs of circle, quadratic curves or triangles. For this reason, 3D skeleton curves are typically approximated by a series of planar curve segments. Depending on the number of those planar primitives, either the computational efficiency or the visual quality of the shape is impaired.

In this paper we introduce for the first time an analytical implicit primitive around the simplest proper space curve, the circular helix; it has constant curvature and constant torsion. It is a promising high level primitive for tessellating arbitrary 3D curves and can be used by itself for modeling realistically a wide range of organic shapes, from hair locks [BAC * 06] to natural branching structures such as trees or antlers, see Figures 1 and16.

Related Work

Convolution surfaces, defined as the level sets of a scalar field obtained by integrating a kernel function along a skeleton, were introduced to provide bulgefree implicit modeling over complex skeletal elements such as graphs of branching curves [START_REF] Bloomenthal | Convolution surfaces[END_REF]. The numerical evaluation used there for the parametric integrals involved left space for improvement. By introducing a different kernel function, so called Cauchy kernel, Mc Cormack and Sherstyuk [START_REF] Mccormack | Creating and rendering convolution surfaces[END_REF] provided the first closed-form formulae for primitives based on line-segments, arcs of circle and triangles. The new profile of computational efficiency [START_REF] Sherstyuk | Kernel functions in convolution surfaces: A comparative analysis[END_REF] called for further investigation on the technique. Angelidis and Hornus [START_REF] Angelidis | Adaptive implicit modeling using subdivision curves and surfaces as skeletons[END_REF][START_REF] Hornus | Implicit modelling using subdivision curves[END_REF] introduced convolution surfaces of varying radius, using linear weight function along line-segments for power-inverse kernels of degree 2 and 3, together with a subdivision strategy. In addition to providing similar formulae for the Cauchy kernel, Jin [START_REF] Tai | Convolution surfaces for arcs and quadratic curves with a varying kernel[END_REF][START_REF] Tai | Analytical methods for polynomial weighted convolution surfaces with various kernels[END_REF] extended the method to polynomial weight up to degree 3 over all quadratic skeleton curves. Recently, a recurrence formula was introduced in [START_REF] Hubert | Convolution surfaces based on polygonal curve skeletons[END_REF] to obtain closed-form primitives along linesegments for power inverse and generalized Cauchy kernels of arbitrary order, with polynomial weight of any degree.

An alternative solution for getting complex analytical implicit primitives, first used in [START_REF] Wyvill | Warping as a modelling tool for csg/implicit models[END_REF], is to warp simpler implicit primitives through a transformation such as Barr's bend, twist or taper [START_REF] Barr | Global and local deformations of solid primitives[END_REF]. As we shall see, these transformations do not preserve the circular quality of the normal cross-section of the primitive (see Figure 10, first row), and thus cannot be applied as is for our purpose.

Overview

Our work builds on the two approaches presented above: we introduce specific space warps to get artefactfree analytical implicit primitives along circular helices. The latter are based on closed-form convolution with varying radius over line segments or arcs of circles.

Our first contribution, described in Section 2, is the introduction of convolution primitives for arcs of circles with polynomial weights and families of infinite support kernel. A new piecewise defined warp then provides analytical expressions for helical implicit primitives.

Our second contribution, presented in Section 3, is the introduction of artefact-free space warps to generate helical primitives from line-segment ones.

We show in Section 4 that these two alternatives are complementary and provide methods for the whole range of helical skeletons. We additionally show that, for a lower computational cost, the shapes based on our helical primitives are visually more appealing than those obtained with a standard tessellation into line segments.

Notations: Without loss of generality we consider that the helical skeleton to be processed is given by:

         x = R cos(u) y = ǫ R sin(u) z = S u (1)
where ǫ = ±1 and u ∈ I ⊂ R. I is the interval defining the part of the helix used as skeleton. The length of the latter is thus √ R 2 + S 2 |I|. The twis is defined as the ratio S R ; we shall see that it dictates the warp to be used. When this ratio tends to 0 (respectively to ∞) the helix tends to a circle (respectively to a line segment). Figure 2 gives some examples of the shape of the helix according to the twist. The thickness of the primitive is assumed to be smaller than the radius of curvature of the helix, as otherwise the warps run into singularities. The Cartesian coordinates of a point P in 3D space are noted (x, y, z). Its cylindrical coordinates are (r, θ, z) where θ is the planar angle with the x-axis (see Fig. 3(a)).

S R = 0.0 S R = 0.2 S R = 0.4 S R = 1.0 S R = ∞
! " # $ % & (a) (b)
Figure 3: (a) The z-axis helix studied with the virtual segment (green) and virtual circle (blue) we use for warping. (b) Principle of a warping for implicit primitive as used in [START_REF] Wyvill | Warping as a modelling tool for csg/implicit models[END_REF].

Our further derivations are applicable to common superset of Cauchy and power inverse kernels of arbitrary orders. Those laters are defined as:

K(r) = 1 (1 + s r 2 ) i/2 and K(r) = 1 r i .
For a skeleton given by a parametrized curve Γ : [a, b] → R 3 with a polynomial weight function g the associated convolution function is given by

f (P) = b a g(t) K(d(Γ(t), P)) |Γ ′ (t)|dt.
The warps we make use of are space transformation W : R 3 → R 3 that maps the circular helix to a simpler skeleton curve (see Fig. 3(b)). The helical primitive function f we look for is then defined from the simpler skeleton primitive f by:

f (P) = f (W(P))

From circle to helical primitives

In this section we first introduce closed form convolution primitives for arcs of circle with a polynomial weight, a case never solved so far. Our approach applies uniformly to both Cauchy and power inverse kernels of even order; we include as an example the closed form expression for a linear weight in the case of the classical (i.e. order 4) Cauchy kernel. We then present a warp that maps the helix onto a circle. This provides a first method for computing an analytical helical primitive.

Closed-form primitive for arcs of circles

When it comes to integration, rational functions are the dependable class [START_REF] Bronstein | Symbolic integration. I[END_REF]. The main ingredient in obtaining closed-form convolution primitives for arcs of circle is to introduce an appropriate rational parameterization.

We assume that the points O, A and B (see Fig. 4) are not aligned and such that |OA| = |OB| = r. They define a plane in space and two arcs of the same circle, one of angle α the other of angle π + α, where 0 < α < π

is given by α = arccos --→ OA• --→ OB r 2
. According to which of the two arcs we deal with T = tan α 4 or T = tan π+α 4 . Momentarily we consider the coordinate system (x, y, z) where the origin is the center of the circle, the x-axis is the bisector of the chosen angle defined by O, A and B and the (x, y) plane is the plane of the circle. A parametrization of the arc of circle is then given by

Γ : [-T, T ] -→ R 3 t → r 2t t 2 + 1 , r t 2 -1 t 2 + 1 , 0 .
When t → ±∞, Γ(t) tends to the point diametrically opposite to the mid-point of the arc. The other points of the circle are obtained for a parameter t ∈ R. In terms of this parameter t the infinitesimal arc length is Consider a point P(x, y, z) in space. We have

|Γ ′ (t)| = 2 r t 2 +1 . α 2 y = -T ( x + r ) A B O 2rt 1+t 2 ,r t 2 -1 t 2 +1 y x y = T ( x -r ) y = t( x -r) α 2
|PΓ(t)| 2 = αt 2 -2 β t + γ t 2 + 1 where α = x 2 +(y-r) 2 +z 2 , β = 2 r x, γ = x 2 +(y+r) 2 +z 2 . Note that γ + α = 2 (|OP| 2 + r 2 ), (2)
α T 2 + 2 β T + γ = (T 2 + 1) |AP| 2 , (3)
γ T 2 -2 β T + γ = (T 2 + 1) |BP| 2 , (4)
so that (α, β, γ) is actually the solution of a linear system that depends on T and the squares of the distances of P to O, A and B. There is a unique solution provided that A, O and B are not aligned, i.e. T (T 2 -1) 0. An explicit expression for (α, β, γ) is obtained by Cramer's rules.

Hence the convolution primitive for the arc of circle with weight g : [a, b] → R is given by:

C OAB (P) = 2 r T -T g(t) (t 2 + 1) i 2 -1 (a t 2 -2 b t + c) i 2 dt where a = 1 + s α, b = s β, c = 1 + s γ
for Cauchy kernels and a = α, b = β, c = γ for power inverse kernels (of order i). Closed form expressions for the following integrals are given in [START_REF] Hubert | Convolution surfaces based on polygonal curve skeletons[END_REF].

I k,i = t k (a t 2 -2 b t + c) i 2 dt
as they appear in line-segment primitives. When using even order Cauchy or power inverse kernels those expressions also provide closed form primitives for arcs of circle with polynomial weight function g. It is just a matter of expanding the numerator. For odd order kernels, the closed form primitives call on elliptic functions. Consider the particular case of an arc of circle with a linear weight defined by the two endpoints weights g A and g B , that is: The explicit expressions for I 4,0 , . . . , I 4,3 are given in [START_REF] Hubert | Convolution surfaces based on polygonal curve skeletons[END_REF]. The left hand sides of Equations 2-4 arise as subexpressions and it is worth using those latter for a faster evaluation. Figure 5 illustrates the level sets of this primitive in the case where g A = g B and in the case where g A g B .

g(t) = u t+v where u = g A -g B 2 T , v = g B +g A

Warp based extension to helical primitives

A first way to generate an analytical, convolution-like field along a helical skeleton is to use a warping method, based on a piece-wise map of the helix onto arcs of a circle. More precisely, we use a translation along the z-axis, where the amount of translation depends on the θ coordinate of the query point P:

W(r, θ, z) =          r θ z -τ(θ)          with τ(θ) = ǫ S θ
For τ to be continuous, we use more than two arcs of circle for mapping a turn of the helix (using a different warp per arc of circle): this way, the warp is injective and O, A, B are not aligned (see Section 2.1). More precisely, we define a warp of an arc of helix using τ(θ) = ǫ.S θ on [-3π 4 , 3π 4 ] which includes all the points of the helix arc. We use a cubic spline junction to have a continuous warp defined everywhere in space (see Fig. 6). Results are depicted on Figure 8.

Handling of thick helical primitives

The warp introduced in previous section is not well defined along the helix axis and bears no continuous extension. As we go across the axis, from θ to θ + π, τ takes a non zero step unless θ = 0, ±π. As a result the scalar field for the primitive is singular along the axis. Plotting its level sets in the vicinity of the axis is thus The improved warp allows us to generate implicit primitives for highly twisted helices. This is illustrated in Figure 8. For the top row we used a constant weight function on the circle. The helical primitive has then a constant radius as is expected. For the bottom row we used a linear weight function. That reflects in a varying radius along the helix that we can wish for.

The deformation introduced by the above circle based warp on the normal cross-section to the helix becomes noticeable as S R increases, i.e. when the helix becomes closer to a line-segment. Typically, when S R is greater than 0.6 the ratio between the incircle and excircle of the section becomes smaller than 0.8. We thus offer next a helical primitive using a warp onto a line-segment. Section 4 shows how to combine them to adequately span the full range of circular helices.

Helical primitives by warping to a line segment

In this section we shall introduce warps of a helix to a line segment providing helical primitives of increasing quality. Our starting point is the inverse of Barr's twist, which was already used in [START_REF] Wyvill | Warping as a modelling tool for csg/implicit models[END_REF] to create implicit primitives. We exhibit the imperfections of this warp for our purpose and offer new transformations to compose it with to obtain the desired results. Those new warps form an essential part of our contribution in this article.

As a convention for the rest of this section we use the line segment parallel to the axis of the helix and shown in green in Figure 3(a). In contrast to the method of previous section, any number of turns in the helix can be handled in a single warp.

Naive twist-based warps

Twisting back a helix of parameter S to a linesegment can be done by setting v to -ǫ S in the equation of a twist:

twist(v, (x, y, z)) =          x cos(v.z) -y sin(v.z) x sin(v.z) + y cos(v.z) z         
We therefore define our basic warp as:

W twist (x, y, z) = twist(- ǫ S , (x, y, z))
As shown in the top row of Figure 10, the normal sections of the resulting primitives are not circular. To get some insight we consider a helix with R = 1, ǫ = 1 and examine the linear part, W simple , of the map W twist around (1, 0, 0):

W simple (x, y, z) =          x y -z S z          .
The preimage of the vector (0, 0, 1) is the tangent vector to the helix at the point (1, 0, 0), that is (0, 1/S , 1). Figure 9(a) depicts a circular cylinder, the axis of which is given by this tangent vector, and its image under W simple . This latter is a cylinder with an elliptic section. Indeed the intersection of the left hand side cylinder with a plane z = c is an ellipse E and the restriction of W simple to that plane is a translation within this plane. Therefore the section of the image cylinder is the same ellipse E. As explained on Figure 9(b) the ratio of the major and minor radius of this ellipse is:

r min r max = sin(α) = S √ R 2 + S 2
(5) Scaling the warp along the y-axis will bring the horizontal section closer to a circle. The resulting transformation is:

! ! ! ! !"#$%&' (!)%*$+, - . / !"#$ !"%& ' ' (!) (!"*+!,-),. (a) (b)
W(x, y, z) =            x cos( ǫ S z) -y sin( ǫ S z) S √ R 2 +S 2 (x sin( ǫ S z) + y cos( ǫ S z)) z           
However, this transformation still does not work for highly twisted helices: our additional scaling component neglected the effect of the rotation (if we take again the analogy with an oblique cylinder, the latter should rotate along the helix). It is more appropriate to apply the scaling correction to the angle in the cylindrical co-

ordinates: W(r, θ, z) = r, S √ R 2 +S 2 (θ -ǫ S z), z
. This is equivalent to applying a rotation around the helix axis after the twisting. The angle of rotation depends on the polar coordinate after twisting. However the rotation to be considered for a helix is a rotation around its center of curvature.

Correctly handling the central part

The warps we described so far gave us some insight on the corrections required to get a reasonably circular cross-section: a scaling should be applied after a rotation around the local center of curvature of the helix. The center of curvature at a parameter t along the helix is given by:

(- S 2 R cos(t), - S 2 R sin(t), S t).
When the helix tends to a line segment the centers of curvature wander off to infinity whereas, when the helix tends to a circle, the centers of curvature are attracted toward the axis of the helix. The correction we define thus varies smoothly from the value of W to that of W according to the amount of twist of the helix.

Our new transformation can be written as:

W(x, y, z) = g(twist(- ǫ S , (x, y, z)))
where

g(x, y, z) =           
x cos(µ ψ(x, y, z))y sin(µ ψ(x, y, z))

x sin(µ ψ(x, y, z)) + y cos(µ ψ(x, y, z))

√ R 2 +S 2 S z            , (6) µ
= S √ R 2 + S 2 -1, and
ψ(x, y, z) = atan2(y, x + S 2 R ).
The restriction of g to a plane orthogonal at some point to the helix axis is a rotation around the center of curvature of the helix at this point. This is equivalent to multiplying the polar coordinates, with respect to the center of curvature, of the point by the ratio S √ R 2 +S 2 (Equation 5). Another important feature is to ensure that the helical primitive has the same thickness as the associated line segment primitive. The length of the skeleton needs to be preserved to ensure this. As the last coordinate of g in Equation 6discloses it, this is obtained by scaling along the z-axis.

The primitives obtained with this new warp are illustrated in the second row of Figure 10. To obtain a helical primitive with varying radius, we can simply assign a polynomial weight function on the line-segment skeleton the helix is warped onto and use the results of [START_REF] Hubert | Convolution surfaces based on polygonal curve skeletons[END_REF]. Results are shown in Figure 11.

Final solution for segment-based primitives

Figure 10 provides a comparison between the results obtained with the original twist and those obtained with the presented new warp. A closer look at the leftmost image in the second row reveals an unwanted artefact toward the end points of the shape. This artefact furthermore prevents the seamless blends that are sought when the helix is part of a more complex skeleton.

!"#$%&'%"#('%) *+,,'-$%./%$"'%0#./%1#,$%+2%$"'%-34./5', !"#$%&'%&#/$ *.,-64#,%7'-$.+/ 8'2+,0'5%7'-$.+/ The artefacts observed around the tips of the helical primitive can be explained through the linearized warp W simple already studied in Section 3.1. Figure 12 illustrates the cause of the problem at the end points and how to measure the unwanted deformation in order to introduce an appropriate correction. A translation along the z-axis, the norm of which depends linearly on y in the deformed space, is enough to correct the tips in this linearized version. As before, this correction is adapted to our main transformation through a translation that depends on the polar coordinate with respect to the center of curvature. This is given explicitly by:

W correct (x, y, z) = τ(W(x, y, z)) where τ(x, y, z) =          x y z -ǫ R 2 +S 2 S ψ(x, y, z)         
The map τ introduces a discontinuity of the field function when ψ = ±π. The latter can be easily overcome through a linear or spline junction as was done for warps onto arcs of circles (Figure 6).

The improvement introduced by this last adjustment can be easily appreciated by comparing the second and third row in Figure 10.

Evaluation and combination of our solutions

We presented two ways of obtaining a helical primitive by warping the helix either on arcs of a circle or onto a line segment. We lead an experimental evaluation of the results obtained through those two warps. On one hand, the warp to circle solution presented in Section 2 leads to good results when S R ∈ [0, ≈ 0.5] (see Figure 2 to have an idea of the shape of the helix). Beyond that the shape of the cross-section is noticeably deformed. We may think of improving the situation by introducing corrections as was done for line segments in Section 3. On the other hand, the solution based on warp to line segment developed in Section 3 gives good results for all kinds of helices, except when the helix becomes highly twisted: for a helix close to a circle, our solution still locally flattens the primitive around the end points of the helix. In our experiments, this is negligible until S R becomes lower than ≈ 0.35 (instead of ≈ 0.7 without the final correction for the tips). Thus, the whole range of helices can be handled nicely by at least one of the two methods we proposed, with good results.

Combination of the two methods

Since the quality of the result depends on the same ratio S R , automatically choosing the method to apply is very easy.

A first solution is to use a segment based primitive, which is computationally cheaper, when S R > 0.4, a circle based primitive when S R < 0.3 and an intermediate solution in-between. This intermediate solution is constructed by interpolation, for instance a cubic interpolation. In this first approach, the evaluation of the helical primitive when 0.3 < S R < 0.4 requires both the evaluation of the circle based primitive and the line segment based primitive. A more efficient way to handle the transition zone of twisting, i.e. when 0.3 < S R < 0.4, is to work with a warp of a line segment for the middle part of the helix and only two warps onto arcs of circles for the tips of the shape. Then interpolation between the two different primitives occurs only within a limited range of the z coordinate (see figure 13). Table 15 compares the timings for the evaluations of our two analytical helical primitives and of a linesegment primitive. This shows that when 3 segments or more are needed to approximate a helical skeleton, our new helical primitive is much more efficient. The fact that the warp-to-circle solution used for highly twisted helices requires two primitives per turn does not change this result, since more than 6 segments per turn would be needed then. To make this comparison more visual, Figure 16 depicts the quality we would get, at constant computational cost, by tessellating a skeleton into linesegments or into helices. For this level of zoom, getting the same visual quality with a tessellation into line segments would respectively multiply computational time by factors 2.71 for the octopus, which includes non helical parts, and 5.84 for the antlers, which is only made of helices.

Conclusion and Future work

We have introduced the first analytical formulation for helical implicit primitives. The method handles primitives of varying radius and can be used with arbitrary convolution kernels such as power inverse and Cauchy kernels of arbitrary order. Indeed, a side, yet important, contribution is a new general formulation for arc of circle convolution primitives, which allows for varying radius.

Our solution for helices can be considered as a pseudo-convolution since it built on arcs of circle and line-segment convolution primitives by warping. Similarly to convolution, the resulting helical primitives can be used in complex configurations where the skeleton is tessellated into several helices: as with direct closedform convolution, the resulting curve is bulge-free.

We furthermore have set up the bases of a methodology to design advanced space warps that progressively build complex primitives from simpler ones. That opens the door to further extensions: the same methodology could be applied to get implicit primitives along 3D spiral skeletons, which were shown to be very useful to represent a larger variety of natural shapes such as seashells and horns [START_REF] Harary | The natural 3D spiral[END_REF]. Modifying our deformation to the sweeping of more general cross-sections around the helix skeleton would also be a useful exten-sion of this work, made possible by our section preserving warp.

Figure 1 :

 1 Figure 1: Implicit modeling of a squid. Each tentacle is made of two helical primitives.

Figure 2 :

 2 Figure 2: Shape of the helix for increasing value of S R .

Figure 4 :

 4 Figure 4: Parameterization of the arc of circle

Figure 5 :

 5 Figure 5: The level sets of the convolution primitive for an arc of circle for constant (right) and linear (left) weight.

  2 so C AOB (P). For the classical (order 4) Cauchy kernel the primitive is given by 2r u(I 4,3 (a, b, c) + I 4,1 (a, b, c) + v(I 4,2 (a, b, c) + I 4,0 (a, b, c)) T -T .

Figure 6 :

 6 Figure 6: Amount of translation along the z-axis according to the angle θ in cylindrical coordinates

Figure 7 :

 7 Figure 7: Right: for a thicker volume around the helix, a pinched blend occurs in the vicinity of the axis if we use the warp of Section 2.2. Left: the lift of Section 2.3 erases this unwanted singularity.

Figure 8 :

 8 Figure 8: Helical primitives based on warps to the circle. Top: constant weight. Bottom: varying weight.

Figure 9 :

 9 Figure 9: (a) Transformation W simple applied to an inclined cylinder, (b) Lateral view of that cylinder

Figure 10 :

 10 Figure 10: Comparison, for different values of S R , between the classic twist (first row), the solution with the end point problem (second row) and our final solution (third row).

Figure 11 :

 11 Figure 11: Helical primitives with varying radius.

Figure 12 :

 12 Figure 12: Idea justifying the correction of both ends of the primitive.

Figure 13 :

 13 Figure 13: Transition between the two kind of primitives in function of S R Figure 14 emphasizes the quality of our global helical primitive in the context of a more complex skeleton. The helix is prolonged by a line segment with continuous tangency. The resulting shape remains smooth around the transition between the helical primitive and the line segment primitive.

Figure 14 :

 14 Figure 14: Helical primitive prolonged by a segment primitive for different values of S R Computational efficiency

Figure 16 :

 16 Figure 16: Comparison of model quality using a tessellation of 3D skeleton curves into line-segments (left) compared to our helical primitives (right) when a given computational time is allowed.

Table 15 :

 15 Computation time for a single primitive. CPU: 2.4 GHz Intel Core 2 Duo (only one core used).