IEEE TRANSACTIONS ON VISUALIZATION & COMPUTER GRAPHICS 1077-2626/97/$10.00 © 1997 IEEE
Vol. 3, No. 1: JANUARY-MARCH 1997, pp. 39-50
Animation of Deformable Models Using Implicit Surfaces
*![[PDF ICON]](admis_files/pdficon.gif)
Marie-Paule Cani-Gascuel
Mathieu Desbrun
In traditional animation systems based on key-framing, specifying the
motion and the successive shapes of objects interacting with a simulated world
requires a great amount of specialized knowledge and intuition from the
animator. Models based on simplified physical laws have been proposed for
automating these tasks. They generate motion and deformation from initial
conditions and from a set of externally applied forces over time, and
automatically detect and respond to collisions. These models are particularly
appropriate for facilitating the animation of deformable objects. They can
either be used alone for the simulation of inanimate bodies, or be combined
with user-controlled structures as has been done for instance in character
animation [25], [7].
1.1 Related Work
Deformable models give a method for computing the alteration of an object's
shape due to a set of externally applied forces. Deformations may be
elastic or inelastic depending on whether the original shape
is restored when external forces are removed. Most deformable models in
Computer Graphics result from "nodal approaches," in the sense that they
approximate deformations by the displacements of elementary nodes inside a
flexible body. Some of them derive from the elasticity theory. Differential
equations of motion are discretized in space, and then integrated over time by
resolving a matrix equation at each time step. This scheme has been used for
modeling both elastic [37], [39], [19] and inelastic [36] deformations. However, since the
topology of the network of nodes does not vary over time, this approach is
restricted to the animation of structured objects. A solution for modeling
soft inelastic bodies that absorb deformations and may separate into pieces or
melt during an animation is a physically based particle system [26], [38], [40], [24]. In this case, a set of elementary
masses called "particles" interact by means of forces that vary with the
distance, such as Lennard-Jones forces that combine short range repulsion with
long range attraction. Motion is computed by independently solving the
equations of motion for each particle.
1.2 Overview
This paper presents an integrated set of methods that use implicit surfaces
for animating a wide variety of deformable models. Implicit surfaces will be
used as an extra layer that coats a base structure, deformable or not, with
some smooth elastic flesh. While the base structure controls the large scale
behavior, the implicit layer performs collision processing and generates local
deformations due to contacts. It also handles constant volume deformations and
topological changes such as separation or fusion. This allows simulation of
behaviors that would be extremely difficult to treat with other methods.
2.1 Implicit Surfaces Generated by
Skeletons
Implicit isosurfaces such as "distance surfaces" [47], [5] allow the design of free form
shapes through the manipulation of "skeletons" generating potential fields.
Because they are simple to define and to control, they constitute a good
alternative to traditional implicit surfaces defined by analytical equations.
(1) (2) where d(., si) is the distance to
si, and Fi can be
defined for instance by pieces of polynomials [47] or by more sophisticated
anisotropic functions [21], [3]. Most implicit contributions
associate a restricted scope of influence to each skeleton in order to provide
local control of the surface and to optimize the computations. In practice, we
use piecewise polynomial field contributions that are parametrized by three
parameters ti, ki,
and Ri, called, respectively, thickness,
stiffness, and radius of influence, as sketched in Fig. 1. Fig. 1. A typical field contribution, with its three
parameters ti, ki,
and Ri. 2.2 Embedding Implicit Surfaces into a Layered
Construction
As emphasized in [7], a layered construction is a very
efficient tool for creating complex models for animation. It often provides
the user with parameters that are more intuitive and easier to use. Implicit
surfaces generated by skeletons are especially well suited to this kind of
approach.
2.3 Benefits of the Approach
As stressed before, the coherence between the representation of an object
and the model used for collision processing is very important for generating
convincing motion. Using an implicit representation of the surface brings
several benefits, such as a precise yet efficient collision detection
mechanism, and a solution to precise contact modeling between deformable
bodies.
2.4 Animation Algorithm
The general scheme for animating the resulting layered model develops as
follows:
In previous deformable models, when a collision is detected, response
forces are approximated first. These forces are then used to compute
subsequent deformation and motion of the objects, without producing an exact
contact surface when collision endures in time. The implicit layer defined in
this section uses a different approach. When a collision is detected, it first
performs precise contact modeling with the other object with local
deformations. Compression along contact surfaces is then used to compute
collision response.
3.1 Collision Detection
Collision detection is performed between each pair of objects by first
testing interpenetration between axis-aligned bounding boxes. When boxes
intersect, the implicit representations of the surfaces and a set of sample
points on them are used for a more precise detection: The surface points of
each object that lie within the bounding box of the other object are evaluated
by the implicit function of the other object. When an interpenetration is
detected, the purely geometric contact modeling process described in the
following section is applied. Issues for maintaining samples on the implicit
surface will be discussed in Section 3.4.
3.2 Modeling Contact Between Objects
Once detected, an interpenetration must be avoided by locally deforming the
implicit layer of each object. Both exact contact surfaces and deformations in
propagation regions that model the transverse propagation of
deformation must be generated (see Fig. 2). We accomplish this by adding
new local terms, called "deformation terms," to the field functions defining
the implicit layers of each object. Fig. 2. Modeling contact consists in applying different
deformation fields in the interpenetration region and in the propagation
regions associated with each object (cross-sectional views). 3.2.1 Deformation in the Interpenetration
Region
The deformation field terms (3) (4) must have a common solution. A simple and symmetric solution is to define
the compression field terms by: (5) (6) (7) (8) This makes the implicit layer of the deformable body exactly fit with the
rigid object in the whole interpenetration region.
3.2.2 Deformation in the Propagation Areas
Our aim is to optimize the contact modeling process by directly computing
deformed shapes in contact situations rather than simulating deformations that
progressively propagate over the implicit layer. The bulge generated in the
propagation region (see Fig. 2) must be computed such that
there is a smooth junction between the interpenetration region and the region
where the object remains undeformed.
We then define the propagation field term pji
to be applied in the propagation zone of the object
si as: (9) where P0 is the closest point to
sj in (10) where p1 = 4 (wk - 4
a0)/w3 and p2 = 4
(3a0 - wk)/w2. Fig. 3. Attenuation function defining the field in
propagation areas. The slope at x = 0 is k, the maximum is
a0, and the function has zero values and derivatives for
x 3.3 Response to Collision
Now that correctly deformed shapes are generated beween colliding objects
as depicted in Fig. 4, the deformation of the implicit
layer must be used to deduce response forces. To achieve this, we describe in
this section a model designed to approximate elastic behavior, first
introduced in [15]. Fig. 4. (left) Contact between two colliding deformable
objects. (right) Cross-sectional view showing the exact contact modeling. A
step-by-step description of this animation is described in the
Appendix. 3.3.1 Modeling Elasticity with Implicit
Surfaces
A deformable model is defined by a given correspondence between applied
forces and deformations. Both linear [39], [19], [45] and nonlinear [37] elasticity have been used in
previous models. Linear elasticity states that stiffness near a given point
P of a solid remains constant during deformations. The displacement
of P from an initial position X0 =
(x0, y0, z0) to a
final position X(P) = (x(P),
y(P), z(P)) is then a linear function of
the applied force R(P): (11) In nonlinear models, the stiffness k is not only a function of the
point P, but may also depend on its current location inside the
solid. The force applied during a displacement of P from
X0 = (x0, y0,
z0) to X(P) = (x(P),
y(P), z(P)) is then: (12) For generality, the implicit layer we are defining should be able to
exhibit both linear and nonlinear behaviors.
(13) (14) (15) Let (16) In practice, we will use a rewritten version of this latter equation: (17) It can be noted that this formulation only approximates elastic behavior
since only the radial component of forces is computed: The correspondence
between applied forces and deformations will only be used to integrate radial
response forces during collisions.
3.3.2 Stiffness Control
As shown in [15], defining stiffness with the
field's gradient has a simple geometric interpretation since field
contributions are decreasing functions of the distance to the associated
skeleton: the user defines local stiffness as the opposite of the slope of the
field function. Both linear and nonlinear elastic behaviors are easily
designed, as depicted in Fig. 5. Fig. 5. Examples of field functions for the implicit
layer. (a) Linear elasticity: stiffness, represented by the opposite of the
slope, is constant during deformations. (b) Nonlinear elasticity: stiffness
increases during compressions. (18) This parameter K enables us to adjust the object stiffness
according to its mass and inertia tensor without modifying its geometry.
3.3.3 Radial Response Forces
We have defined a way to model contact and a correspondence between
deformations of the implicit layer and radial applied forces. Thus evaluating
the resulting normal reaction force R(P) at a point
P on the contact surface during collision is straightforward.
Equations (6) and (17) give us a value for the normal response force: (19) where Ni(P) is the normal vector to
the deformed surface of object i at point P. Since 3.3.4 Friction and Damping Forces
To model both tangential friction in contact areas and damping due to the
progressive compression of the solids, we include a friction coefficient (20) where Vi(P) (respectively,
Vj(P)) is the speed of P, a
point on the surface of the object i (respectively, j).
Fig. 6. Flexible clover falling on a rigid
staircase. 3.4 Implementation
3.4.1 Sampling the Isosurface
Sample points on the implicit surface are needed for both collision
detection and numerical integration of response forces. Points that appear to
be inside another object are moved to the deformed isosurface (for instance,
by performing search along the gradient direction), and the value of the
deformation field at each new location directly gives the intensity of radial
contact force along the small surface area sampled. These forces are added to
friction forces and stored to be integrated by the base structure at the next
time step.
3.4.2 An Adaptive Sampling Technique
The central idea is the following: Each skeleton
si contributing to the implicit layer emits a set
of sample points in directions that are fixed in its local coordinate system,
and are well distributed around it as illustrated in the stages 1 to 3 of Fig. 7. Those of the points that reach
the isosurface without going through an area already sampled by another
skeleton are said to be "valid" and will be used as samples on the isosurface
at a current time step (see stage 4 in Fig. 7). Fig. 7. Different stages of sampling
initialization. (21) 3.4.3 Interactive Visualization
A further benefit of the sampling method above is that the implicit surface
can be displayed, during an interactive animation process, as a set of
polygonal meshes built on the sample points (valid or not) that belong to the
same skeleton. This provides a legible solid representation of the surface
with no extra cost [10] as depicted in Fig. 8. Note that we do not use the
sample points for final high quality rendering of an animation. Storing the
parameters of the isosurface such as the positions of the skeletons and the
lists of colliding objects offers a much more compact representation that can
then be used for computing direct ray-tracing of the implicit surfaces [13]. Fig. 8. Left: visualization of sample points as
scales on the surface. Right: visualization of the same sampling but
with a piecewise polygonization.
The preservation of constant volume of deformable objects is desirable in
animation [22]. Additionally, the user may desire
precise control over volume variation in order to emphasize certain motion.
The problem of volume preservation has been solved by methods based on
Lagrange multipliers in the specific case of objects discretized into lattices
of fixed topology [33], [34]. We present in this section the
only solution to our knowledge that has been proposed for controlling the
volume of bodies that undergo large deformations and topological changes such
as separation and fusion [9].
4.1 Local Volume Variations
The first problem is the detection of volume variation. The volume of an
implicitly-defined object is given by: This expression cannot be computed analytically for most field functions. A
simple method for volume approximation consists of discretizing space into
voxels, and expressing the volume as the sum of voxels that lie inside the
object. However, this technique would not provide a solution to the problem,
since we also need to know near which skeleton the volume is
changing.
Fig. 9. (a) Volume variations of an implicit surface
generated by point skeletons between two steps of animation. (b) Volume
controlled locally in step 2. (22) where is sufficient for avoiding volume variations. Fig. 10. Particle territories and sample points used for
volume approximation. 4.2 Control of Volume
We control local volume variations by associating a proportional-derivative
controller to each skeleton. This controller can be seen as a black box that,
given the current local volume Vi,t and the value
Vi,0 to reach or to maintain, outputs an adequate
adjustment of the field function fi of this
particular skeleton.
(23) Since we need regular shape variations, we control the time derivative (24) Its output is: (25) where Fig. 11. Preserving volume during a blend between two
point-skeletons. The leftmost picture shows the initial configuration and the
speed vector: (a) shows the blending without any control, whereas (b) depicts
a controlled blending ensuring constant volume.
A very simple use of the implicit layer consists of combining it with a
rigid internal structure, as was done in [15]. This leads to animations of
elastic objects that locally deform during contacts, but return to their
initial shape when no external force is applied. Snapshots from such an
animation [12] are displayed in Fig. 12. Fig. 12. Four frames of the animation Simply
Implicit. 5.1 Modeling Simplified Characters
Specifying complex motion is greatly simplified when the animation system
is able to abstract basic shapes from the representation of an object. Motion
can then be refined with these shapes, and the animator only switches to the
detailed representation when necessary. This is particularly true in character
animation, where animators often spend a lot of time specifying motion and
deformations of an articulated "skeleton" representing the character. Much
less time is spent on animating the skin deformations, which may be generated
automatically from the skeleton motion. Layered models are particularly well
adapted to this context. Various approaches, either purely geometric [6], [11] or physically based [7], [17], [31], [41] have already been proposed for the
automatic animation of the skin from the motion of the underlying skeleton.
5.1.1 Structure Used
In the terminology we have developed in Section 2, a character will be
represented by: The animation of this model is straightforward in the general framework we
have defined: The algorithm described in Section 2.4 is used. However, two
problems due to the relative motion of skeletons inside the implicit layer
have to be discussed: The two next paragraphs explain how we deal with these problems.
5.1.2 Avoiding Unwanted Blending Effects
The unwanted blending case is a difficult problem that has been known for a
long time [46]. When we implicitly model
characters for instance, we want their arms to blend with their shoulders, but
not with another part of the body, as illustrated by Fig. 13. Fig. 13. The unwanted blending problem. This algorithm avoids surface discontinuity during the controlled blending
process, as explained in [20]. However, the method can not
guarantee 5.1.3 Processing Intercollisions
Instead of processing collisions between pairs of objects, we use the
algorithm described in Section 3.2 for processing collisions between pairs of
skeleton territories that do not blend. The sampling method described in
Section 3.4 provides us with a set of sample points, so we can easily compute
local bounding boxes from the position of valid sample points and associate
them with each skeleton territory. These boxes (enlarged by the maximal
distance between sample points) cover the isosurface, allowing for precise
collision detection even when the surface separates into several components.
5.1.4 Results
An example of animation performed with this model is depicted in Fig. 14. A simulation method based on
displacement constraints [14] is used for animating the base
structure of the characters. We can observe several intercollisions as the
character falls. Fig. 14. Collisions with the ground and intercollisions
of a simple articulated implicit object falling on its side. 5.2 Animating Soft Substances
This section presents a quite different application of the techniques we
have described. The soft substance model we develop here, first introduced in
[9], is particularly interesting since
it benefits from the specific capability of implicit surfaces to model
separation and fusion. The constant volume deformations generated by our model
are very important in this case, otherwise an important increase of volume
would be produced during fusion.
5.2.1 Structure Used
As emphasized in the introduction, a simple and unified way of modeling a
large variety of behaviors, including inelasticity and fractures, is to use
physically based particle systems. The main drawback of these systems, when
used alone, is the lack of a method for defining a smooth surface for the
objects. This is not a problem when several thousand particles are used for
accurately simulating fluids for instance. For animation purposes, the use of
far fewer particles is sufficient for producing sufficient deformations. A
surface, defined around the particles, should be used for visualization and
for processing contact with other objects. The general framework we have
defined provides a solution to this problem since the implicit layer seems
well adapted for coating the particles. A piece of soft substance will then be
composed of: (26) (27) where The general animation algorithm we have developed also applies to this
case. The next paragraphs explain how we handle separation and fusion of the
substance.
5.2.2 Modeling Separation
When the particle system moves and deforms, a piece of substance may
separate into several components, due to the relative motion of point
skeletons defining the surface. However, if these disconnected chunks come
back close to each other, they will blend, as in Fig. 11, rather than collide, since
they are considered to be parts of the same surface.
Fig. 15. The influence graph and its connected
components. Particles A and B lie in the same component, and
thus their fields will blend if they come close to each other. 5.2.3 Modeling Fusion Under Compression
Blocks of soft substance, such as clay or dough, merge under compression
forces that exceed a specified threshold. This behavior can easily be
simulated with our model.
Fig. 16. Progressive fusion under compression of two
soft substances. Fig. 17. Soft substance grabbed away by pliers and
released. Fig. 18. Four frames of the animation Kitchen
Fiction.
This paper has presented a general framework for building layered
deformable models with implicit surfaces. The implicit formulation is
particularly well adapted to a layered construction. It can be used for
coating any reference component as, for instance, a rigid solid, an
articulated structure, a mass-springs network, or a particle system. It
defines a smooth surface around the object that can be used for rendering, and
offers simple yet precise processing of collisions and contacts. The implicit
inside-outside function facilitates collision detection, while the deformation
of the implicit layer generates exact contact surfaces between colliding
bodies. The physically based model associated with the implicit layer
approximates elasticity and allows the computation of response forces due to
compression and friction. Moreover, preservation of deformed objects' volume
is possible, even when the objects undergo significant changes such as
separation or fusion.
We show next how to create animations and what our typical data structure
is. It will focus on a simple animation of two colliding balls as depicted in
Fig. 4. The lower ball is considered as
motionless while both are deformable. The data structure for each ball is
defined as follows:
where
We wish to thank Jean-Dominique Gascuel and Nicolas Tsingos for their
contributions to the development of the animation software. Many thanks to
Agata Opalach and Jules Bloomenthal for fruitful discussions, to reviewers for
very helpful comments, and to Andrew Hanson and George Drettakis for carefully
rereading this paper. E-mail: {Marie-Paule.Gascuel,
Mathieu.Desbrun}@imag.fr. Marie-Paule Cani-Gascuel is a graduate of the Ecole Normale
Superieure (Paris, France) in both mathematics and computer science and
received the PhD degree in computer science from the University of Paris XI in
1990. She is an assistant professor of computer science at ENSIMAG. She
received the "Habilitation à diriger des recherches" degree from INPG
(Grenoble, France) in March 1995. Her research interests include physically
based simulation, motion control of linked figures, and implicit surfaces
applied to interactive modeling and animation. Dr. Gascuel has served on the
program committee of several Eurographics workshops and conferences since
1993. Mathieu Desbrun was awarded an engineering degree in computer
science at ENSIMAG with distinction and a graduate degree in computer graphics
and vision, both in 1994. He is currently a doctoral candidate in computer
science at the INPG (Grenoble, France). His research interests include
physically based animation and implicit surfaces.![[mdash]](admis_files/mdash.gif)
This paper presents a general approach for
designing and animating complex deformable models with implicit surfaces.
Implicit surfaces are introduced as an extra layer coating any kind of
structure that moves and deforms over time. Offering a compact definition of a
smooth surface around an object, they provide an efficient collision detection
mechanism. The implicit layer deforms in order to generate exact contact
surfaces between colliding bodies. A simple physically based model
approximating elastic behavior is then used for computing collision response.
The implicit formulation also eases the control of the object's volume with a
new method based on local controllers.
Animation, implicit surfaces, deformable
models, collision detection, collision response, inelasticity.
![[ge]](admis_files/ge.gif)
c), which may have several disconnected components. Normal vectors
are directed along the field's gradient.
1) An internal physically based model
which will be used as a base structure during the animation. This model may
be, for instance, a rigid solid defined by a mass and an inertia tensor, an
articulated structure made of several such solids, a mass/spring network, a
particle system, or any other model.
2) An implicit layer that "coats" the
base structure. This layer is built by defining skeletons that generate the
implicit surface in local coordinate systems animated by the base structure.
Skeletons may be points, line segments, triangles, or any graphic primitive.
1) Animate the base structure by integrating the equations of
motion according to the set of applied forces. This computes new positions for
the skeletons that generate the implicit layer.
2) Process interactions
between objects:
a) Use the implicit layer for detecting interpenetrations.
b)
Model contact by locally deforming the implicit layer in order to generate
exact contact surfaces between colliding bodies.
c) Integrate reaction and
friction forces along contact surfaces. Add them to the set of external
actions to be applied to the base structure at the next time step.
modeling compression is added in the interpenetration region in order to
generate a contact surface with the other object.
and
to be added to the equations of the objects i and j should
generate an exact contact surface. Thus the two equations
![[agr]](admis_files/agr.gif)
i giving the ratio between the
maximal value desired for pji and the current
maximal compression term in the interpenetration area.
gradient direction (see Fig. 2),
a0 is the maximal propagation value equal to
i times the
maximal compression field value, and
is a piecewise polynomial function as depicted in Fig. 3. P0 is
computed in practice by iteratively performing small steps from P in
the opposite direction of the object j's gradient. Our choice for the
slope k ensures that the shape of the implicit object stays
C1 at the border of the interpenetration zone. One can use
the following equation for
:
w.
be the deformation field term associated at equilibrium with the radial force
R. If the normal vector N(P) has remained constant
during the deformation (which is then said to be "radial"), the formula giving
the correspondence between applied forces and deformations, obtained by
integrating (15), is:
which models local compression, is negative, Ri
has the same orientation as the normal vector, and models the internal force
that tends to restore the initial shape of the object.
![[lgr]](admis_files/lgr.gif)
i in the
description of each object. When a collision occurs, the friction and damping
force Fi at a point P of the contact
surface between two objects i and j is expressed by:
to each skeleton si:
is the part of the implicit object where
field contribution is the highest. This is equivalent to splitting the
implicit volume into Voronoï regions defined by the skeletons, the "distance"
from a point P to a skeleton si being
defined by the field function. Sample points sent from
si stop at
boundary, and are valid if they lie on the isosurface.
local coordinate system at each time step to meet the surface again. Thus,
using temporal coherence increases efficiency. In addition, sample points that
were located between two territories at a given animation step may be brought
to the surface during later deformations. The sampling of the isosurface
automatically adapts to large deformations and changes in topology.
The total volume of the implicit object can then be expressed as the sum of
local volumes.
is the set of sample points sent by a skeleton
si, d is the distance, and
bi only depends on the angular distribution of
samples for si. In practice, the factor
bi can be omitted in volume computations, since
controlling the value of
gradient gives the object local stiffness (see Section 3.3.1). In order to
adjust the volume of skeleton territories without modifying the object's
physical properties, we combine the original field function with a translation
At each time step, the field originally defined by the decreasing function of
the distance fi(P) =
Fi(d(P,
si)) is replaced by:
of the translation parameter rather than its value. The inputs of the
controller are then the normalized volume variation
![[Dgr]](admis_files/Dgr.gif)
i,t and its time derivative :
and ![[bgr]](admis_files/bgr.gif)
are appropriately chosen parameters. A
simple example of volume control is given in Fig. 11.
1) Unwanted blending effects must be avoided
during deformations.
2) Intercollisions should be detected between the
different parts of a character.
1)
Compute all the field contributions at point P,
2) Select the
predominant contribution from those of groups of skeletons that blend
together,
3) Return this value without summing the other field
contributions.
continuity everywhere.
![[ggr]](admis_files/ggr.gif)
is the stiffness
parameter,
stands for the speed vector of particle Pi, and
![[mgr]](admis_files/mgr.gif)
is a decreasing continous
function with finite support.
1) For each pair of point skeletons that blend together, we check
if their spheres of influence, defined by the radius of influence of their
fields, intersect. This relation defines an "influence graph." We then use
the transitive closure of this graph for computing the blending graph we are
looking for. For instance, in Fig. 15, point A is detected
to be in the same components than point B, while the separation
with the other part is detected.
2) Collisions are detected between
skeleton territories not connected in the blending graph, as was done in
Section 5.1.3. As a result, pieces of substance that separate from the same
body collide instead of blending when they come back close to each
other.
continuity everywhere, as observed in [20]: Tangent discontinuities may
appear locally at some intermediate steps of the fusion.
Ball #1:
Base structure: motionless mass-point
constants
mass 10.0
position P1 (0, 0, 0)
Implicit Layer
skeletons
point-skeleton s1 located at P1
field with linear elasticity
thicknes t1 = 0.03
stiffness k1 = 1.0
radius R1 = 0.1
constants
stiffness scale K = 1.0
sampling rate
![[sgr]](admis_files/sgr.gif)
= 10-2
surface friction = 0.1
Ball #2:
Base structure: mass-point
constants
mass 2.96
time-varying parameters
position P2 (0, 0, 1)
speed (0, 0, 0)
Implicit Layer
skeletons
point-skeleton s2 located at P2
field with linear elasticity
thickness t2 = 0.01
stiffness k2 = 1.0
radius R2 = 0.04
constants
stiffness scale K = 0.8
sampling rate = 10-2
surface friction = 0.2
General data:
dt 0.04
gravity 9.8
is the accumulation of all collision forces computed on the sample points from
ball #2 during the time step and of gravity. Collision detection and
computation of response forces are performed at each time step, on each sample
point, as described in Section 3
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First
Eurographics Workshop Implicit Surfaces, pp. 21-32, Grenoble, France,
Apr. 1995.
First Eurographics
Workshop Implicit Surfaces, pp. 171-185, Grenoble, France, Apr. 1995. An
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1996.
First Eurographics Workshop
Implicit Surfaces, pp. 143-159, Grenoble, France, Apr. 1995.
First Eurographics Workshop Implicit
Surfaces, pp. 107-112, Grenoble, France, Apr. 1995.
First Eurographics Workshop Implicit
Surfaces, pp. 223-232, Grenoble, France, Apr. 1995.
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185-203. Academic Press, June 1995.
