Seashells

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Seashells
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• This presentation presents a method for modeling
  seashells .
• Why seashells you ask ?
• Two main reasons
• The beauty of shells invites us to construct
  their mathematical models .
• The motivation to synthesize realistic images
  that could be
• incorporated into computer-generated scenes and
  to gain a better
• understanding of the mechanism of shell
  formation .
• this presentation propose a modeling technique
  that combines two key
• components
• A model of shell shapes derived from a
  descriptive characterization .
• A reaction-diffusion model of pigmentation
  patterns .
• The results are evaluated by comparing models
  with real shells .

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Modeling Shell Geometry (part 1)
The surface of any shell may be generated by
the revolution about a fixed axis of a closed
curve , which , Remaining always geometrically
similar to itself , and increases its dimensions
continually .

• A shell is constructed using these steps
• the helico-spiral .
• the generating curve .
• Incorporation of the generating curve into the
  model.
• Construction of the polygon mesh
• modeling the sculpture on shell surfaces .

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The helico spiral
The modeling of a shell surface starts with the
construction of a logarithmic helico-spiral H
. In a cylindrical coordinate system it has the
parametric description ? t , r r0?rt , z
z0?zt . t ranges from 0 at the apex of the
shell to tmax at the opening . Given the initial
values ?0 , r0 , z0 ?i1 ti ? t ?i
?? ri1 r0?rti?r?t ri? r ? r
?r?t zi1 z0?zti?z?t zi? z ? z
?z?t In many shells , parameters ? r , ? z are
the same .
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The generating curve
The surface of the shell is determined by a
generating curve C , sweeping along the helico-
spiral H . The size of the curve C increases
as it revolves around the shell axis . In order
to capture a variety and complexity of
possible shapes , C is constructed from one or
more segments of Bezier curves .
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Examples of seashells created from different
generating curve .
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Incorporation of the generating curve into the
model
The generating curve C is specified in a local
coordinate system uvw . Given a point H(t) of the
helico-spiral , C is first scaled up by the
factor ?ct with respect to the origin O of this
system , then rotated and translated so that the
point O matches H(t) . The simplest approach is
to rotate the system uvw so that the axis v and u
become respectively parallel and perpendicular to
the shell axis z , if the generating curve lies
in the plane uv . However , many shells exhibit
approximately orthoclinal growth markings , which
lie in planes normal to the helico-spiral H .
This effect can be captured by orienting the axis
w along the vector e1 , aligning the axis u with
the principal normal vector e2 .

H '(t) e1 x
H''(t) e1 e3
e2 e3 x e1 H '(t)
e1 x H''(t)
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Vector e1, e2, e3 define a local orthogonal
coordinate system called the Frenet-frame ,
where the opening of the shell and the ribs on
its surface lie in planes normal to
the helico-spiral . This is properly captured
in the model in the center which uses
frenet-frame . The model on the right incorrectly
aligns the generating curve with the shell axis .
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Construction of the polygon mesh
In the mathematical sense , the surface of the
shell is completely defined by the generating
curve C, sweeping along the helico-spiral H. The
mesh is constructed by specifying n1 points on
the generating curve , and connecting
corresponding points for consecutive positions of
the generating curve . The sequence of polygons
spanned between a pair of adjacent generating
curves is called a rim .
For pigmentation patterns equations (which will
be explained later on) , it is best if the space
in which they operate is discretized uniformly
. This corresponds to the partition of the rim
into polygons evenly spaced along the generating
curve .
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A method for achieving discretized uniformly
space .
Let C(s) ( u(s) , v(s) , w(s) ) a parametric
definition of the curve C in cordinates uvw ,
with s ? smin , smax .
The length of an arc of C is related to an
increment of parameter s by the equations
dl f (s) , ds
1)
2)
du 2 dv 2 dw 2 f(s)
ds
ds ds
The total length L of C can be found by
integrating f (s) in the interval smin , smax
smin L ? f (s) ds smax
3)
ds 1 dl f (s)
4)
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Given the initial condition s0 smin , the first
order differential equation describes parameter s
as a function of the arc length l. By numerically
integrating (4) in n consecutive intervals of
length ?l L/N we obtain a sequence of
parameter values , of s , Representing the
desired sequence of n 1 polygon vertices
equally spaced along the curve C .
Here you can see the effect of the
reparametrization of the generating curve .
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Modeling the sculpture on shell surfaces

• Many shells have a sculptured surface which
  include ribs .
• There are two types of ribs
• ribs parallel to the direction of growth .
• ribs parallel to the generating curve .
• Both types of ribs can be easily reproduced by
  displacing the
• vertices of the polygon mesh in the direction
  normal to the shell
• surface .

In case of ribs parallel to the direction of
growth ,the displacement d varies periodically
along the generating curve . the amplitude of
these variations is proportional to the actual
size of the curve , thus it increases as
the shell grows .
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ribs parallel to the direction of growth .
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Ribs parallel to the generating curve are
obtained by periodically varying the value of the
displacement d according to the position of the
generating curve along the helico-spiral H .
The ribs parallel to the generating curve could
have been incorporated into the curve definition
. But this approach is more flexible
and can be easily extended to other sculptured
patterns.
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Generation of pigmentation patterns (part2)

• Pigmentation patterns constitute an important
  aspect of shell
• appearance because they show enormous diversity ,
  which
• may differ in details even between shells of the
  same species .
• In this presentation pigmentation patterns are
  captured using
• a class of reaction-diffusion models .
• Generally , we group our models into two basic
  categories
• Activatorsubstrate model .
• Activatorinhibitor model .

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Activatorsubstrate model
Activator
? a a2
? 2a ?s( ?0 )
µa Da ? t 1 ka2
? x2
Substrate
? s a2
? 2s s - ?s(
?0 ) ?s Da ? t 1 ka2
? x2
a concentration of activator . Da rate
of diffusion along the x-axis . µ the decay
rate . s concentration of the substrate . Da
rate of diffusion along the x-axis . ? -
the decay rate . s the substrate is produced
at a constant rate s . ? the coefficient of
proportionality . k controls the level of
saturation . ?0 represents a small base
production of the activator , needed to
initiate the reaction process .
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An example using the Activatorsubstrate model .
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Activatorinhibitor model .
As you can see in the picture colliding waves is
essential. Observation of the shell indicates
that the number of traveling waves is
approximately constant over time , this suggests
a global control mechanism that monitors the
total amount of activators in the system and
initiartes new waves when its concentration
becomes too low .
? a ? a2
? 2a ( ?0
) µa Da ? t hh0 1 ka2
? x2
? h a2 ? ?
2h s ? - h Dh ? t
1 ka2 c ? x2
dc ?' xmax
? adx - ?c dt xmax - xmin xmin
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Conclusion

• This presentation presents a comprehensive model
  of seashells ,
• There are still some problems for further
  research
• proper modeling of the sea shell opening .
• modeling of spikes
• capturing the the thickness of shell walls .
• alternative to the integrated model
• improved rendering

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Real seashells
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The end