Download - 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 :
θ = t , r = r0ξrt , z = z0ξz
t .
t ranges from 0 at the apex of the shell to tmax at the opening .
Given the initial values θ0 , r0 , z0 :
θi+1 = ti + Δ t = θi + Δθ
ri+1 = r0ξrtiξr
Δt = riλ r λ r = ξrΔt
zi+1 = z0ξztiξz
Δt = ziλ z λ z = ξzΔt
In many shells , parameters λ r , λ 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 ξc
t 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 meshIn 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 n+1 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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Let C(s) = ( u(s) , v(s) , w(s) ) a parametric definition of the curve
C in cordinates uvw , with s [smin , smax ] .
dl = f (s) , ds
du 2 dv 2 dw 2
f(s) = + + ds ds ds
smin
L = ∫ f (s) ds smax
ds 1 = dl f (s)
A method for achieving discretized uniformly space .
The length of an arc of C is related to an increment of
parameter s by the equations :
The total length L of C can be found by integrating f (s) in the interval [smin , smax ] :
1(
2(
3(
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
Δl = 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 :
• Activator–substrate model .
• Activator–inhibitor model .
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= ρs( + ρ0 ) – μa + Da ∂ t 1 + ka2 ∂ x2
= σ - ρs( + ρ0 ) – νs + Da ∂ t 1 + ka2 ∂ x2
Activator–substrate model
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 .
ν - the decay rate .
σ – the substrate is produced at a constant rate σ .
ρ – the coefficient of proportionality .
k – controls the level of saturation .
ρ0 – represents a small base production of the activator ,
needed to initiate the reaction process .
Activator:
Substrate:
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An example using the
Activator–substrate model .
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Activator–inhibitor 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 ρ a2 ∂ 2a = ( + ρ0 ) – μa + Da ∂ t h+h0 1 + ka2 ∂ x2
= σ + ρ - h + Dh ∂ t 1 + ka2 c ∂ x2
= ∫ adx - ŋc 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