Seashells

1. Seashells

2. 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 .

3. 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 .

5. 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 = z0ξzt . 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 = r0ξrtiξrΔt = riλr λr = ξrΔt zi+1 = z0ξztiξzΔt = ziλz λz = ξzΔt In many shells , parameters λr , λz are the same .

7. 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 .

8. Examples of seashells created from different generating curve .

9. 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 ξct 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 xe1 |H'(t)||e1 x H''(t)|

10. 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 .

11. 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 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 .

12. 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 : 1) dl = f (s) , ds 2) du 2 dv2 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)

13. 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 .

14. 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 .

15. ribs parallel to the direction of growth .

16. 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.

17. 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 .

18. Activator–substrate model Activator : ∂ a a2∂2a = ρs( +ρ0 ) – μa + Da ∂ t 1 + ka2∂ x2 Substrate : ∂ s a2∂2s = σ - ρs( +ρ0 ) – νs + 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 . 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 .

19. An example using the Activator–substrate model .

21. 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 ∂ h a2ν∂2h = σ + ρ - h + Dh ∂ t 1 + ka2 c ∂ x2 dc ρ'xmax = ∫ adx - ŋc dt xmax - xminxmin

23. 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

24. Real seashells

25. The end