Seashells

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.

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.

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 = r 0 ξ r t, z = z 0 ξ z t. t ranges from 0 at the apex of the shell to t max at the opening. Given the initial values θ 0, r 0, z 0 : θ i+1 = t i + Δ t = θ i + Δθ r i+1 = r 0 ξ r ti ξ r Δt = r i λ r λ r = ξ r Δt z i+1 = z 0 ξ z ti ξ z Δt = z i λ z λ z = ξ z Δt In many shells, parameters λ r, λ z are the same.

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.

Examples of seashells created from different generating curve.

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 ξ c 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 e 1, aligning the axis u with the principal normal vector e 2. H '(t) e 1 x H''(t) e 1 = e 3 = e 2 = e 3 x e 1 |H '(t)| |e 1 x H''(t)|

Vector e 1, e 2, e 3 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.

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.

Let C(s) = ( u(s), v(s), w(s) ) a parametric definition of the curve C in cordinates uvw, with s  [s min, s max ]. 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 [s min, s max ] : 1) 2) 3) 4)

Given the initial condition s 0 = s min, 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.

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.

ribs parallel to the direction of growth.

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.

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.

∂ a a 2 ∂ 2 a = ρs( + ρ 0 ) – μa + D a ∂ t 1 + ka 2 ∂ x 2 ∂ s a 2 ∂ 2 s = σ - ρs( + ρ 0 ) – νs + D a ∂ t 1 + ka 2 ∂ x 2 Activator–substrate model a – concentration of activator. D a – rate of diffusion along the x-axis. μ – the decay rate. s – concentration of the substrate. D a – 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. ρ 0 – represents a small base production of the activator, needed to initiate the reaction process. Activator : Substrate :

An example using the Activator–substrate model.

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 a 2 ∂ 2 a = ( + ρ 0 ) – μa + D a ∂ t h+h ka 2 ∂ x 2 ∂ h a 2 ν ∂ 2 h = σ + ρ - h + D h ∂ t 1 + ka 2 c ∂ x 2 dc ρ' x max = ∫ adx - ŋc dt x max - x min x min

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

Real seashells

The end