REPRESENTING SEASHELLS SURFACE

G. Lucca

Italy

vanni_lucca@inwind.it

 Abstract: The paper describes a general algorithm for representing seashells surface of very different shape. The method is based on simple geometric transformations that lead to the surface equations expressed in parametric form; different examples of application considering real species (living or fossil) are presented.

## 1. INTRODUCTION

Seashells may be considered a connection point among geometry, mathematics and art. In fact, since ancient times, seashells have always fascinated human kind for their beauty, elegance and, in many cases, symmetry; in particular, this last aspect can be an invite to describe seashells surface by a means of a suitable mathematical model and to visually represent it by the aid of the computer.

The first studies on seashells model are due to Reverend H. Moseley [1] and to D'Arcy W. Thompson [2] who emphasized the gnomonic properties and the role of the equiangular (logarithmic) spiral of many seashell species during the growth process. More recent contributions are the ones of D.M. Raup [3], M. B. Cortie [4], [5], C. Illert [6], [7] and D. Fowler et al. [8].

On the basis of those works, the next paragraphs are devoted to shortly describe the algorithm for the seashells surface representation and to present some examples of applications.

## 2. THE MODEL

2.1 Structural curve and Frenet frame

The first step is to define a Cartesian coordinate system xyz in the space whose origin is O and characterized by unit vectors i, j, k respectively (see Fig.1); let us define, with respect to that system, a curve in the space whose name is structural curve, which is strictly related to the overall shape of the seashell. (In many seashells species the structural curve is represented by the equiangular spiral in the space i.e. the helico-spiral). It is convenient to represent the structural curve in parametric form where the parameter is the azimuth angle q that is:

(1)

Related to the structural curve, we introduce the relevant Frenet frame i.e. an orthogonal coordinate system characterized by the unit vectors t, n, b (see Fig.1) given by the equations:

(2)

Fig.1: Structural curve (in this case an helico-spiral) and associated Frenet frame.

2.2 Generating curve

The second curve that is necessary to introduce is the generating curve which is strictly related to the shape of the seashell aperture; in most of cases, an ellipse or an arc of ellipse, with center in O' (the origin of the Frenet frame) can be a convenient choice. By expressing the ellipse in parametric form with the angular parameter s as shown in  Fig.2, with respect to the Frenet frame, we have:

(3)

being (s) given by:

(4)

with a and c major and minor semi-axes of the ellipse respectively.

Fig.2: Generating curve expressed with respect to the Frenet frame.

For the following,  it is  convenient to represent the generating curve with respect to the fixed system xyz with origin in O; thus, by considering the linear transformation which transforms the unit vectors b into i, t into j and n into k respectively, we have:

We emphasize that in this case the generating curve is also function of the parameter q just because the orientation of the Frenet frame along the structural curve is a function of  q.

2.3 Generation of the surface

The seashell surface is generated by simply translating the generating curve along the structural curve and, at the same time, by dilating it, in order to model the growth process; such a process occurs, in most of cases, according to exponential law. Therefore,the seashell surface equations given in function of the parameters s and q are:

(6)

In (6), w is a parameter controlling the exponential growth rate.

It is also possible to model the typical ornamentations present on seashells surface like ribs, bumps and spikes by introducing a perturbation function y=y(s,q) to be added to the function r=r(s) expressed by formula (4). In such a way, the new following expression for r(s) is obtained:

(7)

Without entering into details, we only add that suitable forms for y=y(s,q) are sinusoidal functions (for modelling ribs) or bell-shaped functions (for modelling bumps); the parameters characterising such functions clearly influence the aspect, not only of the ornamentations themeselves, but also of the whole seashell surface.

3. EXAMPLES

This paragraph is devoted to show some examples of real species of seashells represented by applying the algorithm above described; some among them are relevant to living molluscs (see Figs. (3), (4), (5), (6), (7), (8)) but some other to extinct molluscs i.e. fossils. (See Figs. (9), (10), (11))

Fig.3

Fig.4

Fig.5

Fig.6

Fig.7

Fig.8

Fig.9

Fig.10

Fig.11

It is useful to mention that Figs. from 3 to 8 are based on a structural curve that, in all the cases, is represented by an helico-spiral (or, as a limit case, a plane equiangular spiral); the consequence is  that the seashells exhibit a self similarity property during the growth i.e. the shape of the seashell does not change during the mollusc life but only changes its size.

On the contrary, Figs from 9 to 11 represents seashells of heteromorphic molluscs; in those cases the structural curve is not an helico-spiral and thus the self similarity property is not present; this is particularly evident by looking at Figs 9 and 11.

REFERENCES

[1] H. Moseley: On the geometrical forms of turbinated and discoid shells, Philosophical Transactions of the Royal Society of London, Pt I, p. 351-370, 1838.

[2] D'Arcy W. Thompson: On Growth and Form, University Press, Cambridge, 2nd Edition, 1942.

[3] D. M. Raup: Geometrical analysis of shell coiling: general problems, Journal of Paleontology, vol. 40, no. 5, p. 1178-1190 September 1966.

[4] M. B. Cortie: Models for mollusc shell shape, South African Journal of Science, vol. 85, p. 454-460, July 1989.

[5] M. B. Cortie: Digital Seashells, Computer & Graphics, vol.17, no. 1, p. 79-84, 1993.

[6] C. Illert: Formulation and Solution of the Classical Seashell Problem. II Tubular Three-Dimensional Seashell Surfaces, Il Nuovo Cimento, vol. 11 D, no. 5, May 1989.

[7] C. Illert: Nipponites Mirabilis-A Challenge to Seashell Theory?, Il Nuovo Cimento, vol. 12 D, no.10, October 1990.

[8] D. R. Fowler, H. Meinhardt, P. Prusinkiewicz: Modeling seashells, Proceeding of SIGGRAPH '92 (Chigago, Illinois, July 26-31, 1992), In Computer Graphics, 26, 2 (July 1992), ACM SIGGRAPH, New York, p. 379-387 (available on line at the address: http://www.cpsc.ucalgary.ca/Redirect/bmv/papers/shells/sig92.pdf).

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