REPRESENTING SEASHELLS
SURFACE
## 1. INTRODUCTIONSeashells 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 MODEL2.1
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
(see Fig.1) given by the equations:b
(2)
The
second curve that is necessary to introduce is the (3) being
( (4)
with
For
the following, it is convenient to represent the generating
curve with respect to the fixed system , iinto t
and j into
n respectively, we have:
kWe 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.
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), It
is also possible to model the typical ornamentations present on seashells
surface like ribs, bumps and spikes by introducing a perturbation function
(7)
Without
entering into details, we only add that suitable forms for y=y(
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))
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.
[1]
H. Moseley: On the geometrical forms of turbinated and discoid shells,
[2]
D'Arcy W. Thompson: [3]
D. M. Raup: Geometrical analysis of shell coiling: general problems, [4]
M. B. Cortie: Models for mollusc shell shape, [5]
M. B. Cortie: Digital Seashells, [6]
C. Illert: Formulation and Solution of the Classical Seashell Problem.
II Tubular Three-Dimensional Seashell Surfaces, [7]
C. Illert: Nipponites Mirabilis-A Challenge to Seashell Theory?, [8]
D. R. Fowler, H. Meinhardt, P. Prusinkiewicz: Modeling seashells, Proceeding
of SIGGRAPH '92 (Chigago, Illinois, July 26-31, 1992), In |