Technical SectionModelling mollusc shells with generalized cylinders
Introduction
Almost everyone, who carefully looks at a mollusc shell, will be impressed by the complexity and beauty of it. Scientists have therefore already studied their properties for centuries, and in particular mathematicians have derived formulas to describe their shape. More recently, computer graphics researchers have tried to generate realistic images of such shells.
The basis for rendering a shell is, of course, a geometric model that describes the shape of the shell accurately enough. In addition, there should be a model that realistically describes the surface properties of the shell. In this paper, the use of generalized cylinders and textures is proposed for modelling shells. Images can be generated with ray tracing, which guarantees realistic results.
In Section 2, mathematical models, geometric models and textures for mollusc shells used until now are discussed, and the proposed method is introduced. In Section 3, a definition of generalized cylinders is given. In Section 4, it is shown how generalized cylinders can be used for modelling shells, and how textures can be mapped onto the surfaces of such models. In Section 5, some images of shells generated with ray tracing, and some conclusions, are given.
Section snippets
Mollusc shell models
In the field of conchology, the science of molluscs and the shells they form, one can gain better understanding of shell formation by using computer simulations. A mathematical model can be used to obtain properties such as volume and surface area, but images and animations are more suitable to provide insight into the structure and development of a shell.
Molluscs grow their shells by depositing material on the shell's aperture (mouth). It has been found that for most molluscan species, the
Generalized cylinders
A generalized cylinder is a 3D object defined by moving an arbitrary 2D closed contour along an arbitrary 3D trajectory, while simultaneously transforming, i.e. scaling, translating and rotating, the contour in its plane at each point on the trajectory8, 9, 10.
The 2D closed contour, c, is defined as a parametric curve:
The 3D trajectory, t, is likewise defined as a parametric curve:
The parametric curves can, in principle,
A shell model based on generalized cylinders
In order to use a generalized cylinder as a geometric model for a shell, a conversion must be made from the mathematical model for the shell shape to the curves defining the generalized cylinder. The mathematical model that will be used here is the Cortie model (see Section 2), but the methods described would also work when applied to models such as described in[5], as long as the trajectory and contour shapes can be treated separately.
The conversion from the Cortie model to generalized
Results and conclusions
We present four images of shells modelled with generalized cylinders and textures. The images have been generated with a ray tracing algorithm that directly renders the surface of a generalized cylinder, i.e. without first converting it into, for example, a polygon mesh or a NURBS surface[9].
Fig. 3 shows a Turritella Terebra. The shell's outer surface is coloured according to an existing specimen. Fig. 4 shows a Thatcheria Mirabilis. The shell's surface is coloured according to photos from Dance
References (15)
Digital seashells
Computers & Graphics
(1993)- et al.
A method for converting the surface of a generalized cylinder into a B-spline surface
Computers & Graphics
(1992) - et al.
Visible light and X-ray ray tracing of generalized cylinders
CVGIP: Graphical Models and Image Processing
(1994) Models for the mollusc shell shape
South African Journal of Science
(1989)- Cortie, M. B., Modelling the surface bumps and spikes of mollusc shells. Proceedings First International Conchology...
- et al.
Modeling seashells
Computer Graphics (Proceedings Siggraph `92)
(1992) Formulation and solution of the classical seashell problem
Il Nuovo Cimento
(1989)