Raup's Coiling Model applet

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This applet lets you vary the parameters W, D and T from Raup's Coiling Model, in order to construct shells of various shapes. David M. Raup stated this model in 1962, and the parameters' meaning was expressed by him as follows:

"A simple model of variation in the rate of growth in several dimensions accounts for variation in the form of gastropod shells. The model specifies the shape of the aperture, or generating curve, the axis of coiling, the size ratio (W) of successive generating curves, the distance (D) of the generating curve from the axis, and the proportion (T) of the height of one generating curve that is covered by the successive gyres."

In our model, this description is formulated in the more precise definitions, as follows:

• W is the factor by which the generating curve C is magnified at each revolution. So, for instance, if W = 2 and C is a circle, the diameter of C is doubled after a revolution.
• D is the length, at θ = 0, from the center of C to the orthogonal projection, onto the radius vector r₀, of the closest point on C to the z-axis. So, if again C is a circle, D is the radius of C at θ = 0. It is not difficult to figure out that, with this definition, we can indeed vary the distance from C to the z-axis, since we thereby vary the slope of the inner tangent to the shell's surface.
• T is the ratio between z(θ) and r(θ).

You may vary these parameters in two ways: either by entering their exact value in the form below, or by moving the three control points on the graphic. The red point modifies W, the yellow D and the blue T. This option can be disabled by unchecking the case "Show control points". Some shapes have already been created, and can be loaded by checking the "Load shape" box and then choosing the wanted shaped. Only when the box is checked will the shape be loaded; if not, it is the parameters W, D and T in the left column that will take effect. When options and parameters have been entered, you can see the result of the changes by pressing the button "Show the shell". In the graphic window, you can rotate the shell by dragging the mouse (left button pressed) over it. For further instructions on using the applets on this site, please visit Wolfram's LiveGraphics3D user instructions page. There is another good applet which demonstrates Raup's Coiling Model on this page, created by Raymond F. Gildner.

NB! Depending on your internet connection speed, the load time of the applet might be very long, especially when the "Growth type" option is set to "Orthoclinal"; please read the note below.

The parameter W must be a strictly positive number. The value of W has been substituted by 2."; $W = 2; $errflag = 1; } elseif ($W <= 0) { $errstr .= "