Generator Controls ` Whelk 1-2-1`

The spiral is generated by three independent parameters R, L, W.

R is the rate of radial expansion, L the rate of Longitudinal expansion, and W the rate of whorl expansion.

Additionally, R, L and W can be modified by factors Rs, Ls and Ws, respectively. Initially, R = L = W, giving an isometric form.

NOTE: The controls R, L and W are only active when ‘Custom’ is selected as the spiral choice in the Spiral panel.

You may also specify a spiral by modifying its polar equation directly. See Spiral Controls for details.

In the diagram above, K0 is the position of the generating curve after one revolution; K1 is the position of the generating curve after two revolutions; and K2 is the position of the generating curve after three revolutions. The radii r0, r1, r2, measured from the central axis, also correspond to three revolutions, i.e., the angle between r0 and r1 is 360 °, and the angle between r1 and r2 is also 360°.

Whelk versions prior to 1-2-1, only used a single radial expansion rate. This was modified by a trochoidal shift value (now called Ls), and a profile radius factor (now termed Ws). Using three individual expansion rates makes it easier to create concave and convex spires without using growth curves. I realised this after reading the paper “Spiral Growth: The ‘Museum of all Shells’ Revisited” by Bernard Tursch 1997. This is why Whelk 1-2-1 is code-named ‘Tursch’.

The modus operandi of some generator controls is not immediately obvious. These controls are documented here. (TO DO: Update screenshots for version 1-2-1).

NOTE: The best way to understand how Whelk operates is to Load the sample `WLK` files in the Generators directory.

Mobius

This applies a trochoidal twist such that the coil doubles back on itself, rather like a Mobius strip. As the value moves toward `HALF_PI` (90°), it creates two apparent coils - an inner and an outer - though both are formed from a single mesh. This resuls in two apertures, each facing in opposite directions. The normals of the inner aperature face inward, whilst the normals of the outer aperture face outward.

Tropism

This ‘bends’ the axis of the trochoidal shift and is useful when generating horns.

Tropism A applies pitch and Tropism B applies roll.

Generate Slice Profiles on Trochoidal Axis   `Whelk-1-0-2`

This legacy feature has been removed in `Whelk-1-2-1` which achieves the same result by setting the Radial expansion scale factor Ws to `ZERO`.

When this option is `ON`, profiles are built along the trochoidal axis, rather than the generated spiral radius. The trochoidal shift must be greater than `ZERO` for the horn to form - otherwise profiles will be simply stacked one on top of the other in the same plane, resulting in a 2D shape.

Generate Slice Profile on Trochoidal Axis `ON`

Note that when the option is `ON`, and a Trochoidal shift is appplied, the form will resemble that of a Mobius Inversion. At first this may seem counter-intuitive, but by changing just a few additional parameters, it becomes possible to create many types of horn found in Nature…

Profile Yaw ε, Pitch λ, Roll υ`  Whelk-1-0-2`

Combinations of Yaw ε, Pitch λ, Roll υ are used to change the angle of the slice profile plane and so modify the entire shape. Initially, the profile plane is coincident with the Trochoidal axis. As the profile plane tilts away from this axis, the form begins to open up and the Mobius Inversion property dissapears. When the profile plane is perpendicular to the Trochoidal shift axis, a simple cone is formed…

Note that at 90°, the profile plane is perpendicular to the Trochoidal shift axis, resulting in a cone; the spiral generator then has no effect because at 90°, the section through the cone is a circle, rather than ellipse. This is when to use 1) a custom profile; 2) a peturbation - (which can reveal the underlying morphology in the mesh; or 3), use a Mobius Inversion, which will twist the entire cone around the trochoidal axis…

Profile Pitch λ 90° Mobius Inversion `ON`

Note that when 'Generate slice profiles along trochoidal axis' is `ON`, the Mobius inversion property coils the form around the trochoidal axis (which is Y in this example). The following examples also use mesh peturbations in various layers…

Simple Horn example

This example uses tropism to curve the form.

Profile Yaw ε

In order to stop umbos protuding on bivalves, you can increase the ε value which lifts the umbo away from the seam of the shell:

Profile Yaw ε Pitch λ `ON`

Combinations of Yaw ε and Pitch λ can be used to reorient the whorl aperture and change the form of the columella:

Profile Rotate υ

This only changes the form when using a non-circular custom profile, and rotates the entire profile around the profile center.