Example Morphospaces

Raupian Coiling

Introduction

Raupian coiling is a mathematical model developed by David M. Raup to describe the three-dimensional geometry of mollusk shells. It assumes a cylindrical coordinate space.

Calculation of the Radius from the Coiling Axis

$$ r_\theta = r_0 W^{\frac{\theta}{2\pi}} $$

Where:

• $r_\theta$ = Distance after $\theta$ revolutions
• $r_0$ = Initial distance of point A from the axis
• $W$ = Rate of whorl expansion
• $\theta$ = Angle in radians

Calculation of the Angle of the Coiling Axis in Planispiral Forms

$$ y_\theta = y_0 W^{\frac{\theta}{2\pi}} $$

Where:

• $y_\theta$ = y-value of the point after $\theta$ revolutions in planispiral shells where $T = 0$
• $y_0$ = Initial y-value of the point
• $W$ = Rate of whorl expansion
• $\theta$ = Angle in radians

Calculation of the Angle of the Coiling Axis in Helicoid Forms

$$ y_\theta = y_0 W^{\frac{\theta}{2\pi}} + r_c T \left( W^{\frac{\theta}{2\pi}} - 1 \right) $$

Where:

• $y_\theta$ = y-value of the point after $\theta$ revolutions in helicoid forms
• $y_0$ = Initial y-value of the point
• $W$ = Rate of whorl expansion
• $\theta$ = Angle in radians
• $r_c$ = The r-value of the center of the initial generating curve
• $T$ = Rate of translation defined as $\frac{dy}{dr}$, with respect to the center of the generating curve

Figure 1a from Gerber et al 2017

References

1. Gerber, S. The geometry of morphospaces: lessons from the classic Raup shell coiling model. Biological Reviews 92, 1142–1155 (2017).