l_r

ktch.coiling.l_r(theta, w_r, t_r, d_r, r0=1.0)

Arc length of growth trajectory at coiling angle theta.

Maps the coiling angle \(\theta\) of the Raup’s model to the arc length \(l_R\) of the reference-point trajectory ([Noshita_2014]):

\[l_R(\theta) = r_0\,(W_R^{\theta / 2\pi} - 1)\, \frac{\sqrt{\Lambda}}{(1 - D_R)\,\ln W_R},\]

where \(\Lambda\) is the trajectory discriminant. The relation is closed-form for constant parameters; theta may be array-like.

Parameters:

thetaarray-like

Coiling angle \(\theta\) (radians).

w_r, t_r, d_rfloat

Raup parameters (w_r > 1, -1 < d_r < 1).

r0float, default = 1.0

Initial tube radius (arc length scales with r0).

Returns:

l_rfloat or ndarray

Trajectory arc length at theta.

See also

theta_r

Inverse function (arc length to coiling angle).

References

[Noshita_2014]

Noshita, K., 2014. Quantification and geometric analysis of coiling patterns in gastropod shells based on 3D and 2D image data. Journal of Theoretical Biology 363, 93–104.