map_to_torus_sineZ Subroutine

pure subroutine map_to_torus_sineZ(this, c, R, twist_turns, Az, nwaves_z, phase_z)

Map a straight pipe (ring extruded in $z$) onto a toroidal pipe with: • cross-section twist: $\phi' = \phi + 2\pi\,(\texttt{twist\_turns})\,s$, • vertical wobble: $z_\text{off}(s) = A_z \sin(2\pi\,n_\text{waves}\,s + \varphi)$.

Parameterization: For the axial (k) index, define $$s(k) = \begin{cases} \dfrac{k-1}{n_c^{(z)}-1}, & n_c^{(z)} > 1,\\[4pt] 0, & \text{otherwise}, \end{cases}$$ and the torus angle (here one full loop since $\theta=2\pi s$): $$\theta(s) = 2\pi\,s .$$

For each control point $\mathbf{X}_c=(x,y,z)$, relative to $\mathbf{c}=(c_x,c_y,c_z)$, set $\rho=\sqrt{(x-c_x)^2+(y-c_y)^2}$, $\phi=\mathrm{atan2}(y-c_y,x-c_x)$, then $\phi'=\phi+ 2\pi(\texttt{twist\_turns})\,s$.

Mapping onto a torus of major radius $R$: $$\begin{aligned} X &= c_x + \bigl(R + \rho\cos\phi'\bigr)\cos\theta,\\ Y &= c_y + \bigl(R + \rho\cos\phi'\bigr)\sin\theta,\\ Z &= c_z + \rho\sin\phi' + z_\text{off}(s), \qquad z_\text{off}(s) = A_z \sin\!\bigl(2\pi\,n_\text{waves}\,s + \varphi\bigr). \end{aligned}$$

Knots/weights are preserved; only the control points are updated.

Arguments