Based on the answer on my previous question I managed to come up with the following equations: $$\begin{eqnarray} k &=& 1 \\ r_\Delta &=& r_b - r_t \\ r(\theta) &=& r_t * e^{\theta/360*k} \\ x(\theta) &=& r(\theta) \cos(\theta) \\ y(\theta) &=& r(\theta) \sin(\theta) \\ z(\theta) &=& (\frac{r_b}{r_\Delta} - \frac{r(\theta)}{r_\Delta}) * h_s \\ d(\theta) &=& diameter * \frac{r(\theta)}{r_b} \\ \end{eqnarray}$$
that give almost the desired spiral.
The problem is that with $k = 1$ the spiral goes beyond the stump (defined by the top radius $r_t$, the bottom radius $r_b$ and the height $h_s$).
The question is what does $k$ need to be so that the spiral ends at the bottom of the stump (at $z = 0$ and $r(\theta) = r_b$)