So I have a parametric curve $\bf{r}=${$x(n),y(n),z(n)$} such that the functions $x(n)$, $y(n)$ and $z(n)$ are polynomials of $4$-th degree. I have several of these curves, and I want to calculate the torsion $\tau(n)$ and the curvature $\kappa(n)$ at a particular point $n_0$ in each of these curves.
So what I am doing is, I am defining the vectors ${\bf r^{(m)}} =\lbrace \frac{d^mx}{dn^m},\frac{d^my}{dn^m},\frac{d^mz}{dn^m} \rbrace$ and I am calculating these things according to the formulae I saw in Wolfram World both for Torsion as well as Curvature.
$\tau(n)= \frac{[\bf{r}^{(1)} \bf{r}^{(2)} \bf{r}^{(3)}]}{|\bf{r}^{(1)} \times \bf{r}^{(2)}|^2}$
and
$\kappa(n)= \frac{|\bf{r}^{(1)} \times \bf{r}^{(2)}|}{|\bf{r}^{(1)}|^3}$
Where $[\bf{r}^{(1)} \bf{r}^{(2)} \bf{r}^{(3)}]$ is the Scalar Triple Product. and $| \bf v |$ denotes the norm of $\bf v$.
So I made the calculations but some numbers are just too weird. There are reasonable curves with $\tau(n_0) \approx 1000$. Nothing in the curves would suggest such a strange number, but then again, I'm a newbie here so the first thing I thought was that I had done something wrong along the way and that the computation is screwed up.
I verified the reference again but it seems like everything is in place, I think I'm missing something pretty basic here. Is that a typical value for Torsion in any known case? Can someone point up the mistake in that? Are the formulae wrong?
At least if these are right I can focus all my attentions to implementation mistakes or numerical instabilities.
[EDIT]
Examples of curves where the magnitude of the torsion $|\tau|$ is around the thousands for $n_0=4$. The curves are represented in matrices such that the first line of each matrix represents the polynomial coefficients for $x$, the second for $y$ and the third for $z$.
Curve 1, $\tau(4) = -2 025.22815027$
\begin{matrix} 0.147791666667 & -1.30558333333 & 3.36670833333 & -2.05691666667 & 1.21 & \\ 0.046 & -0.2195 & 0.1335 & -0.406 & -0.884 & \\ -0.11225 & 0.945333333333 & -2.48525 & 2.90216666667 & 4.102 & \\ \end{matrix}
Curve 2, $\tau(4)=-32705.0075194$
\begin{matrix} 0.18625 & -1.48266666667 & 3.62875 & -2.18033333333 & 1.21 & \\ -0.0775833333333 & 0.618166666667 & -1.51441666667 & 0.527833333333 & -0.884 & \\ -0.168583333333 & 1.34033333333 & -3.27591666667 & 3.35416666667 & 4.102 & \end{matrix}
Curve 3, $\tau(4)=-3341.5324268$
\begin{matrix} 0.154625 & -1.36408333333 & 3.494375 & -2.13291666667 & 1.21 & \\ 0.08 & -0.492 & 0.713 & -0.747 & -0.884 & \\ -0.0382916666667 & 0.399583333333 & -1.36570833333 & 2.25441666667 & 4.102 & \end{matrix}
Curve 4, $\tau(4)=-1026.39066614$
\begin{matrix} 0.146125 & -1.29758333333 & 3.379875 & -2.13241666667 & 1.21 & \\ 0.0777916666667 & -0.58175 & 1.31270833333 & -1.95175 & -0.884 & \\ 0.145291666667 & -1.13441666667 & 2.82470833333 & -2.51958333333 & 4.102 & \end{matrix}