Determine the condition makes the curve helix

Question

(Twisted cubic) Show that $\alpha(t)=(at,bt^2,ct^3)$ is a helix $\iff$ $4b^2 = 9a^2c^2$.
I think $\frac{\tau}{\kappa}$ must be constant. I've solved for them. However, I got really mixed equations which I cannot simplify.

Answer 1

Helices are defined by requiring that the unit tangent makes a constant angle with a fixed line in space. It is possible to prove that this implies that \begin{equation} \frac{\tau(t)}{k(t)}=cot \alpha \end{equation} and thus, for any helix,the ratio of curvature to torsion (the pitch) is constant. When calculating the curvature and torsion I got: \begin{equation} \frac{2 \sqrt{9 c^2 a^2 t^2+b^2 a^2+9 b^2 c^2 t^4}}{\left(4 b^2 t^2+9 c^2 t^4+ a^2\right)^{3/2}} \end{equation} And for the torsion: \begin{equation} \frac{3 a b c}{b^2 a^2+9 c^2 a^2 t^2+9 b^2 c^2 t^4} \end{equation} The ratio is: \begin{equation} \frac{\tau(t)}{k(t)}=\frac{3 a b c \left(4 b^2 t^2+(a^2+9 c^2 t^4)\right)^{3/2}}{2 \left(9 a^2 c^2 t^2+b^2 \left(a^2+9 c^2 t^4\right)\right)^{3/2}} \end{equation} In the above computation I assumed that a, b, c and t are real values. Except if I did a computational mistake (which is possible) I don't think the curve is an helix when the condition $4 b^2 = 9 a^2 c^2$ is satisfied. In principle you should differentiate the ratio $\frac{\tau(t)}{k(t)}$ and impose that the results is zero.

Answer 2

As Upax already calculated \begin{equation} \frac{\tau(t)}{k(t)}=\frac{3 a b c \left(4 b^2 t^2+(a^2+9 c^2 t^4)\right)^{3/2}}{2 \left(9 a^2 c^2 t^2+b^2 \left(a^2+9 c^2 t^4\right)\right)^{3/2}}. \end{equation}

This can be simplified to \begin{equation} \frac{\tau(t)}{k(t)}=\frac {3abc}{2b^2} \left(\frac{4 b^2 t^2+(a^2+9 c^2 t^4)}{\frac{9 a^2 c^2}{b^2} t^2+\left(a^2+9 c^2 t^4\right)} \right)^{3/2}. \end{equation} Since you have quotient of two polynomials of degree 4 it is a constant iff they are proportional. Since the coefficients of the highest order are equal ($9c^2$ in both cases) they have to be equal, therefore the curve is helix iff $$4b^2 = \frac {9a^2c^2}{b^2}$$ which simplifies to what you need. Btw, there is a typo in the question.