# Thread: What curve is this?

1. ## What curve is this?

Hello!

Can somebody tell me what curve is this:

(a t; b sin t; c cos t), t e R

2. ## Re: What curve is this?

Hello, mboricgs!

Can somebody tell me what curve is this:

. . $(a t,\; b\sin t,\; c\cos t),\; t \in R$

It is an ellipsoidal helix centered on the x-axis.

3. ## Re: What curve is this?

what is equation of this curve without t?

4. ## Re: What curve is this?

Originally Posted by mboricgs
Hello!

Can somebody tell me what curve is this:

(a t; b sin t; c cos t), t e R
\displaystyle \begin{align*} x &= a\,t \\ \\ y &= b\sin{t} \\ \frac{y}{b} &= \sin{t} \\ \frac{y^2}{b^2} &= \sin^2{t} \\ \\ z &= c\cos{t} \\ \frac{z}{c} &= \cos{t} \\ \frac{z^2}{c^2} &= \cos^2{t} \\ \\ \frac{y^2}{b^2} + \frac{z^2}{c^2} &= \sin^2{t} + \cos^2{t} \\ \frac{y^2}{b^2} + \frac{z^2}{c^2} &= 1 \end{align*}

So that means you have ellipses on the \displaystyle \begin{align*} y-z \end{align*} plane, and as you change \displaystyle \begin{align*} t \end{align*}, the x values increase, which means it wraps around the \displaystyle \begin{align*} x \end{align*} axis, creating an elliptical helix.

5. ## Re: What curve is this?

Thank you. What about flexion and torsion? Are they constant?

6. ## Re: What curve is this?

Originally Posted by mboricgs
Thank you. What about flexion and torsion? Are they constant?
Flexion: || cx c||/ (||c||^3), torsion: det(c,c,c)/(||c x c`||)^2.. do they depend on t?

7. ## Re: What curve is this?

Can someone tell me hot to do that in wolframalpha?