What curve is this?

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May 12th 2012, 12:08 PM
mboricgs

What curve is this?

Hello!

Can somebody tell me what curve is this:

(a t; b sin t; c cos t), t e R
May 12th 2012, 01:42 PM
Soroban

Re: What curve is this?

Hello, mboricgs!

Quote:

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.
May 27th 2012, 06:51 AM
mboricgs

Re: What curve is this?

what is equation of this curve without t?
May 27th 2012, 06:56 AM
Prove It

Re: What curve is this?

Quote:

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.
May 27th 2012, 07:00 AM
mboricgs

Re: What curve is this?

Thank you. What about flexion and torsion? Are they constant?
May 27th 2012, 07:14 AM
mboricgs

Re: What curve is this?

Quote:

Originally Posted by mboricgs

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

Flexion: || c`x c``||/ (||c`||^3), torsion: det(c`,c``,c```)/(||c` x c``||)^2.. do they depend on t?
May 28th 2012, 07:26 AM
mboricgs

Re: What curve is this?

Can someone tell me hot to do that in wolframalpha?