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Math Help - What curve is this?

  1. #1
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    What curve is this?

    Hello!

    Can somebody tell me what curve is this:

    (a t; b sin t; c cos t), t e R
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  2. #2
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    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.
    Thanks from HallsofIvy
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  3. #3
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    Re: What curve is this?

    what is equation of this curve without t?
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  4. #4
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    Re: What curve is this?

    Quote Originally Posted by mboricgs View Post
    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.
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  5. #5
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    Re: What curve is this?

    Thank you. What about flexion and torsion? Are they constant?
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  6. #6
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    Re: What curve is this?

    Quote Originally Posted by mboricgs View Post
    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?
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  7. #7
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    Re: What curve is this?

    Can someone tell me hot to do that in wolframalpha?
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