Results 1 to 10 of 10
Like Tree3Thanks
  • 1 Post By romsek
  • 1 Post By romsek
  • 1 Post By romsek

Thread: Curvature & Radius of Curvature

  1. #1
    Senior Member
    Joined
    May 2016
    From
    NYC
    Posts
    368
    Thanks
    5

    Curvature & Radius of Curvature

    Find the curvature and radius of curvature.
    This question involves encapsulation.

    y = 3x - 2 given x = a

    r (t) = < t, y, 0 >

    r (t) = < t, 3t - 2, 0 >

    r (t) = (t)i + (3t - 2) j

    r'(t) = i + 3 j

    ||r'(t)|| = root {10}

    T(t) = (i + 3 j)/[root {10}]

    I am stuck here.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,691
    Thanks
    2394

    Re: Curvature & Radius of Curvature

    do you know what the formula for curvature of a space curve is?
    Thanks from USNAVY
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    May 2016
    From
    NYC
    Posts
    368
    Thanks
    5

    Re: Curvature & Radius of Curvature

    Quote Originally Posted by romsek View Post
    do you know what the formula for curvature of a space curve is?
    There are two formulas:

    1. ||T'(t)||/||r'(t)||

    2. ||r'(t)Xr"(t)||/||r'(t)||^3

    I find that the second formula works best for polynomials. How do I rewrite the given equation as a vector r (t)? What are the steps? When do I plug x = a? Here "a" is constant, right?
    Last edited by USNAVY; Jan 3rd 2017 at 04:10 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    May 2016
    From
    NYC
    Posts
    368
    Thanks
    5

    Re: Curvature & Radius of Curvature

    y = 3x - 2 given x = a

    r (t) = < t, y, 0 >

    r (t) = < t, 3t - 2, 0 >

    r (t) = (t)i + (3t - 2) j

    r'(t) = i + 3 j

    r'(a) = 0

    ||r'(a)|| = root (0^2) = 0

    T (t) = r'(a)/||r'(a)||

    T(t) = 0/0 = 0

    T'(t) = 0

    ||T'(t)|| = root (0) = 0

    K = ||T'(t)||/||r'(t)||

    K = 0/0

    K = 0

    Radius = 1/k

    Radius = undefined when k = 0.

    Is this correct?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,691
    Thanks
    2394

    Re: Curvature & Radius of Curvature

    start with

    $r(t) = (t, 3t-2,0)$

    $r^\prime(t) = (1, 3, 0)$

    $\| r^\prime(t) \| = \sqrt{10}$

    $T(t) = \dfrac{r^\prime(t)}{\|r^\prime(t)\|} = \dfrac{(1,3,0)}{\sqrt{10}}$

    $T^\prime(t) = (0,0,0)$

    $\kappa=0$

    as expected. A straight line has no curvature. And yes, the radius is infinite as you'd expect for a straight line.

    you shouldn't plug $a$ until the very end and as seen in this example the curvature is constant.
    Thanks from USNAVY
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    May 2016
    From
    NYC
    Posts
    368
    Thanks
    5

    Re: Curvature & Radius of Curvature

    Quote Originally Posted by romsek View Post
    start with

    $r(t) = (t, 3t-2,0)$

    $r^\prime(t) = (1, 3, 0)$

    $\| r^\prime(t) \| = \sqrt{10}$

    $T(t) = \dfrac{r^\prime(t)}{\|r^\prime(t)\|} = \dfrac{(1,3,0)}{\sqrt{10}}$

    $T^\prime(t) = (0,0,0)$

    $\kappa=0$

    as expected. A straight line has no curvature. And yes, the radius is infinite as you'd expect for a straight line.

    you shouldn't plug $a$ until the very end and as seen in this example the curvature is constant.
    We can also say that the T'(t) =(0,0,0) = zero vector, right?

    If so, then the magnitude of T'(t) = 0.

    K = 0/root{10}

    K = 0

    Because K = 0, the radius of curvature R = 1/K is undefined.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,691
    Thanks
    2394

    Re: Curvature & Radius of Curvature

    Quote Originally Posted by USNAVY View Post
    We can also say that the T'(t) =(0,0,0) = zero vector, right?

    If so, then the magnitude of T'(t) = 0.

    K = 0/root{10}

    K = 0

    Because K = 0, the radius of curvature R = 1/K is undefined.
    yes, I didn't state the completely obvious. Apologies.
    Thanks from USNAVY
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member
    Joined
    May 2016
    From
    NYC
    Posts
    368
    Thanks
    5

    Re: Curvature & Radius of Curvature

    Very good.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,250
    Thanks
    2838

    Re: Curvature & Radius of Curvature

    Did you not realize that this is a straight line so the curvature is 0 without having to do any computation?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Senior Member
    Joined
    May 2016
    From
    NYC
    Posts
    368
    Thanks
    5

    Re: Curvature & Radius of Curvature

    Quote Originally Posted by HallsofIvy View Post
    Did you not realize that this is a straight line so the curvature is 0 without having to do any computation?
    No.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Radius of Curvature
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Sep 5th 2012, 02:40 PM
  2. Radius of Curvature
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Apr 11th 2010, 12:26 PM
  3. What is the instantaneous radius of curvature?
    Posted in the Math Topics Forum
    Replies: 0
    Last Post: Nov 11th 2009, 03:37 AM
  4. Radius of curvature problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Mar 14th 2009, 02:48 AM
  5. radius of curvature
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Feb 8th 2007, 11:07 AM

/mathhelpforum @mathhelpforum