Results 1 to 6 of 6
Like Tree2Thanks
  • 1 Post By TheEmptySet
  • 1 Post By TheEmptySet

Math Help - Curvature of an ellipse

  1. #1
    Newbie
    Joined
    Oct 2012
    From
    United States
    Posts
    12

    Curvature of an ellipse

    I'm asked to calculate the curvature of an ellipse:

    (x^2/a^2)+(y^2/b^2)=1

    The professor is setting x(t) = acos(t) and y(t) = bsin(t)

    but I don't know why.

    I think I can calculate the curvature from there.

    Any help is appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78

    Re: Curvature of an ellipse

    Quote Originally Posted by MajMinor View Post
    I'm asked to calculate the curvature of an ellipse:

    (x^2/a^2)+(y^2/b^2)=1

    The professor is setting x(t) = acos(t) and y(t) = bsin(t)

    but I don't know why.

    I think I can calculate the curvature from there.

    Any help is appreciated.

    You can use this defintion to to calculate the curvature or a parametrically defined curve.

    \kappa = \frac{|x'y''-y'x''|}{((x')^2+(y')^2)^{\frac{3}{2}}}

    This gives
    \kappa = \frac{|(-a\sin(t))(-b\sin(t))-(b\cos(t))(-a\cos(t))|}{((-a\sin(t))^2+(b\cos(t))^2)^{\frac{3}{2}}}=\frac{2ab  }{(a^2\sin(t)+b^2\cos(t))^{\frac{3}{2}}}
    Thanks from MajMinor
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2012
    From
    United States
    Posts
    12

    Re: Curvature of an ellipse

    Why is x(t)= acos(t) and y(t)=bsin(t), though?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78

    Re: Curvature of an ellipse

    Quote Originally Posted by MajMinor View Post
    Why is x(t)= acos(t) and y(t)=bsin(t), though?
    That is the parametric represention of an ellipse.

    Note that

    \frac{x}{a}=\cos(t) \quad \frac{y}{b}=\sin(t)

    Now if square both equations and add them you get

    \left( \frac{x}{a} \right)^2=\cos^2(t) \quad \left( \frac{y}{b}\right)^2=\sin^2(t)

    \ \frac{x^2}{a^2}+\frac{y^2}{b^2}=\cos^2(t) +\sin^2(t)=1
    Thanks from MajMinor
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Oct 2012
    From
    United States
    Posts
    12

    Re: Curvature of an ellipse

    Got it. Thanks a lot!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Sep 2013
    From
    Oregon
    Posts
    1

    Re: Curvature of an ellipse

    The empty set made a couple of small mistakes reducing the parameteric equation. I'm a newbie here, so I may get the markup wrong, but:



    which, because , simplifies to:



    Test: if a = b = r for a circle,



    For a circle, the curvature becomes 1/r as expected.

    Thank you "Mr. Set" for showing me how to solve this, and an opportunity to help.
    Last edited by keithl; September 24th 2013 at 11:10 AM. Reason: fixed sign error
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. curvature
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 20th 2009, 07:29 PM
  2. curvature
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 17th 2009, 01:26 PM
  3. Curvature
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: May 4th 2009, 04:29 PM
  4. Replies: 0
    Last Post: December 9th 2008, 06:41 PM
  5. Curvature
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 21st 2007, 06:13 AM

Search Tags


/mathhelpforum @mathhelpforum