Results 1 to 6 of 6

Math Help - Osculating circle at given points

  1. #1
    Member mybrohshi5's Avatar
    Joined
    Sep 2009
    Posts
    230

    Osculating circle at given points

    Find the osculating circle at the given points.

     r(t) = <t,t^2> at  t=0

    I really have no idea where to even start this one. My professor has not gone over anything about osculating circles

    If i could get any tips on how to do this one that would be great

    Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,829
    Thanks
    123
    Quote Originally Posted by mybrohshi5 View Post
    Find the osculating circle at the given points.

     r(t) = <t,t^2> at  t=0

    I really have no idea where to even start this one. My professor has not gone over anything about osculating circles

    If i could get any tips on how to do this one that would be great

    Thank you!
    To start have a look here:

    Osculating Circle -- from Wolfram MathWorld

    or here:

    Osculating circle - Wikipedia, the free encyclopedia
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,326
    Thanks
    1298
    The "osculating circle" to the graph of a fuction, f, at a given point, is the circle that passes thorugh that point, whose center lies on the normal line to that graph at that point, and whose radius is 1 over the curvature.

    So the first thing you want to do is to identify the normal vector to the curve and find the curvature at that point.

    By the way, you might find it intersting to look up "osculate" in a dictionary.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member mybrohshi5's Avatar
    Joined
    Sep 2009
    Posts
    230
    Ok i found the normal vector using N(t) = \frac{T'(t)}{||T'(t)||} to the curve (it is very messy so i am not going to type it up).

    The curvature k is just k(t) = \frac{||T'(t)||}{||r'(t)||}

    Not sure where to go from here...
    Last edited by mybrohshi5; September 29th 2010 at 03:35 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member mybrohshi5's Avatar
    Joined
    Sep 2009
    Posts
    230
    so, this is what i have came up with....

    r(0) = <0,0> so the circle is at the origin
    N(0) = <0,1>
    k(0) = \frac{2}{\sqrt(5)}
    p = radius = \frac{\sqrt(5)}{2}

    i know the equation of the circle is (x-h)^2 + (y-k)^2 = r^2

    i am just not sure how to finish it up now
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Mar 2010
    Posts
    107
    To find the center of the osculating circle, you need a line to the center. You have a direction (N(t)) and you have the magnitude (the radius). Make a vector line out of it and find the point @ t = 1.

    for r^2, you can simply put 5/4.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Find the Osculating Circle
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 24th 2010, 12:23 AM
  2. determining radius of osculating circle
    Posted in the Geometry Forum
    Replies: 2
    Last Post: October 22nd 2009, 11:59 AM
  3. Osculating Circle for a Curve
    Posted in the Calculus Forum
    Replies: 4
    Last Post: June 27th 2009, 11:16 AM
  4. Vector Calculus (osculating circle)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 21st 2009, 04:29 AM
  5. osculating circle problem please!!!
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 26th 2009, 12:26 PM

Search Tags


/mathhelpforum @mathhelpforum