Results 1 to 2 of 2

Math Help - Find equation of curve, by rotation of ellipse ?

  1. #1
    Newbie
    Joined
    Dec 2010
    From
    United Kingdom
    Posts
    15

    Find equation of curve, by rotation of ellipse ?

    Find the equation of the curve obtained by rotation of the ellipse
    x^2 + 4y^2 = 4,
    by π/3 clock wise around the point (1, 2).
    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 abrahamtim View Post
    Find the equation of the curve obtained by rotation of the ellipse
    x^2 + 4y^2 = 4,
    by π/3 clock wise around the point (1, 2).
    A) I hope this reply doesn't come too late ...

    B) I'm not very practiced in mapping - so there must be a more elegant way to do your question.

    1. Since the rotation has to be done clockwise the angle of rotation is \alpha = -\frac{\pi}3

    2. I assume that you know how to rotate the coordinate system by:

    \left|\begin{array}{l}x = x' \cdot \cos(\alpha)-y' \cdot \sin(\alpha) \\ y = x' \cdot \sin(\alpha)+y' \cdot \cos(\alpha) \end{array}\right.

    where x', y' are the new coordinates after the rotation.

    3. With \sin\left(-\frac{\pi}3\right)=-\frac12 \cdot \sqrt{3} and \cos\left(-\frac{\pi}3\right)=\frac12
    you'll get:

    x'=\frac12 (x-y\cdot \sqrt{3})

    y'=\frac12 (x\cdot \sqrt{3} + y)

    Replace the variables x, y in the original equation by x', y' and you have rotated the ellipse around the origin.

    4. The new midpoint of the ellipse is M' \left(\frac12 - \sqrt{3},\ 1+\frac12 \cdot \sqrt{3}\right)

    5. Now translate the rotated ellipse by \vec t = \left\langle\frac12 - \sqrt{3},\ 1+\frac12 \cdot \sqrt{3}\right\rangle and you'll get:

    13x^2 + 6 \sqrt{3} \cdot xy + x(20 \sqrt{3} - 22) + 7y^2 + 2y(2 - 5 \sqrt{3}) = 12 \sqrt{3} - 25

    6. I've attached a sketch.
    Attached Thumbnails Attached Thumbnails Find equation of curve, by rotation of ellipse ?-gedrehteellipsen.png  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: May 29th 2011, 06:48 AM
  2. find the equation of the normal on the ellipse
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 30th 2009, 02:01 AM
  3. Find Equation of Ellipse
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: October 16th 2008, 06:11 PM
  4. Find the equation of an ellipse
    Posted in the Pre-Calculus Forum
    Replies: 12
    Last Post: March 18th 2008, 08:36 PM
  5. find the parametric equation of an ellipse?
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: October 28th 2007, 01:38 PM

Search Tags


/mathhelpforum @mathhelpforum