Results 1 to 3 of 3

Math Help - How to graph a lemniscate in polar coordinates?

  1. #1
    Newbie
    Joined
    May 2010
    Posts
    1

    How to graph a lemniscate in polar coordinates?

    Hey guys,

    The basic form of a lemniscate is r^2=sin(theta)      or r^2=cos(theta)
    if taking root on both sides of the equation,than it will become
      r= sqrt(cost theta)

    now since square root only has positive values ,than how can a lemniscate be drawn if the value of cos theta becomes negative inside the square root??
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member AllanCuz's Avatar
    Joined
    Apr 2010
    From
    Canada
    Posts
    384
    Thanks
    4
    Quote Originally Posted by Maha211 View Post
    Hey guys,

    The basic form of a lemniscate is r^2=sin(theta) or r^2=cos(theta)
    if taking root on both sides of the equation,than it will become
     r= sqrt(cost theta)

    now since square root only has positive values ,than how can a lemniscate be drawn if the value of cos theta becomes negative inside the square root??
    You should note that the basic form is r^2 = a^2 cos 2 \theta

    This equation, as you mentioned, is only defined for certain angles.

     - \frac{ \pi }{4} \le \theta \le \frac{ \pi }{4} and  \frac{ 3 \pi }{4} \le \pi \le \frac{ 5 \pi }{4} this should be sufficient to draw the lemniscate.

    Alternatively, you could turn back into x and y and get

     x = \frac{ acost } { 1 + sin^2 t }

    and

     y = \frac{ asintcost }{ 1 + sin^2 t }

    More can be found here: Lemniscate -- from Wolfram MathWorld
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,861
    Thanks
    742
    Hello, Maha211!

    The basic form of a lemniscate is: . r^2\:=\:\sin\theta\,\text{ or }\,r^2\:=\:\cos\theta

    If taking root on both sides of the equation, than it will become: .   r\:=\: \sqrt{\cos\theta} . . . . no

    Now since square root only has positive values, then how can a lemniscate be drawn
    if the value of \cos\theta becomes negative inside the square root?

    When we take the square root, we get: . r \;=\:{\color{red}\pm}\sqrt{\cos\theta}


    You are right: . \cos\theta \:\geq\:0

    If \cos\theta is negative, \tfrac{\pi}{2}<\theta <\tfrac{3\pi}{2}, there are no points to plot.

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 10
    Last Post: September 24th 2010, 05:33 AM
  2. Polar Coordinates
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 18th 2009, 05:04 PM
  3. Replies: 6
    Last Post: February 4th 2009, 01:12 AM
  4. Replies: 9
    Last Post: March 21st 2008, 10:27 PM
  5. Polar Coordinates and Polar Graphs
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: October 2nd 2007, 05:44 PM

Search Tags


/mathhelpforum @mathhelpforum