Results 1 to 3 of 3

Math Help - Hypocycloid

  1. #1
    MHF Contributor
    Joined
    Jul 2008
    From
    NYC
    Posts
    1,489

    Hypocycloid

    The hypocloid is a curve defined by the parametric equations x(t) = cos^3 (t), y(t) = sin^3 (t), where
    0 < or = to t < or = to 2pi.

    Given the information above, find rectangular equations of the hypocycloid.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,915
    Thanks
    779
    Hello, magentarita!

    The hypocloid is a curve defined by the parametric equations: .**

    . . \begin{array}{ccc} x(t) &=& \cos^3\!t \\ y(t) &=& \sin^3\!t\end{array}\;\;\text{ where } 0 \leq  t \leq 2\pi

    Given the information above, find rectangular equations of the hypocycloid.

    **
    .There are many many hypocycloids. .This is just one of them.
    This requires a clever trick . . .


    We have: . \begin{array}{c}x\:=\:\cos^3\!t \\ y \:=\:\sin^3\!t \end{array}


    Raise both equations to the power \tfrac{2}{3}

    . . \begin{array}{ccccccc}x^{\frac{2}{3}} \;=\;\left(\cos^3\!t \right)^{\frac{2}{3}} & \Rightarrow & x^{\frac{2}{3}} \:=\:\cos^2\!t & {\color{blue}[1]}\\<br />
y^{\frac{2}{3}} \;=\;\left(\sin^3\!t\right)^{\frac{2}{3}} & \Rightarrow & y^{\frac{2}{3}} \:=\:\sin^2\!t & {\color{blue}[2]} \end{array}


    \text{Add {\color{blue}[1]} and {\color{blue}[2]}: }\;x^{\frac{2}{3}} + y^{\frac{2}{3}} \;=\;\underbrace{\cos^2\!t + \sin^2\!t}_{\text{This is 1}}

    Therefore: . x^{\frac{2}{3}} + y^{\frac{2}{3}} \;=\;1

    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Jul 2008
    From
    NYC
    Posts
    1,489

    Soroban...

    Quote Originally Posted by Soroban View Post
    Hello, magentarita!

    This requires a clever trick . . .


    We have: . \begin{array}{c}x\:=\:\cos^3\!t \\ y \:=\:\sin^3\!t \end{array}


    Raise both equations to the power \tfrac{2}{3}

    . . \begin{array}{ccccccc}x^{\frac{2}{3}} \;=\;\left(\cos^3\!t \right)^{\frac{2}{3}} & \Rightarrow & x^{\frac{2}{3}} \:=\:\cos^2\!t & {\color{blue}[1]}\\ y^{\frac{2}{3}} \;=\;\left(\sin^3\!t\right)^{\frac{2}{3}} & \Rightarrow & y^{\frac{2}{3}} \:=\:\sin^2\!t & {\color{blue}[2]} \end{array}" alt="
    y^{\frac{2}{3}} \;=\;\left(\sin^3\!t\right)^{\frac{2}{3}} & \Rightarrow & y^{\frac{2}{3}} \:=\:\sin^2\!t & {\color{blue}[2]} \end{array}" />


    \text{Add {\color{blue}[1]} and {\color{blue}[2]}: }\;x^{\frac{2}{3}} + y^{\frac{2}{3}} \;=\;\underbrace{\cos^2\!t + \sin^2\!t}_{\text{This is 1}}

    Therefore: . x^{\frac{2}{3}} + y^{\frac{2}{3}} \;=\;1

    Your replies are simply amazing.

    Thanks
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Hypocycloid
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 30th 2010, 09:07 AM
  2. Green's Theorem and a Hypocycloid
    Posted in the Calculus Forum
    Replies: 0
    Last Post: December 16th 2009, 11:05 AM
  3. Hypocycloid Problem
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: April 14th 2009, 11:38 PM
  4. Circumference of a Hypocycloid
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 30th 2009, 12:15 PM
  5. Hypocycloid. Oh so fun!
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 2nd 2008, 09:06 PM

Search Tags


/mathhelpforum @mathhelpforum