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Math Help - Hypocycloid Problem

  1. #1
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    Hypocycloid Problem

    "A hypocycloid is a curve traced by a fixed point P on a circle C with radius b as C rolls on the inside of a circle with the center on the origin and radius a. If the initial point P is (a,0) and the parameter is theta, then the parametric equation is

    x = (a-b)cos(theta) + bcos((theta)(a-b)/b))
    y = (a-b)sin(theta) + bsin((theta)(a-b)/b))

    Question 1: Show that the parametric equation is correct"


    This is the picture I drew out to clarify..



    But I don't even know how to start explaining why it is correct... any suggestions?
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  2. #2
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    Quote Originally Posted by coolguy99 View Post
    "A hypocycloid is a curve traced by a fixed point P on a circle C with radius b as C rolls on the inside of a circle with the center on the origin and radius a. If the initial point P is (a,0) and the parameter is theta, then the parametric equation is

    x = (a-b)cos(theta) + bcos((theta)(a-b)/b))
    y = (a-b)sin(theta) + bsin((theta)(a-b)/b))

    Question 1: Show that the parametric equation is correct"

    ...
    1. In my opinion the given equation describes an epicycloid. If you want to get a hypocycloid the equation should read:
    <br />
\left|\begin{array}{l}x = (a-b) \cos(\theta) + b \cos\left(\dfrac{\theta(a-b)}{b} \right) \\ y = (a-b) \sin(\theta) {\color{red}\bold{-}} b \sin\left(\dfrac{\theta(a-b)}{b} \right) \end{array} \right.

    2. The distance in your sketch which is labeled a - b is actually a - 2b

    3. To start:
    - Determine where you can find \theta in your sketch.
    - Determine where you can find \left(\dfrac{\theta(a-b)}{b} \right) in your sketch.
    - Calculate the corresponding distances which form the coordinates of the points placed on the cycloid.
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