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Math Help - Weird Parametric Problem

  1. #1
    Junior Member
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    Weird Parametric Problem

    This isn't a homework problem that's due, it's an application problem I found in the nether-regions of the questions for the section in my book on parametric equations, and I thought it would be fun to try. I couldn't figure it out, so I thought I'd post it here. I think it requires a bit of geometry which may be why I'm so confused.. It reads:

    "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
    Question 2: If b=1, show that we can simplify the parametric equations to:
    x = acos^3(theta)
    y = asin^3(theta)
    [Only show that this holds for a = 3]."
    ----------------------------------------

    ...Yeah. So, I drew out a picture, looks a little something like this:



    Where the circle C is at an arbitrary point in the original circle to make the image more clear.


    I'm still wrapping my brain over how to even begin either of the questions.

    For Q1, I think I have to find angle theta (the angle between a-b and the x-axis in the first quadrant). But the fact that there are no numbers is really confusing me. And I don't even know why they gave you point P.

    For Q2, I tried plugging in b=1 and a=3, but I got as far as..

    x = 2cos(theta) + cos(2theta) [y is basically the same so I'm not going to type that out]

    And I can't figure out how to get that to look like the equations it said it had to look like.

    soo... that's my dilemma right now. any help or insight is appreciated.
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  2. #2
    MHF Contributor Calculus26's Avatar
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    answer

    See Hypocycloid


    Based on your question I developed an animation to show the generation
    on my web site Calculus Animations

    Go to the special topics page--I am also experimenting with various ratios of a to b .
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