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.
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 .