Parametrization of Hypocycloid

[Alfredoz]

Homework Statement

Hi,
Refer to: http://press.princeton.edu/books/maor/chapter_7.pdf ( Page 2 & 3)

How do we derive the x-coordinate to be (R-r)cosθ + r cos[(R-r)/r]θ

Homework Equations

Let 'r' & 'R' be radius of small & big circles respectively; Let the angle by which a point on the small circle rotates about its center, C be β

The Attempt at a Solution

x-coordinate of point P relative to O = (R-r)cosθ + r cosβ = ?

The text states that the arcs of the fixed & moving circles that come into contact must be of EQUAL LENGTH. May I know how do we prove it mathematically?

 

[maCrobo]

Maybe the picture on the second page (FIG. 40) can be helpful.
You should imagine that in a given period of time the little circle has moved across a certain path, which is the arc of the big circle, but in the same period of time the little circle has rounded as much as it needs to go across a path whose length is the arc length. It means that the length of the arc of the little circle that has touched the great circle is exactly the same of the length of the path covered by the little circle. In particular the arc of the little circle is described by r(ø+θ). Since this last is equal to the arc of the path, which is Rθ, we have r(ø+θ)=Rθ. From here you can take ø and substitute it in the former equation.

 

[Alfredoz]

Maybe the picture on the second page (FIG. 40) can be helpful.
You should imagine that in a given period of time the little circle has moved across a certain path, which is the arc of the big circle, but in the same period of time the little circle has rounded as much as it needs to go across a path whose length is the arc length. It means that the length of the arc of the little circle that has touched the great circle is exactly the same of the length of the path covered by the little circle. In particular the arc of the little circle is described by r(ø+θ). Since this last is equal to the arc of the path, which is Rθ, we have r(ø+θ)=Rθ. From here you can take ø and substitute it in the former equation.

Dear Sir,

Thanks for reply. Yes, I've referred to FIG 40. Sorry, I find it hard to visualize. How do we prove it mathematically instead?

 

[maCrobo]

Well, the x-coordinate is (R-r)cosθ + r cosø.
Let's focus on the part rcosø. Since, Rθ=r(ø+θ), we have ø=(Rθ-rθ)/r=(R-r)θ/r, so we get to x-coordinate= (R-r)cosθ+ r cos[(R-r)θ/r] because ø=(R-r)θ/r.
This sort of proof comes from intuition. There isn't anything more to say about it, it's just that, unfortunately.

Anyway, you can also imagine the wheel of a bike. Suppose this situation: you have a plane surface full of red paint and a new completely white wheel. You put the wheel on the surface, so only the point of contact has been colored by the red paint on the surface. Then you push the wheel to make it move and then stop it after half a turn, so you have half the circumference of the wheel red colored. It means your wheel has covered a distance whose length is half the circumference of the wheel, right? That's it!
In this way r(ø+θ) is the same of the length of arc covered, which is Rθ, so you have Rθ=r(ø+θ) (see the FIG.40).

 

[Alfredoz]

Well, the x-coordinate is (R-r)cosθ + r cosø.
Let's focus on the part rcosø. Since, Rθ=r(ø+θ), we have ø=(Rθ-rθ)/r=(R-r)θ/r, so we get to x-coordinate= (R-r)cosθ+ r cos[(R-r)θ/r] because ø=(R-r)θ/r.
This sort of proof comes from intuition. There isn't anything more to say about it, it's just that, unfortunately.

Anyway, you can also imagine the wheel of a bike. Suppose this situation: you have a plane surface full of red paint and a new completely white wheel. You put the wheel on the surface, so only the point of contact has been colored by the red paint on the surface. Then you push the wheel to make it move and then stop it after half a turn, so you have half the circumference of the wheel red colored. It means your wheel has covered a distance whose length is half the circumference of the wheel, right? That's it!
In this way the length of the path covered by the little circle is the same of r(ø+θ) (see the FIG.40).

Appreciate your painstaking efforts to explain:) I'm an undergraduate majoring in Math and yet I can't even visualize:(

 

[maCrobo]

Mmh...

Take in account my paint example. Take a completely white wheel, put it on the red surface, make it roll for half a turn so that it makes an angle of PI, then you want to measure the distance covered, how? You take the wheel and measure how much circumference is red colored, exactly πR (where R in this case is the radius of the wheel). That's to say the distance covered is exactly the portion of circumference that had touched the ground.

 

[Alfredoz]

Mmh...

Take in account my paint example. Take a completely white wheel, put on the red surface, make it roll for half a turn, so that it makes an angle of PI, then you want to measure the distance covered, how to do? You take the wheel and measure how much circumference is red colored. That's to say that the distance covered is exactly the portion of circumference that had touched the ground.

Yes, I understand your analogy. Thank you:) May I know what is your major in university?

 

[maCrobo]

I'm an undergraduate attending courses in Engineering Sciences, but I noticed a lot of math is based just on intuitions ;)

 

[Alfredoz]

I'm an undergraduate attending courses in Engineering Sciences, but I noticed a lot of math is based just on intuitions ;)

I agree. I believe it's possible to frame our intuitions mathematically. For eg. Limits & Continuity(Rolle's Theorem). Sometimes, confusion arises when we try to question our intuition.