Could someone explain this hypocycloid question?

[Seung Tai Kang]

Homework Statement

A circle of C of radius b rolls on the inside of a larger circle of radius a centered at the origin. Let P be a fixed point on the smaller circle, with initial position at the point (a, 0).

Homework Equations

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

The Attempt at a Solution

I understand part of it. Exactly what I don't understand is how thata of the big circle is related to phi of the smaller circle. Some other explanations say the arclength of the smaller circle is b(θ+Φ) when I think it should be just bΦ. Why add theta to phi all of sudden? Shouldn't the distance the smaller circle travel be bΦ and not b(θ+Φ)?

 

[issacnewton]

You have to think about the distance traveled by the point $P$ as the inner circle rolls inside the outer circle. Using the distance formula, we can say that $a\theta = b\phi$ And your x coordinate formula seems wrong. If you put $\theta = 0$, you should get $x=a$

 

[Seung Tai Kang]

That is what I think too. But Stewart's Precalc textbook and all the other guys on the internet says they are the answers.
And aθ = b (Φ +θ) not aθ=b(Φ). That is what people say. And that is exactly what throws me off.

 

[issacnewton]

Any source on internet for these answers ? Any answer should match with the initial conditions.

 

[Seung Tai Kang]

Any source on internet for these answers ? Any answer should match with the initial conditions.

The equations are given in the textbook. So I think they are legitimate enough. Maybe you can look it up yourself.
Below is the link.
https://math.stackexchange.com/ques...ations-for-hypocycloid-and-epicycloid/1128217

 

[issacnewton]

The coordinates (for point $P$) which I got for hypocycloid are $$x = (a-b)\cos\theta + b \cos\left[\frac{(a-b)}{b} \theta\right] $$ $$y = (a-b)\sin\theta - b \sin\left[\frac{(a-b)}{b} \theta\right]$$ But the way I have measured $\phi$ is different from the way its measured by the answer given by Upax. I have measured $\phi$ from the point of contact of the two circles and Upax has measured $\phi$ from the horizontal line. So my relationship between $\theta$ and $\phi$ differs from Upax's relationship. But it does not affect the final answers. And your answers don't match even Upax's answer for the hypocycloid.

 

[Seung Tai Kang]

yes. I can see that I wrongly put - sign instead of + sign, if that is what you mean.

But could you elaborate more on Upax starting from the horizontal line, and you starting from the pont of contact?
Isn't starting point same as the horizontal line(x-axis)?

 

[issacnewton]

I am busy right now, I will explain it in few hours

 

[haruspex]

yes. I can see that I wrongly put - sign instead of + sign, if that is what you mean.

But could you elaborate more on Upax starting from the horizontal line, and you starting from the pont of contact?
Isn't starting point same as the horizontal line(x-axis)?

If P is the point on the smaller circle initially at A, when the smaller circle has rotated φ on its own axis, as shown, it has moved φb clockwise around from the horizontal position. At the same time, the point of contact has moved tb anticlockwise from that horizontal. So the arc length from the new position of P to the new point f contact is b(t+φ).
Since it is roling contact, this arc length must equal ta.

 

[Seung Tai Kang]

If P is the point on the smaller circle initially at A, when the smaller circle has rotated φ on its own axis, as shown, it has moved φa clockwise around from the horizontal position. At the same time, the point of contact has moved tb anticlockwise from that horizontal. So the arc length from the new position of P to the new point f contact is b(t+φ).
Since it is roling contact, this arc length must equal ta.

Wait... are your a referring to the bigger circle? and b to the smaller circle?
I still don't get it... shouldn't φ incorporate all the angle difference since it is the distance that the smaller angle travelled... why suddenly add t... Is it because it is the inside of the circle... still it is DISTANCE the small circle travelled.
Does it mean then if it were a flat line the angle difference the circle would travel is φ, and that the circle would travel less distance...

 

[haruspex]

Wait... are your a referring to the bigger circle? and b to the smaller circle?

Sorry, I did write that wrongly. I will edit my previous post.

 

[haruspex]

shouldn't φ incorporate all the angle difference since it is the distance that the smaller angle travelled...

For reference, label as S the point on the upper small circle directly to the right of its centre B. So the angle SBA' is t.
Do you agree that the distance around the arc A'SP must equal the distance around the large circle arc AA'?
Do you agree that the original position of the point P was at A, and that since the small circle has rotated angle φ on its own axis that point has moved, relative to the centre of the small circle, a distance bφ along its circumference, and that this equals the arc from S to P?

Sometimes it helps to break motions like this into two parts considered separately. Consider the small circle sliding along the inside of the large circle without rotating on its own axis. The distance along its circumference from its original point of contact to its new point of contact is tb. Now add the rotation φ about its own axis. That moves the original contact point a further φb from the new point of contact.

 

[Seung Tai Kang]

For reference, label as S the point on the upper small circle directly to the right of its centre B. So the angle SBA' is t.
Do you agree that the distance around the arc A'SP must equal the distance around the large circle arc AA'?
Do you agree that the original position of the point P was at A, and that since the small circle has rotated angle φ on its own axis that point has moved, relative to the centre of the small circle, a distance bφ along its circumference, and that this equals the arc from S to P?

Sometimes it helps to break motions like this into two parts considered separately. Consider the small circle sliding along the inside of the large circle without rotating on its own axis. The distance along its circumference from its original point of contact to its new point of contact is tb. Now add the rotation φ about its own axis. That moves the original contact point a further φb from the new point of contact.

You must be saying that the distance the small circle traveled is actually more than the angle has travelled... I kinda get a sense that it is because it's within the big circle...
((a−b)cos(t)+bcos(ϕ),(a−b)sin(t)−bsin(ϕ))
https://math.stackexchange.com/questions/1123421/parametric-equations-for-hypocycloid-and-epicycloid

But in the link above has above parameters in the process of deriving the answers same as yours. I wonder shouldn't it be φ+Φ not φ in the first place then?

 

[haruspex]

the distance the small circle travelled

How are you defining the distance the circle has travelled? Do you mean the angle through which it has rotated?

shouldn't it be φ+Φ

I don't know what you intend by that. It seems to be phi+phi, but with the two phi characters written in different fonts.

 

[Seung Tai Kang]

For reference, label as S the point on the upper small circle directly to the right of its centre B. So the angle SBA' is t.
Do you agree that the distance around the arc A'SP must equal the distance around the large circle arc AA'?
Do you agree that the original position of the point P was at A, and that since the small circle has rotated angle φ on its own axis that point has moved, relative to the centre of the small circle, a distance bφ along its circumference, and that this equals the arc from S to P?

Sometimes it helps to break motions like this into two parts considered separately. Consider the small circle sliding along the inside of the large circle without rotating on its own axis. The distance along its circumference from its original point of contact to its new point of contact is tb. Now add the rotation φ about its own axis. That moves the original contact point a further φb from the new point of contact.

Alright. I don't know if I understand, so verify me I am correct or wrong. The smaller circle travels, but the circumference of bigger circle that is curved inward wraps around the the smaller circle so point of contact is farthered by angle θ, which means the distance it traveled is not only contribited by φ but also contributed by θ, since the bigger circle kinda wraps around the smaller circle. This is how I understand it. Is this correct?

 

[haruspex]

the distance it traveled is not only contribited by φ but also contributed by θ

Not the distance traveled by some point, no.
The point on the smaller circle that was originally the point of contact has moved clockwise through an angle φ from its initial 3 o'clock position, so you could say it has traveled φb. At the same time, the locus of the point of contact has moved an angle θ anticlockwise from the 3 o'clock position. So the angular distance from where the original point of contact is now around to the new point of contact is φ+θ.

 

[Seung Tai Kang]

Not the distance traveled by some point, no.
The point on the smaller circle that was originally the point of contact has moved clockwise through an angle φ from its initial 3 o'clock position, so you could say it has traveled φb. At the same time, the locus of the point of contact has moved an angle θ anticlockwise from the 3 o'clock position. So the angular distance from where the original point of contact is now around to the new point of contact is φ+θ.

But if it were on the flat line then the distance it traveled would be represented only by φ. But since the line wraps around the circle as much as θ, then θ is added. Isn't this right? Maybe I meant to say the circle travel only by φ but the curved line also affects the distance it travels as much as θ.
By the way I meant to say φθ not φΦ. sorry about that

 

[haruspex]

But if it were on the flat line then the distance it traveled would be represented only by φ. But since the line wraps around the circle as much as θ, then θ is added. Isn't this right?

I think we're saying the same thing in different words.

 

[Seung Tai Kang]

I think we're saying the same thing in different words.

Thanks a lot. You have been of great help.