Parametric Equation Trochoid Explanation

[brojas7]

The third line is the type of problem I have:
Derive the parametric equation for a circle with a distance 'b' from the circle with a radius 'r'.

So from the edge of the circle to the red dot is a distance 'b' but from the center to the edge of the circle is 'r'



I know the parametric equation of a circle is x = rθ+rsinθ and y=r+cosθ when it is only dealing with a distance r. but what about when it is further from the circle than just a distance r?



I think the answer is
x=rθ-bsinθ
y=r=bcosθ
but I don't understand why it is.

Can someone please explain why?

 

[HallsofIvy]

The circle has radius r and so every point on the circle can be written as variations on (cos(t), rsin(t)) with t depending on the angle at which you start. In particular, If we take t= 0 at the beginning with the radius pointing straight down, we have (p sin(t), -p cos(t)). If we want to take y= 0 at the straight line, x= 0 at the left end, we can write (p sin(t), -p cos(t)+ r).

 

[brojas7]

The circle has radius r and so every point on the circle can be written as variations on (cos(t), rsin(t)) with t depending on the angle at which you start. In particular, If we take t= 0 at the beginning with the radius pointing straight down, we have (p sin(t), -p cos(t)). If we want to take y= 0 at the straight line, x= 0 at the left end, we can write (p sin(t), -p cos(t)+ r).

Im sorry, I understand some if this but basically if thr radius is point straight up it should be [rsin (t) , bcos (t) +r]?

 

[haruspex]

Let the radius of the circle be r and the distance from centre to point be d. What are the coordinates of the circle's centre when it has turned through angle θ? What are the coordinates of the point relative to the centre of the circle?