Circle Ellipse Tangent

[mdb]

I'm trying to figure out an equation for the distance between the midpoints of a circle and an ellipse while they are tangent to each other as the ellipse rotates. The ellipse and the circle are centered on the same vertical axis and can only move relative to each other along that axis. See attached picture.

Thanks...

[attachment=0:7cz2es13]Circle Ellipse.jpg[/attachment:7cz2es13]

 

[DrMike]

I think this might be quite messy. What do you need it for?

 

[Denis]

No, needs not be messy:
http://math2.org/mmb/thread/42377

 

[Deleted member 4993]

I do not agree with the solution provided in the web-site.

The parametric equation x = aSin? and y = bCos? will work with respect to the center of the ellipse. In this case the ellipse is rotating hence the x and y axes (related to the parametric equation) is rotating. The angle ? - I think is referring to the stationary x-axis attached to the circle.

Unlike two tangent circles - the center of the circle, the center of the ellipse and the point of tangency, need not be co-linear.

I might be missing something - but I vote with DrMike - it could be messy....

 

[mdb]

DrMike-
This is an engineering problem involving a grinding wheel (circle) that needs to polish the outside edge of an elliptical part (ellipse).

Denis-
The equation provided on the link doesn't seem to give the correct answer. When I plug values of a=2.5, b=1.25, R=3, and Theta = 30 deg, I get a value for the distance between centers of 5.25. The actual value should be closer to 4.5 (see attached diagram).

Subhotosh Khan-
I think you are correct.

 

[galactus]

Using the values you have in your last post, I get 4.65. That seems reasonable.

 

[mdb]

R+ ((a*cos(theta))^2 + (b*sin(theta))^2)^0.5

R = 3

(a*cos(theta))^2 = (2.5 * cos(30))^2 = 4.69

(b*sin(theta))^2 = (1.25 * sin(30))^2 = 0.39

D = 3 + (4.69 + 0.39)^0.5 = 5.25

 

[galactus]

By Jove you're right. Sorry, I entered in a wrong number.

 

[Denis]

The equation provided on the link doesn't seem to give the correct answer. When I plug values of a=2.5, b=1.25, R=3, and Theta = 30 deg, I get a value for the distance between centers of 5.25. The actual value should be closer to 4.5

Do you not agree with this (given at the other site):

theta = 90: 4.25
theta = 60: ~4.65359
theta = 30: ~5.25347
theta = 00: 5.5

 

[mdb]

No-

The values should be:

theta = 0: 4.25
theta = 30: ~4.5
theta = 60: ~5.14
theta = 90: 5.5

 

[Denis]

No-

The values should be:

theta = 0: 4.25
theta = 30: ~4.5
theta = 60: ~5.14
theta = 90: 5.5
/quote]
BUT BUT you're using the FULL lengths of the major and minor axis;
shouldn't your 4.25 be 3 + 1.25/2, your 5.5 be 3 + 2.5/2 ?

 

[mdb]

semi major axis (a) = 2.5
semi minor axis (b) = 1.25

 

[Denis]

OK...my oversight: I can now see that in your last diagram.
And 2a, 2b are standards for the axes: I should have remembered that! Thanks, Mdb.