# Ellipse in relation to a circle

• Sep 29th 2006, 01:55 AM
Optiminimal
Ellipse in relation to a circle
An ellipse can be formed in several ways, here are two examples:

- A circle viewed from an angle looks like an ellipse.

- While a circle is a horisontal cut through a cone, an ellipse can be taken from a cone with an angular cut.

Now, assume that a point moves around the circle at constant speed. What speed will the same point have on an ellipse which has the relationship to the circle as described above (for example, it is the circular movement viewed from an angle)?

Let's think about speed as angular speed relative to the center of the ellipse.
• Sep 29th 2006, 07:30 AM
JakeD
Quote:

Originally Posted by Optiminimal
An ellipse can be formed in several ways, here are two examples:

- A circle viewed from an angle looks like an ellipse.

- While a circle is a horisontal cut through a cone, an ellipse can be taken from a cone with an angular cut.

Now, assume that a point moves around the circle at constant speed. What speed will the same point have on an ellipse which has the relationship to the circle as described above (for example, it is the circular movement viewed from an angle)?

Let's think about speed as angular speed relative to the center of the ellipse.

I suggest you view this problem in terms of 3D graphics and rotation matrices.

Think of the circle as centered at the origin in the x-y plane. In 3-space the x-y plane has z coordinate zero. Suppose you are viewing the circle from point (x,y,z) = (0,0,10). Now tilt and/or rotate the x-y plane in 3-space. Keeping your viewing point the same, the tilt/rotation will transform the circle into an ellipse. That transformation can be described using a 3x3 rotation matrix M. Coordinates on the circle will be transformed to coordinates on the ellipse by multiplying by the matrix M. I am not familiar with the physics, but I would think since the transformation is linear, the transformations of physical quantities such as velocity should be straightforward to calculate.

Here are two links that will introduce you to 3D graphics in a more general way: Brian Wyvill. Wikipedia on 3D projection.

I think these may be helpful to you because your other post on this subject says you have a photo of a tilted clock face. When you have a photo taken by a camera (or drawn by computer graphics) there is another transformation that comes into play: the perspective transform. That is, objects that are farther away from the camera appear smaller than objects nearer the camera. This means a tilted circle will not transform exactly into an ellipse. Ignoring the perspective transformation will introduce another set of errors into your calculations besides measurement errors. The above links describe the transformations involved and how to apply them using matrix methods.

The above links give you the full generality of 3D graphics, which can be daunting when you are first introduced to it. But the transformations turn out not so bad for this problem and if you are interested in this approach, I can help you work out an example.
• Sep 30th 2006, 05:05 AM
Optiminimal

• Sep 30th 2006, 12:17 PM
JakeD
Here are the transformations between points (X,Y) on the circle and points (X',Y') on the ellipse.

Assume a rotation matrix of this form is given.

Call this matrix M and call the 3 parameters A,B and C. Denote the elements of M by Mxx, Mxy, Mxz, etc.

Four other parameters are needed for a total of 7: Tx and Ty to locate the center of the ellipse, R for the radius of the circle, and D for the distance from the origin to the camera looking down on the X-Y plane. D is the parameter for the perspective transform I talked about above.

Given a point (X,Y) on the circle satisfying X^2 + Y^2 = R^2, the points (X',Y',Z') on the ellipse are

X' = (Mxx.X + Mxy.Y + Tx)/(D - Z')
Y' = (Myx.X + Myy.Y + Ty)/(D - Z')
Z' = Mzx.X + Mzy.Y.

In the 2D picture of the ellipse, the Z' coordinates are not measurable. But this transformation is invertible. So given a point (X',Y') on the 2D ellipse, the corresponding points (X,Y) on the circle are found by calculating

Z' = [Mxz(DX' - Tx) + Myz(DY' - Ty)]/[Mxz(X' - Tx) + Myz(Y' - Ty)]
X" = (D - Z')X' - Tx
Y" = (D - Z')Y' - Ty

and

X = Mxx.X" + Myx.Y" + Mzx.Z'
Y = Mxy.X" + Myy.Y" + Mzy.Z'.

X" and Y" are just intermediate variables for calculating X and Y.

Now to the topic of your other posts. To determine the 7 parameters given a set of observed points (X',Y') on the ellipse, I would use an iterative numerical search method on the parameters to get the corresponding points (X,Y) to lie as closely as possible on a circle where X^2 + Y^2 = R^2. To do this I would search for the parameter values that minimize the sum of squares of the quantities X^2 + Y^2 - R^2.

PS: To get a good set of starting values for the parameters, I might use the other form of a rotation matrix: