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From: "Jan Simon" <matlab.THIS_YEAR@nMINUSsimon.de>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Normal vector of circles
Date: Sat, 23 Oct 2010 23:13:05 +0000 (UTC)
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Dear Paulo,

> For the first case, the plane of the great circle can be created arbitrarily due to the infinity of possible choices. There is no restriction with that. Then, the next task I will need to save the direction of the normal vector during the sphere rotations.

Now it is getting easier to answer:
You *cannot* calculate a normal vector of a great circle, if you do not define this circle. Did you really expect that there is a method to calculate n = [a,b,c] with well defined values a, b and c, as long as the orientation of the circle is "arbitrary"?!
As said already, a specific normal vector is needed to define the circle - not the other way around.

Then you want to rotate the sphere? How? Do you apply a rotation matrix? Then you can apply this matrix to the normal vector also. "Save the direction of the normal vector during the sphere rotation" means the same problem: The rotation itself means the change of the local coordinates. You do not have to save the directions, the "rotation" contains all information you can get from the system!

Let me translate your problem to the 1D case. Then your question is equivalent to:
1. How can I calculate the length of an arbitrary distance. Answer: This depends on the distance between the end points.
2. How does this length change after a transformation of the coordinates. Answer: If the transformation dilates the coordinates by a factor x, the length is dilated by a factor x.

As far as I can understand your problem, you can omit the sphere, the great circle and the normals. All you need is inlcuded in the "rotation" already.

Kind regards, Jan