Response of a dynamical system forms a circle along a plane in 3D?

Question

I know the output of a linear system. The system is having one pair of eigenvalues on the imaginary axis and remaining on the left half of the plane. So the output of the system will be circle in 2D and in 3D it will be along a plane. How do one verifies the response forms a circle in 3D?(say I know by simulation) If so how do I find the center and the radius. We can assume we know a normal vector to the plane of the circle.
My situation is similar as Statement 2 of Theorem 3.2 in this Ren's paper. In the proof, I couldn't verify or see the radius as derived(pg. 3). Also it is not quite evident to me that the Euler axis is perpendicular to the circle.

Answer 1

If you know four different points on the circle, lets call them $p_1,p_2,p_3$ and $p_4$. Lets call the center of the circle $c$.

Then we know that $$|p_1-c|^2=R^2,$$ $$|p_2-c|^2=R^2,$$ $$|p_3-c|^2=R^2,$$ $$|p_4-c|^2=R^2.$$

Which is the same as the following equations:

$$|p_2-c|^2=|p_1-c|^2,$$ $$|p_3-c|^2=|p_1-c|^2,$$ $$|p_4-c|^2=|p_1-c|^2.$$

This system has 3 unknowns in 3D as $c$ and 3 equations. After having found $c$ use one of the previous equations to calculate $R$.

If the data is not exact because its numerical. You will need as many points as possible on the curve lets call them $p_i$, with $i=1,\dots,n$. The aim will be to determine $c$ and $R$ in such a way that a specific criterion function is minimized. A very common method is to minimize the squared error. For every equations we get an individual error $\varepsilon_i$. So each point estimation gives the residual error

$$\varepsilon_i=|p_i-c|^2-R^2.$$

We define the function

$$F(c,R)=\sum_i\varepsilon^2_i=\sum_{i}\left((p_i-c)^2-R^2 \right)^2.$$

Calculate the gradient of this function with respect to $c_1,c_2,c_3,R$ and set it to zero. Solve the nonlinear system with an algorithm like Newton-Raphson. To get your nonlinear Least Squares approximation.