$ U = 9 - 3 \cos(2t) + 8.5 \sin(2 t) $
$ V = 5.5 - 0.5 \cos(2t) + 6 \sin(2t) $
1)How to know the point (U, V) will form an ellipse?
2)How to draw it?
3)Is there any tool which can display this path?
Attempt:
I read the wiki: https://en.wikipedia.org/wiki/Ellipse
And so far, I only know how to draw the path on paper like:
$\frac{\left(x-x_\circ\right)^2}{a^2}+\frac{\left(y-y_\circ\right)^2}{b^2}=1\mathrm{~.}$
Shifted ellipse If the standard ellipse is shifted to have center ${\displaystyle \left(x_{\circ },\,y_{\circ }\right)}$
The axes are still parallel to the x- and y-axes.
If $r(t) = (U(t), V(t))$ then
$ r(t) = C + V_1 \cos(2 t) + V_2 \sin(2 t) $
where
$C = \begin{bmatrix} 9 \\ 5.5 \end{bmatrix}, \ V_1 = \begin{bmatrix} -3 \\ -0.5 \end{bmatrix}, \ V_2 = \begin{bmatrix} 8.5 \\ 6 \end{bmatrix} $
Define the unit vector
$ u = \begin{bmatrix} \cos(2t) \\ \sin(2t) \end{bmatrix} $
Then
$ r = C + A u $
where
$ A = \begin{bmatrix} -3 && 8.5 \\ -0.5 && 6 \end{bmatrix} $
Therefore,
$ u = A^{-1} (r - C) $
And since $u^T u = 1 $ , then this means that
$ (r - C)^T G (r - c) = 1 $
where $G = A^{-T} A^{-1} $
Numerically, we have
$G = \begin{bmatrix} 0.191735537 && -0.27768595 \\ -0.27768595 && 0.429752066 \end{bmatrix} $
Diagonalizing $G$, we can express it as follows:
$ G = R D R^T $
where
$ D = \begin{bmatrix} 0.008630488 && 0 \\ 0 && 0.612857115 \end{bmatrix} $
and
$ R = \begin{bmatrix} 0.834841092 && -0.550491009 \\ 0.550491009 && 8.834841092 \end{bmatrix} $
If we define the vector $w = R^T (r - C) $ then
$w^T D w = 1 $
expanding this expression, we obtain,
$ \dfrac{w_1^2}{a^2} + \dfrac{w_2^2}{b^2} = 1 $
where
$ a^2 = \dfrac{1 }{0.008630488} = 115.8682983 $
$ b^2 = \dfrac{1}{ 0.612857115} = 1.631701706 $
So vector $w$ traces an ellipse of semi-major axis length equal to
$a = \sqrt{115.8682983} = 10.76421378 $
and a semi-minor axis length equal to
$ b = \sqrt{ 1.631701706 } = 1.277380799 $
The $w$-ellipse is in standard position, with its major axis parallel to the horizontal axis, and its minor axis parallel to the vertical axis. From the above definition of $w$, we have that
$ r = C + R w $
Hence, $r$ is on a translated and rotated ellipse in the (U, V) plane. The translation from the origin is by the vector $C$, and the rotation is counter clockwise by an angle of $\theta = \tan^{-1} \left( \dfrac{ 0.550491009}{0.834841092} \right) = 33.40070474^\circ $
Using all this information, we can now attempt to plot the ellipse that represents $r = (U,V)$.
The major semi-axis is of length $10.76421378$ and is rotated by $\theta =33.40070474^\circ$ degrees in the counter clockwise direction. Therefore, the vector representing it is
$ X_1 = 10.76421378 (\cos \theta, \sin \theta) $
And the minor semi-axis is of length $1.277380799$ and it is rotated from the vertical direction by the same angle $\theta$ (counter clockwise). Hence, the vector representing the minor semi-axis is
$ X_2 = 1.277380799 (- \sin \theta, \cos \theta ) $
Now, we have a different parametrization of the ellipse that we started with, and it is as follows
$ r = C + X_1 \cos(s) + X_2 \sin(s), \ s \in [0, 2\pi] $
From this expression we can plot the ellipse, by substituting uniformly spaced values of $s$ over $[0, 2 \pi ]$, then connecting the dots.
This what I did using Excel, and obtained the following graph