This is my first question here, so hopefully I will be able to explain my problem in a coherent way. My ultimate question is: do you see any way I can simplify the following system so I can have a more intuitive solution to it? Let me explain in details what I mean:
I want to characterize $x,y,z$ that solve the following system of non-linear equations:
\begin{align} \Delta x &= \frac{\theta\Delta z\Delta y}{\Delta y-\theta\Delta z}\tag{EQ1}\\ \Delta x+\Delta y+\Delta z&=\overline{Y}\tag{EQ2}\\ \theta\left(\Delta z\right)^{2}+\left(\Delta y\right)^{2}&=\overline{U}\tag{EQ3} \end{align}
where
\begin{align} \Delta x &=x-\overline{x}\\ \Delta y &=y-\overline{y}\\ \Delta z &=z-\overline{z} \end{align}
and $0<\theta<1$, $\overline{Y}$, $\overline{U}$, $\overline{x}$, $\overline{y}$ and $\overline{z}$ are parameters.
The way I've approached this was to parameterize (EQ2): \begin{align} \Delta y+\Delta z&=\alpha \end{align} such that $\alpha=\overline{Y}-\Delta x$
For $\theta=0.8$, $\overline{x}=\overline{y}=\overline{z}=0.5$ and $\overline{Y}=-0.5$ and $\overline{U}=0.2$ this parametric system would look like the following (I apologize I don't have reputation to embed the photo here):
I can characterize $\Delta y$ and $\Delta z$ as a function of $\alpha$: \begin{align} \Delta y=\frac{\theta\alpha \pm\sqrt{\overline{U}(1+\theta)-\theta\alpha^{2}}}{1+\theta}\\ \Delta z=\frac{\alpha \mp\sqrt{\overline{U}(1+\theta)-\theta\alpha^{2}}}{1+\theta} \end{align}
My problem arrives in the characterization of $\alpha$ using (EQ1). Although we can show such an $\alpha$ exists, it is far from an intuitive closed-form characterization, as $\alpha$ solves: \begin{align} \pm\frac{(\overline{Y}-\alpha)(1-\theta)\sqrt{\overline{U}(1+\theta)-\theta\alpha^2}}{\theta\alpha\pm\sqrt{\overline{U}(1+\theta)-\theta\alpha^2}}&=\theta\left(\alpha\mp\sqrt{\overline{U}(1+\theta)-\theta\alpha^2}\right) \end{align}
Am I using the correct approach to this problem? Or do you see anyway I could simplify this? Thank you!
I will reason in terms of the $\Delta$ rather than the non-centered $x,y,z$.
The first equation, which can be written $$\Delta x\Delta y=\theta\Delta z(\Delta x+\Delta y)$$
is that of a cone with apex at the origin. The second is a plane and the third and elliptic cylindre of vertical axis. The latter two define an ellipse, which you can represent by the parametric equations
$$\Delta y=\sqrt{\overline U}\frac{1-u^2}{1+u^2},\\ \Delta z=\sqrt{\frac{\overline U}\theta}\frac{2u}{1+u^2},\\ \Delta x=\overline Y-\Delta y-\Delta z.$$
(The rational expressions can be obtained from the trigonometric representation.)
Plugging these expressions in the first equation, you will get a quartic equation in $u$, which can have zero, two or four solutions (intersection points of the ellipse and the cone).
It is unlikely that there will be a simple expression for these solutions, even though there are only three independent parameters and the quartic equation is not fully general.