Three. (20 points) Draw two perpendicular rays through the origin intersecting the ellipse $\frac{x^{2}}{2}+y^{2}=1$ at points $A$ and $B$. Let the midpoint of segment $AB$ be $P$. Find the equation of the locus of point $P$.

 untranslated part:
 
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This part is a note and not part of the problem statement, so it is not translated.
Let $P(x, y), A\left(x_{A}, y_{A}\right)$. Since $P$ is the midpoint of $A B$, we have $B\left(2 x-x_{A}, 2 y-y_{A}\right)$. Given that $A$ and $B$ lie on the ellipse and $O A \perp O B$, we have
$$
\begin{array}{l}
\left\{\begin{array}{l}
x_{A}=\sqrt{2} \cos \alpha, \\
y_{A}=\sin \alpha ;
\end{array}\right. \\
\left\{\begin{array}{l}
2 x-x_{A}=\sqrt{2} \cos \beta, \\
2 y-y_{A}=\sin \beta,
\end{array}\right. \\
\sqrt{2} \cos \alpha \cdot \sqrt{2} \cos \beta+\sin \alpha \cdot \sin \beta=0 .
\end{array}
$$

First, (3) can be rewritten as
$$
\cos \alpha \cdot \cos \beta+\cos (\alpha-\beta)=0.
$$

From (1) and (2), we have
$$
\left\{\begin{array}{l}
\sqrt{2} x=\cos \alpha+\cos \beta, \\
2 y=\sin \alpha+\sin \beta .
\end{array}\right.
$$

Squaring and adding these two equations, we get
$$
2 x^{2}+4 y^{2}=2+2 \cos (\alpha-\beta)=2-2 \cos \alpha \cdot \cos \beta,
$$

which implies $\cos \alpha \cdot \cos \beta=1-x^{2}-2 y^{2}$.
From (3), we have
$$
\begin{array}{l}
4 \cos ^{2} \alpha \cdot \cos ^{2} \beta=\sin ^{2} \alpha \cdot \sin ^{2} \beta \\
=\left(1-\cos ^{2} \alpha\right)\left(1-\cos ^{2} \beta\right) \\
=1-\left(\cos ^{2} \alpha+\cos ^{2} \beta\right)+\cos ^{2} \alpha \cdot \cos ^{2} \beta \\
=1-(\cos \alpha+\cos \beta)^{2}+2 \cos \alpha \cdot \cos \beta+\cos ^{2} \alpha \cdot \cos ^{2} \beta \\
=1-2 x^{2}+2 \cos \alpha \cdot \cos \beta+(\cos \alpha \cdot \cos \beta)^{2} .
\end{array}
$$

Thus, $3(\cos \alpha \cdot \cos \beta)^{2}-2 \cos \alpha \cdot \cos \beta+2 x^{2}-1=0$.

Substituting (5) into (6), we get
$$
3\left(1-x^{2}-2 y^{2}\right)^{2}-2\left(1-x^{2}-2 y^{2}\right)+2 x^{2}-1=0,
$$

which simplifies to $3\left(x^{2}+2 y^{2}\right)^{2}-2 x^{2}-8 y^{2}=0$.
Removing the origin, we obtain the trajectory equation of point $P$.