---- > [!definition] Definition. ([[ellipsoid]]) > An **ellipsoid** in $\mathbb{R}^{3}$ is given by the set $\left\{ (x,y,z) \in \mathbb{R}^{3}: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} \right\}=1.$ > \ > Ellipsoids are [[differentiable Euclidean submanifold|regular surfaces]], as witnessed by the global [[coordinate patch]] $\begin{align} \alpha: & (0, \pi) \times (0, 2\pi) \subset \mathbb{R}^{2} \to \mathbb{R}^{3} \\ (u,v) \mapsto & (a \sin u \cos v, b \sin u \sin v, c \cos u). \end{align}$ ^063ebd > [!justification] > We need to show that $\im \alpha = \{ \text{ellipsoid} \}$:![[CleanShot 2024-03-04 at [email protected]]] > \ > The curves $u=\text{const.}$ on the ellipsoid look like like ellipse slices across it. Explicitly, once we fix as constants $s:=\sin u$ and $k:\cos u$, we get $\alpha(u,v)=(as \cos v, bs \sin v, ck)$, which is just the parameterization of an ellipse in $\mathbb{R}^{3}$, parallel to, and hovering a distance $ck$ above, the $xy$-plane. ^47f2a7 ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```