📂Geometry

Eigen Decomposition

Definition¹

• Let us assume that $M \subset \mathbb{R}^{3}$ and $\epsilon >0$ are given. Let’s call $d$ the Euclidean distance. The set defined as follows is called the $\epsilon -$ neighborhood of point $P \in M$.

  $$N_{p} := \left\{ Q \in M : d(P,Q) < \epsilon \right\}$$

• Let’s say $M \subset \mathbb{R}^{3}$. Given a function $g : M \to \R^{2}$. For all open sets $U \subset \R^{2}$ containing $g(P)$, if there exists a neighborhood $N_{p}$ of $P$ that satisfies $g(N) \subset U$ for $\epsilon -$, then $g$ is said to be continuous at $P \in M$.

• If the inverse function $\mathbf{x}^{-1} : \mathbf{x}(U) \to U$ of the simple surface $\mathbf{x} : U \to \R^{3}$ is continuous at every point in the domain $\mathbf{x}(U)$, then $\mathbf{x}$ is called a proper patch.

Explanation

The $\epsilon -$ neighborhood is the same as the intersection of an open ball with a radius of $\epsilon$ on $M$ and $\mathbb{R}^{3}$.

Continuity is defined in the same way as continuity in topology, only limiting the domain to surfaces.

Saying that $\mathbf{x}$ is a proper patch means the same thing as saying that $U$ and $\mathbf{x}(U)$ are homeomorphic. It’s like saying in topology that a doughnut and a cup are the same shape, implying that $U$ can be stretched or bent (without cutting or making holes) to match $\mathbf{x}(U)$.