a function $f: X \rightarrow Y$ between two topological spaces is a homeomorphism if:
$f$ is a bijection (i.e. one-to-one and onto) $f$ 是一个双射，i.e.单射且满射
$f$ is a continuous function
the inverse function $f^{-1}$ is continuous
e.g. 咖啡杯和甜甜圈这两个拓扑空间同胚
manifold流形， chart坐标卡，parameterization参数化
流形是一个拓扑空间
2-manifold(two-dimensional manifold)二维流形的定义：
a subset $\mathcal{S}$ of $\mathbb{R}^3$ is a 2-manifold if
for every point $\boldsymbol{p}\in\mathcal{S}$ there is an open set $V$ in $\mathbb{R}^2$ and an open set $W$ in $\mathbb{R}^3$ containing $\boldsymbol{p}$ such that $U=\mathcal{S} \cap W$ is homeomorphic to $V$
A chart for a topological spaceM is a homeomorphism $\varphi$ from an open subset U of M to an open subset of a Euclidean space. 一个拓扑空间的坐标卡，就是这个拓扑空间的一个开子集到一个欧式空间的开子集的同胚
the chart is traditionally recorded as the ordered pair $(U,\varphi)$ 坐标卡一般用有序对$(U,\varphi)$表示
image of an element
If x is a member of X, then the image of x under f, denoted f(x), is the value of f when applied to x.
image of a subset
the image of subset $A \subseteq X$ under f, denoted $f[A]$ is the subset of Y which can be defined as: $f[A] = \lbrace f(x) \vert x \in A\rbrace$ when there is no risk of confusion, $f[A]$ is simply written as $f(A)$
inverse image / preimage原像： the preimage or inverse image of set $B \subseteq Y$ under f , denoted by $f^{-1}[B]$, is the subset of X defined by $ f^{-1}[B]=\lbrace x\in X \vert f(x) \in B\rbrace $
atlas图册
图册是一族坐标卡，一族同胚，一族函数，一族映射
a index family $\lbrace (U_\alpha,\varphi_{\alpha}):\alpha \in I \rbrace$ of charts on M which coversM (that is, $\cup_{\alpha \in I} U_{\alpha}=M$)
an algebraic(polynomial) surface $f(x,y,z)=0$ of degree n that has an $(n-1)-fold$ point (a point of multiplicity n-1) 一个有n-1重点的n次代数曲面(线)即为一个monoidal curve
monoidal surfaces include:
quadrics 二次曲面
cubic surface with a double point 有二重点的三次曲面
quartic surface with a triple point 有三重点的四次曲面
etc.
parameterization: implicit -> parametric
==本质==
这里的参数化，和拓扑学中的参数化，是一回事：从一个欧式空间到一个拓扑空间的同胚(映射)
curve
Noether’s theorem A plane algebraic curve f(x,y)=0 possesses a rational paramtric form iff f has genus 0
surface
没有已知的通用工具来判断一个给定的implicit surface是否可以被参数化，以及if so, how
monoidal curves/ surfaces can be parameterized in a simple manner
参数化时常用方式：parameterization using a pencil of lines
pencil
in geometry, a pencil is a family of geometric objects with a common property
a pencil of lines through a point p is a set of lines each containing p
Bezout's Theorem 贝组定理
Let $\mathcal{C}$ and $\mathcal{D}$ be projective plane curves without common components and degrees n and m, respectively. Then $n \cdot m = \underset{P \in \mathcal{C} \cap \mathcal{D}}{\sum} mult_P(\mathcal{C},\mathcal{D})$
因此，对于monoidal curves/surfaces来说，只要让a pencil of lines共同经过那个(n-1)重点，则这些直线一定与曲线/曲面还剩一个交点，如此便可实现参数化
让直线束经过二次曲线的一个"一重点"来参数化
让直线束经过三次曲线的一个二重点来参数化
a rational parameterization of a surface in affine (x,y,z)-space corresponds to a polynomial parameterization of the same surface in projective (w,x,y,z)-space 一个曲面在(x,y,z)-仿射空间的有理参数化 对应 同样曲面在(w,x,y,z)-射影空间的多项式参数化
implicitization: parametric -> implicit
all curves and surfaces with a rational parametric form can be converted to implicit form
