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: \(f[A] = \{f(x) \vert x \in A\}\) 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 \(f^{-1}[B]=\{x\in X \vert f(x) \in B\}\)
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 \(\{(U_\alpha,\varphi_{\alpha}):\alpha \in I \}\) 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
