代数转几何:应用复数的locus
先上个匪夷所思的东西:
Fractional Linear Transformation
A fractional linear transformation is a function of the form $T(z)=\frac{(az + b)}{(cz + d)}$, where $a, b, c, d$ are complex constants and $ad-bc≠0$. These are also called Mobius transforms or bilinear transforms.
(1) Theorem: A linear fractional transformation maps lines and circles to lines and circles.
(2) Mapping of $\infty$. (the line is a considered as a circle with radius $\infty$)
(3) Composition of linear transformations are like multiplication of matrices.
(4) How to map three points on complex plane to another three points.
原理
就是每个复数可以看成复平面上的点,然后就可以固定不同的东西,比如夹角或长度或比例。
1.
如果对于一个复数$x$,如果$|x-y|$为定值,那么$y$是一个以$x$为圆心的圆。
2.
如果平面上有两个定点$A,B$,那么用$\angle ACB$就能确定由点$C$生成的两段弧。(如图,$\angle AC_1B=\angle AC_2B$,$C_1,C_2$为圆$O_1,O_2$上的点。)
那么,$\cos\angle ACB=\frac{\vec{CA}\cdot\vec{CB}}{|CA||CB|}$,如果把$A,B,C$当成复数$a,b,c$,那么$\frac{c-a}{c-b}$的辐角为定值。
例子
首先,可以思考一下复数的conic section。
椭圆 / ellipse: $|c-a|+|c-b|$为常数
双曲线 / hyperbola: $(|c-a|-|c-b|)^2$为常数