Möbius Transformation 莫比乌斯变换

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中文	English
莫比乌斯变换	Möbius Transformation / Fractional Linear Transformation
变换行列式	Determinant：Δ=ad−bc（核心判别式）
非退化	Non-degenerate
退化	Degenerate

标准表达式

$\boldsymbol{f(x)=\frac{ax+b}{cx+d}},\quad a,b,c,d\in\mathbb{R}$

When $\boldsymbol{\Delta=ad-bc \neq 0}$, the linear polynomials $ax+b$ and $cx+d$ are linearly independent, not scalar multiples of each other. The function cannot reduce to a constant; its graph is a hyperbola, a bijection with another Möbius transformation as its inverse.

$\boldsymbol{\Delta=ad-bc=0}$ means coefficient vectors $(a,b)$ and $(c,d)$ are linearly dependent, so there exists a constant k such that $a=kc,\;b=kd$. Substitute into the rational function: $f(x)=\frac{k(cx+d)}{cx+d}=k$ The expression collapses into a constant horizontal line $y=k$. It loses all features of a Möbius transformation → degenerate case.

行列式不等于 0，一次多项式 $ax+b$ 和 $cx+d$ 线性无关、不成正比例，无法约分消去分母。

函数图像是双曲线，是双射函数，存在同类型分式形式的反函数，是标准有效的莫比乌斯变换。

只有这类非退化双曲线才有可能图像关于 $y=x$ 对称。

$a=1,d=-1 \Rightarrow bc=-1$，取 $b=1,c=-1$： $y=\frac{x+1}{-x-1}=-1$

degenerate to horizontal line (y=-1), invalid for the problem.

2024 AMC 12A Problems/Problem 25

A graph is $\textit{symmetric}$ about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers $(a,b,c,d)$ , where $|a|,|b|,|c|,|d|\le5$ and $c$ and $d$ are not both $0$ , is the graph of $y=\frac{ax+b}{cx+d}$ symmetric about the line $y=x$ ?