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本帖最后由 hbghlyj 于 2023-3-26 21:23 编辑 定义: $T:\Bbb C\to\Bbb C,T(z)=az+b,(a,b\in\Bbb C)$ 称为顺相似变换
Fundamental Theorem of Directly Similar Figures
Let $F_0$ and $F_1$ denote two directly similar figures in the plane, where $P_1$ in $F_1$ corresponds to $P_0$ in $F_0$ under the given similarity. Let $r$ in $(0,1)$, and define $F_r=\{(1-r)P_0+rP_1:P_0\in F_0,P_1\in F_1\}$. Then $F_r$ is also directly similar to $F_0$.
证明:
设 $c$ 为相似中心(不动点), 则变换 $T_r(z)=(1-r)z+rT(z)$ 也是顺相似变换, 满足 $T_r(P_0)=(1-r)P_0+rP_1$, 且 $T_r(c)=(1-r)c+rT(c)=(1-r)c+rc=c$, 故 $T_r$ 的相似中心为$c$. 证毕. |
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