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本帖最后由 hbghlyj 于 2024-3-31 17:04 编辑 直线$L$
圆$C$
透视变换$φ:C\to L$
$L$上的射影变换$g∈\operatorname{Stab}L$
$C$上的射影变换$f=φgφ^{-1}∈\operatorname{Stab}C$
$L$和$C$都(通过有理参数化)同构于$\mathbb{RP}^1$,考虑$PGL(2,\mathbb R)$在$L$和$C$上的作用:
用$\sim$表示在$\mathbb R$上相似,二阶实矩阵有3个real canonical form.
$\mathbb R^2$的等距即旋转和反射,容易想象,以下都用某个等距(和它导出的映射)作例子.
圆上的等距只有关于圆心旋转、关于通过圆心的直线反射.
直线上的等距只有平移、关于垂线反射.
因此,把这4个例子归入3种real canonical form下:
在2个不动点的情况,直线和圆都易举例:$f$为反射和它导出的$g=φ^{-1}fφ$、$g$为反射和它导出的$f=φgφ^{-1}$
在1个不动点的情况,直线较易举例: $g$为旋转和它导出的$f=φgφ^{-1}$, 但不存在$f$为等距的例子.
在0个不动点的情况,圆较易举例: $f$为旋转和它导出的$g=φ^{-1}fφ$, 但不存在$g$为等距的例子.
2个不动点$φ\sim\pmatrix{\lambda_1\\&\lambda_2}$
$f$为等距:
$f$为关于通过圆心的一条直线反射.
$f$的2个不动点是$C$与对称轴的交点.
$g$为等距:
$g$为关于$L$的一条垂线反射.
$g$的2个不动点是$L$与对称轴的交点和$L$的无穷远点.
1个不动点$φ\sim\pmatrix{\lambda_1&1\\0&\lambda_1}$
$g$为等距:
$g$为沿$L$的平移
按$f=φgφ^{-1}$导出的$f$如图所示:
size(14cm); real lsf=0.5; pen ds=blue; real xmin=-2.5272093023255935,xmax=4.548604651162797,ymin=-0.16096083231334624,ymax=2.578806609547126;
pair A_1=(1.,0.), A_2=(0.,0.), A_3=(-1.,0.), A_4=(-2.,0.), B_2=(0.652635372434094,0.6822889572423139), B_3=(0.46005463166603383,1.0224284901719294), B_4=(0.41193377495183836,1.269763462590303), B_1=(1.176177704473976,0.3632155475501865);
draw((xmin,0)--(xmax,0),linewidth(1.5)+black);
draw(circle((1.4097674418604613,1.335550795593634),1.),linewidth(1.5)+black); draw(A_4--(2.028764104876982,2.1209444110339506),linewidth(1.5)+black); draw((2.028764104876982,2.1209444110339506)--A_3,linewidth(1.5)+black); draw((2.028764104876982,2.1209444110339506)--A_2,linewidth(1.5)+black); draw((2.028764104876982,2.1209444110339506)--A_1,linewidth(1.5)+black); defaultpen(fontsize(10)+red);
dot((2.028764104876982,2.1209444110339506),ds); label("$B$",(1.9853488372093016,2.2104345165238692),NE*lsf); dot(A_1,ds); label("$A_1$",(1.033720930232555,0.0769461444308405),S*2); dot(A_2,ds); label("$A_2$",(0.02837209302325015,0.0769461444308405),S*2); dot(A_3,ds); label("$A_3$",(-0.9693023255814035,0.0769461444308405),S*2); dot(A_4,ds); label("$A_4$",(-1.9669767441860573,0.0769461444308405),S*2); dot(B_2,linewidth(4.pt)+ds); label("$B_2$",(0.6806976744186006,0.7446205630354933),NE*lsf); dot(B_3,linewidth(4.pt)+ds); label("$B_3$",(0.48883720930232105,1.0822949816401453),NE*lsf); dot(B_4,linewidth(4.pt)+ds); label("$B_4$",(0.442790697674414,1.3278763769889832),NE*lsf); dot(B_1,linewidth(4.pt)+ds); label("$B_1$",(1.2102325581395321,0.4222949816401437),NE*lsf);
$g$的1个不动点是$A=[0:0:1]$
$A_1,A_2,A_3,A_4\dots$为$L$上的等距离的点.
平移$g:A_1\mapsto A_2\mapsto A_3\mapsto A_4\dots$趋于$[0:0:1]$
$f$的1个不动点是$B=φ(A)$
按$f=φgφ^{-1}$导出的$f:B_1\mapsto B_2\mapsto B_3\mapsto B_4\dots$趋于$B$,但走得“越来越慢”.
0个不动点(不动点是虚点) $φ\sim\pmatrix{a&b\\-b&a}$
旋转$f:B_1\mapsto B_2\mapsto B_3\mapsto B_4\dots$
按$g=φ^{-1}fφ$导出的$g:A_1\mapsto A_2\mapsto A_3\mapsto A_4\dots$从最左端开始走得“越来越慢”后又“越来越快”.
size(11cm); real lsf=1.5; pen ds=blue; real xmin=-2.0744186046511803,xmax=2.5455813953488304,ymin=-0.29142594859241616,ymax=2.402294981640148;
pair B=(2.028764104876982,2.1209444110339506), B_4= (0.41208, 1.26753) ,B_3=(0.5797599631254696,0.7777985340617042), B_2=(0.969836010710851,0.4375194287212648), B_1=(1.4777913299978387,0.3378671029120939), A_1=(1.3733900377016934,0.), A_2=(0.6946223126344564,0.), A_3=(-0.2593397330625953,0.), A_4=(-1.989069078144931,0.);
draw((xmin,0)--(xmax,0),linewidth(1.5)+black);
draw(circle((1.4097674418604613,1.335550795593634),1.),linewidth(1.5)+black);
draw(B--A_1,linewidth(1.5)+black); draw(A_2--B,linewidth(1.5)+black); draw(A_3--B,linewidth(1.5)+black); draw(A_4--B,linewidth(1.5)+black); defaultpen(fontsize(10)+red);
dot(B,ds); label("$B$",B,NE*lsf); dot((0.4120837491789213,1.2675269074562565),ds); label("$B_{4}$",B_4,NE*lsf); dot(B_3,ds); label("$B_3$",B_3,NE*lsf); dot(B_2,ds); label("$B_2$",B_2,NE*lsf); dot(B_1,ds); label("$B_1$",B_1,NE*lsf); dot(A_1,linewidth(4.pt)+ds); label("$A_1$",A_1,S*lsf); dot(A_2,linewidth(4.pt)+ds); label("$A_2$",A_2,S*lsf); dot(A_3,linewidth(4.pt)+ds); label("$A_3$",A_3,S*lsf); dot(A_4,linewidth(4.pt)+ds); label("$A_4$",A_4,S*lsf);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); |
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