本文探讨了111维舞台上的Poincaré盘模型，介绍了如何通过复数定义超曲面距离，如Poincaré距离，并揭示了在双曲几何中直线视觉扭曲的现象。还通过映射和极限性质展示了距离属性。
111-dimensional stage: (−1,1)(-1,1)(−1,1).
If you get a complex stage, you will get Poincaré disc.
I guess its stage is {∣z∣⩽1∣z∈C}\{|z|\leqslant1|z\in\mathbb C\}{∣z∣⩽1∣z∈C}.
If you define routine of shortest distance straight line, then you will find that straight line is not visually “straight” in hyperbolic geometry.
Structure
Distance
Definition:
d(x1,x2)=∣∫x1x2dx1−x2∣.
d(x_1,x_2)=|\int_{x_1}^{x_2}\frac{\text dx}{1-x^2}|.
d(x1,x2)=∣∫x1x21−x2dx∣.
This is called hyperbolic distance or Poincaré distance.
We can prove it satisfies those 3 axioms of distance.
Properties:
1, Define: ωi=xi−x01−xix0\omega_i=\dfrac{x_i-x_0}{1-x_ix_0}ωi=1−xix0xi−x0.
This mapping keeps order: (x1−x2)(ω1−ω2)>0,x1≠x2(x_1-x_2)(\omega_1-\omega_2)>0,x_1\neq x_2(x1−x2)(ω1−ω2)>0,x1=x2.
It keeps distance: d(x1,x2)=d(ω1,ω2)d(x_1,x_2)=d(\omega_1,\omega_2)d(x1,x2)=d(ω1,ω2).
2, limx→±1d(x,x0)=+∞\lim\limits_{x\rightarrow\pm 1}d(x,x_0)=+\inftyx→±1limd(x,x0)=+∞.
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