将点从欧氏 2 空间映射到庞加莱圆盘上
由于某种原因，似乎每个编写有关庞加莱圆盘的网页的人都只关心如何表示线和测量距离。
我想将 2D 点的集合（由欧几里得平面中的 x,y 坐标定义）变形到庞加莱圆盘上，但我不知道该算法应该是什么样的。此时我什至不知道是否可以在欧几里得2空间和庞加莱圆盘之间创建映射...
有任何指针吗？
善意， 大卫
For some reason it seems that everyone writing webpages about Poincare discs is only concerned with how to represent lines and measure distances.
I'd like to morph a collection of 2D points (as defined by x,y coordinates in the Euclidian plane) onto a Poincare disc, but I have no idea what the algorithm is supposed to be like. At this point I don't even know if it's possible to create a mapping between Euclidian 2-space and a Poincare disc...
Any pointers?
Goodwill,
David
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您将数据描述为点的集合。但根据您的评论，您希望使平面中的线仍然映射到磁盘中的线。您似乎想以某种方式保留空间的“结构”，这可能就是您使用术语“变形”的原因。我认为您需要一个共形地图。
磁盘和平面之间不共形双射。在半平面和磁盘，并且它保留“行”，但不是那种不幸的是，你想要的。
您说“我什至不知道是否可以创建映射”...有许多映射供您选择（请参阅 Unit Disk 页面为例），但没有一个具有您似乎想要的所有功能。
You describe your data as a collection of points. But from your comments, you want to make lines in the plane still map to lines in the disk. You seem to want to preserve the "structure" of the space somehow, which is probably why you use the term "morph". I think that you want a conformal map.
There is no conformal bijection between the disk and the plane. There is such a mapping between the half-plane and the disk, and it preserves "lines", but not the kind that you want, unfortunately.
You said "I don't even know if it's possible to create a mapping" ... there are a number of mappings for you to choose from (see the Unit Disk page for an example) but there are none with all the features you seem to want.
如果我理解正确的话，你在其他论坛上得到的答案是贝尔特拉米-克莱因模型。一旦你有了这个，你就可以得到 Poicare' 圆盘中的坐标，其中
p是 Poincare' 圆盘中的坐标向量（即你需要的）和b> 是贝尔特拉米-克莱因模型中的一个（即你从另一个答案中得到的）。If I understand everything correctly, the answer you get on the other forum is for the Beltrami–Klein model. Once you have that, you can get to the coordinates in the Poicare' disk with
Where
pis the vector of coordinates in the Poincare' disk (i.e. what you need) andbis the one in the Beltrami–Klein model (i.e. what you get from the other answer).