Understanding the Hyperbolic Distance Formula: Deriving Log QA.PB/QB.PA

[Kevin_Axion]

I'm currently reading through Roger Penrose's book The Road to Reality and in his Hyperbolic Geometry discussion he introduces the concept of how to define the distance between two points. He defines a Conformal Representation of a Hyperbolic Space bounded by a circle and then he states there are two points A and B (within the hyperbolic space) and there is a hyperbolic line (an arc/Euclidean Circle) That intersects A and B and meets orthogonally to the bounding circle at points P and Q. Where QA etc are the Euclidean distances. The distance between A and B is thus defined by the formula,
log QA.PB/QB.PAWhere log is the natural logarithm and '.' is multiplication, assumingly.Firstly, can you please define a Euclidean distance and secondly can you help me understand how to derive this formula?

Thanks

 

[arkajad]

Firstly, can you please define a Euclidean distance and secondly can you help me understand how to derive this formula?
Thanks

On the plane Euclidean distance between points $$(x,y)$$ and $$(x',y')$$ is
$$\sqrt{(x-x')^2+(y-y')^2}$$

Or, if you set $$z=x+iy$$ it is $$|z-z'|$$

To derive this formula one has to know the assumptions. One possibility is to consder the group of fractional linear transformations of the form

$$z\mapsto\frac{az+b}{cz+d}$$

with real a,b,c,d and $$\det (\{a,b;c,d\})\neq 0$$. (search the net for $$PSL(2,R)$$) These 'conformal' transformations map the (closed) unit disk into itself. Then seek a distance formula that is invariant under these transformations.

 

[Kevin_Axion]

Thanks.