First, is there a relatively simple formula for the midpoint of two points a 1 and a 2 in the disk with respect to the hyperbolic geometry? That is, the point on the bisector of a 1 and a 2 which has minimal distance from a 1 and a 2 .Second, if B denotes the degree 2 finite Blaschke product with zeros a 1 and a 2 , is the critical point of B which is in the disk equal to the midpoint of a 1 and a 2 ?

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First, is there a relatively simple formula for the midpoint of two points       a    1   and       a    2   in the disk with respect to the hyperbolic geometry? That is, the point on the bisector of       a    1   and       a    2   which has minimal distance from       a    1   and       a    2   .Second, if B denotes the degree 2 finite Blaschke product with zeros       a    1   and       a    2   , is the critical point of B which is in the disk equal to the midpoint of       a    1   and       a    2   ?

seraphinod · Accepted Answer

Step 1Critical pointYes, it&#039;s at the midpoint. For Blaschke products B with zeros at   ±  a this is immediate from the formula   B  (  z  )  =  (      z    2    −      a    2    )      /    (  1  −                    a        ¯              2        z    2    ) .If zeros are elsewhere, say       z    1    ,      z    2   , then consider the composition   B  ∘      ϕ          −      1       where   ϕ is a Mobius transformation that sends the hyperbolic midpoint between       z    1    ,      z    2   to 0. This composition is a Blaschke product with zeros   ϕ  (      z    1    )  ,  ϕ  (      z    2    ) which satisfy   ϕ  (      z    1    )  +  ϕ  (      z    2    )  =  0 , reducing the problem to the previous case.MidpointThe connection with Blaschke product might be the easiest approach. Move one of points to zero, the other goes to  c  =                    a        1            −              a        2                    1      −              a        1                                          a            ¯                          2            The midpoint between 0 and c can be found by differentiating   z  (  z  −  c  )      /    (  1  −            c      ¯        z  ) . When c is between 0 and 1 (can be achieved by rotation), it&#039;s  d  =            1      −              1        −                  c          2                      c  Finally, the midpoint is obtained by undoing,  m  =            d      +              a        2                    1      +      m                                    a            ¯                          2

First, is there a relatively simple formula for the midpoint of two points a 1 </m

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