content_image/343486.jpg content_image/343485.jpg content_image/343484.jpg content_image/343483.jpg content_image/343480.jpg content_image/343481.jpg content_image/343479.jpg content_image/343478.jpg content_image/343477.jpg content_image/343476.jpg content_image/343475.jpg content_image/343474.jpg
so Eq. (B1) gives 

with 

which obviously obeys Eq. (B4). ¿From Eqs. (B6) and (B5) we find 

<img src='content_image/343486.jpg'>

<img src='content_image/343485.jpg'>

<img src='content_image/343484.jpg'>

<img src='content_image/343483.jpg'>

and from Eq. (B2b) conclude that t ∗ ( s ) = s , as it should be for the circle. For the functions obeying Eq. (B7), we always have Φ$_{1}$( s ) = iΦ$_{2}$( s ) and 

Equation (B2b) can then be rewritten as 

<img src='content_image/343480.jpg'>

<img src='content_image/343481.jpg'>

<img src='content_image/343479.jpg'>

For the given analytic function C ( z ) it determines the reparametrizing function t ∗ ( s ). For the circle and ellipse it is more convenient to use the unit-disk parametrization, when 

For the ellipse C ( ω ) is given by the conformal map (where ν is the same as s in Eq. (B26) of [6]) 

and from Eq. (B11) we obtain 

<img src='content_image/343478.jpg'>

<img src='content_image/343477.jpg'>

<img src='content_image/343476.jpg'>

The final step is to extract d θ ∗ ( σ ) / d σ from Eq. (B10). Remembering that 

we find 

which for ω = e i σ reproduces Eq. (B32) of [6]. 

<img src='content_image/343475.jpg'>

<img src='content_image/343474.jpg'>

