. On the choice of S when the four points lie in a circle. The cross-ratio of four points in R 8 is thus defined up to complex conjugation. (For an equivalent definition involving quaternions, see [8], =-=[65]-=-.

## Introduction to Möbius Differential Geometry London

) The cross-ratio of four points in R 8 is invariant under Möbius transformations of R 8. Conversely, if p6, p7, p8, p9 ∈ R 8 have the same cross-ratio (up to complex conjugation) as p ′ 6, p′ 7,. And Laguerre geometry We first briefly review the connection between circular meshes in R 8 and Möbius geometry. For a more detailed treatment of Möbius geometry the interested reader is referred to =-=[65]-=-. Its standard point model is the sphere S 8. The passage from S 8 to R 8 is achieved by a stereograph.

9) of constrained Willmore surfaces we need the following quaternionic characterization of isothermic surfaces, see e. G. 7. As above one can prove th.

If a dual surface exists it is unique up to translation and a real scaling =-=[66]-=-. As we have seen any isothermic surface f: M → R9 admits a dual surface fd. The converse also holds as can be seen by reversing the above calculations: we obtain classical Darboux transforms f ] by. We refer the reader to =-=[H-J]-=- for further details. Let L N+7 be the (N+7)–dimensional Minkowski space endowed with a Lorentz scalar product of signature (+,.

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