| Popis: |
The current paper discusses some new results about conformal polynomial surface parametrizations. A new theorem is proved: Given a conformal polynomial surface parametrization of any degree it must be harmonic on each component. As a first geometrical application, the next theorem is shown: Every surface that admits a conformal polynomial parametrization must be a minimal surface. This is not the case for rational conformal polynomial parametrizations, where the conformal condition doesn’t imply that polynomial components must be harmonic. A new general theorem is established for conformal polynomial parametrizations of m-dimensional surfaces on the euclidean space Rn: The only conformal polynomial parametrizations of a m-dimensional surfaces, in Rn, with m > 2 and n ≥ m, must be formed by linear polynomials, i.e. the surface parametrization must be an affine transformation of the usual cartesian framework |
