The conformal mapping as a limiting case of mapping

摘要

Una trasformazione regolare differenziabile di una porzione S di una superficie su una porzione S di un'altra superficie è detta conforme se preserva localmente gli angoli corrispondentisi. Nelle rappresentazioni conformi la scala può dipendere dalla posizione ma è indipendente dalla direzione. In una trasformazione regolare differenziabile, in qualsiasi punto il grafico della scala, dipendente dalla direzione, su S è una curva pedale e su S l'ellisse corrispondente a quella curva pedale. Una rappresentazione conforme può essere considerata come un caso limite di trasformazione in cui le suddette curve si riducono a circonferenze uguali.%A regular differentiable mapping of a portion S of a surface onto a portion S of a surface is denominated conformal if it locally preserves the corresponding angles. In conformal mapping the scale may be dependent upon position but it is independent of direction. In regular differentiable mapping, at any point, the graph of the directional scale on S is a pedal curve and on S the corresponding ellipse to that pedal curve. Conformal mapping may be considered as a limiting case of mapping where the aforementioned curves are reduced to equal circles. Conformal mapping can associate portions of geodetic surfaces of reference and is facilitated by the adoption of planes as intermediate surfaces. Two portions of different planes, representing different maps of the same area derived from the portions of those geodetic surfaces by conformal mapping, are also conformally mapped to each other by means of an analytic function. This is the only kind of complex function of which the limit of variation at each point -equalling the scale of mapping- exists, is finite and independent of direction. In practice, as analytic functions, complex polynomials of various degrees relating known corresponding points and formed via recursive formulae are used, whose coefficients can be determined by Least Squares method employing real numbers only.