Submitted by FERNANDO MARQUES CALISTER null (fcalister@gmail.com) on 2016-10-02T01:33:35Z No. of bitstreams: 1 Representações dos Números Complexos e Transformações de Mobius.pdf: 617587 bytes, checksum: e9bbc4361adf7d335874ab0c7f3fdc3f (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-10-05T16:28:55Z (GMT) No. of bitstreams: 1 calister_fm_me_sjrp.pdf: 617587 bytes, checksum: e9bbc4361adf7d335874ab0c7f3fdc3f (MD5) / Made available in DSpace on 2016-10-05T16:28:55Z (GMT). No. of bitstreams: 1 calister_fm_me_sjrp.pdf: 617587 bytes, checksum: e9bbc4361adf7d335874ab0c7f3fdc3f (MD5) Previous issue date: 2016-08-19 / O objetivo deste trabalho é ampliar os conhecimentos sobre números complexos já adquiridos no ensino médio. Diversas formas de representação e propriedades operatórias são abordadas. Para este fim, primeiramente, os números complexos são definidos a partir do conceito de matrizes quadradas de ordem 2, e portanto, serão definidos como pares ordenados de números reais. Na sequência, a partir da apresentação geométrica dos conceitos e operações, é estudado o plano complexo estendido, as Transformações de Möbius e a Projeção Estereográfica. / The objective of this paper is to extend the concepts of complex numbers already acquired in high school. Many forms of representation and operative properties are used. For that, first, the complex numbers are defined from the concept of square matrices of order 2, and will therefore be defined as ordered pairs of real numbers. Following, from the geometric presentation of concepts and operations, it is studied the extended complex plane, the Möbius Transformations and the Stereographic Projection.

The purpose of this thesis is to study the effects of hyperbolic transformations on the cubic that is determined by locus of centroids of the equilateral triangles in H² whose base coincides with the line y=0, and whose common vertex is at the origin. The derivation of the formulas within this work are based on the Poincaré disk model of H², where H² is understood to mean the hyperbolic plane. The thesis explores the properties of both the untransformed cubic (the original locus of centroids) and the transformed cubic (the original cubic taken under a linear fractional transformation).