GEOMETRIC CLASSI?CATION OF M¨OBIUS TRANSFORMATIONS

• Main Author: RAINAL IHSAN , IDEN
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/22487
• Institution: Institut Teknologi Bandung

Mo¨bius transformation is a geometric transformation that maps a set of circles and lines back to be a set of circles and lines in extended complex plane (C?). Mo¨bius transformations is a composition from a?ne transformations and inversion that maps C? one-to-one and onto itself. A set of M¨obius transformations is a group with respect to composition operation. <br /> <br /> Mo¨bius transformations can be classi?ed based on many ?xed points. A nonidentity Mo¨bius transformations has the most two ?xed points. In extended complex plane, every Mo¨bius transformations has ?xed point. Classi?cation of Mo¨bius transformations can be observed from two cases, that is has one ?xed point or two ?xed points. <br /> <br /> The group of M¨obius transformations is isomorphic with group of PSL2(C). By using the concept about the value of the trace from a connected matrix, Mo¨bius transformations can be classi?ed into conjugacy classes that de?ne its geometric classi?cations. According to the value of the trace from a connected matrix, Mo¨bius transformations can be classi?ed into M¨obius transformations of parabolic, elliptic, hyperbolic, or loxodormic. <br />