On successful completion of this module, students will be able to:

justify with reasoned logical argument basic properties
of triangles, circles and polygons in the Euclidean plane;

describe measures of curvature, both extrinsic and
intrinsic, applicable to smooth surfaces in three-dimensional
Euclidean space;

identify, and justify with reasoned logical argument,
significant geometric features and properties of the
hyperbolic plane, and the corresponding features
and properties of the disk that represent them in the
Poincaré disk model of the hyperbolic plane.

Module Content

Aims to introduce complex variable theory and reach the residue theorem, applications of that to integral evaluation.

Euclidean geometry: an exploration of Euclid's
Elements of Geometry, based on editions
freely available online, with detailed discussion of the
definitions, axioms and propositions of Book 1, followed
by discussion of a selection of significant results
contained in Books 2—6.

Curvature of surfaces: a discussion of some of the
principal results of C.F. Gauss's General
Investigations of Curved Surfaces.

The hyperbolic plane: the Poincaré disk model
of the hyperbolic plane, geodesics and curvature,
homogeneity, the representation of isometries
as Möbius transformations.

Module Prerequisite

None

Assessment Detail

This module will be examined in a 2 hour examination in Trinity term.