The module introduces the study of curves and surfaces in Euclidean space. The geometry of curves involves the concept of torsion (twisting out of a plane) and curvature (twisting away from a line), and the geometry of surfaces involves the mean and gaussian curvatures (the bending away from a plane). The topics covered include arc length, Frenet frames, calculus on curves and surfaces, tangent vectors of curves and surfaces, geodesics on surfaces and their role as the shortest distance between two points, the normal vector of a surface, and integration along surfaces. Examples of surfaces are spheres, tori, ruled surfaces, surfaces of revolution, and minimal surfaces. Examples from mechanics, computer graphics and other areas are used for illustration. The module consists of five parts

* Planar curves: representation, arc-length, parameterisation, curvature

* Space curves: representation, arc-length, parameterisation, curvature, torsion

* 2D surfaces in 3D: representation, tangent space, normal space, metrics, calculus

* Paths in surfaces: geodesics, length and speed, parallel transport

* Curvature of surfaces: mean curvature, gaussian curvature, implications of curvature