# CURVES AND SURFACES - 2022/3

Module code: MAT2047

## Module Overview

The module has three parts. The first part is the study of plane curves in 2D and space curves in 3D and their properties. The second part develops the definition of surfaces in 3D and their properties. The third part is the study of curves such as geodesics within surfaces in 3D.

### Module provider

Mathematics

BRODY Dorje (Maths)

## Overall student workload

Independent Learning Hours: 106

Lecture Hours: 11

Seminar Hours: 11

Guided Learning: 11

Captured Content: 11

Semester 1

## Prerequisites / Co-requisites

MAT1005 Vector Calculus and MAT1034 Linear Algebra

## Module content

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: length and speed, curves with zero geodesic curvature

• Curvature of surfaces: mean curvature, Gaussian curvature, implications of curvature

## Assessment pattern

Assessment type Unit of assessment Weighting
Online Scheduled Summative Class Test ONLINE TEST 20
Examination Online ONLINE EXAM 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of fundamental concepts and ability to develop and apply them to a new context.

• Subject knowledge through recall of key definitions, formulae and derivations.

• Analytical ability through the solution of unseen problems in the test and examination.

Thus, the summative assessment for this module consists of:

• One final examination worth 80% of the module mark.

• One in-semester test worth 20% of the module mark.

Formative assessment and feedback

Students receive written feedback via the marked in-semester tests.  The solutions to the in-semester tests are also reviewed in the lecture.  Two un-assessed courseworks are also given to the students for submission, and complete solutions to these are also provided.  In addition, verbal feedback is provided during lectures and office hours.

## Module aims

• The main aim of this lecture course is to introduce the differential geometry of curves and surfaces in three-dimensional Euclidean space.  A secondary aim is to show how diverse topics, such as vector calculus, linear algebra and differential equations are brought together to advance understanding of a new topic, which has implications for both pure mathematics and applied mathematics.

## Learning outcomes

 Attributes Developed 1 Demonstrate understanding of geometric properties of curves and surfaces, and how first year calculus and linear algebra underpins the new concepts. K 2 Interpret and apply basic concepts and theorems in linear algebra and vector analysis to the new topic. KCT 3 Develop formulae for curvature and apply them to a range of examples, using the theory developed in the module. KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• A detailed introduction to geometric properties of curves and surfaces, extending the ideas learned in calculus and linear algebra in the first year to a new context

• Experience (through demonstration) of the methods used to interpret, understand and solve problems in differential geometry

The learning and teaching methods include:

• 3 X 1 hour lectures per week for 11 weeks,

• Supplementary notes for topics of significant difficulty or special interest

•  Q+A opportunites for students

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.