## 3D MATHEMATICS FOR SIMULATION AND VISUALISATION

 SHE Level 2 SCQF Credit Points 20.00 ECTS Credit Points 10.00 Module Code M2I623008 Module Leader Leonard Scott School School of Engineering and Built Environment Subject Software Engineering Trimester A (September start)

### Pre-Requisite Knowledge

Mathematics for Computer Games or equivalent.

### Summary of Content

This module completes the foundation to the mathematics for games technology started in 'Mathematics for Computer Games 1'. The module will cover the mathematical techniques needed by professionals in the simulation and visualization industries for the understanding and application of routines used in industry standard Physics engines and graphics libraries.

### Syllabus

Trigonometry - review of sine, cosine, tangent and inter-relationships; - application to 3D geometry. 3D Co-ordinate systems - cartesian; - spherical; - cylindrical; - quaternions; - homogeneous coordinate systems; - relative coordinate systems; - yaw, pitch and roll. 3D Geometry - vectors: as models of position, velocity, acceleration and force; - symmetry; - parametric representations of curves and surfaces; - polyhedrons; - light sources; - triangulations. 3D Transformations - translations, rotations, reflections, inversions and scaling. Projections - perspective; - axonometric. Kinematics - position, velocity and acceleration of a moving particle; - rigid body motion. Classical mechanics of particles - force; - Newton's laws; - equations of motion; - potential and kinetic energy. Mechanics of rigid body motion - introduction to multiple integration - inertia, angular momentum, torque; - roll, pitch and yaw; - Euler's theorem on rigid body motion. Numerical methods - Euler's method; - Runge-Kutta methods; - stability - convergence. Further forces - equations of aerodynamic and hydrodynamic lift; - viscosity; - elasticity. Collisions - conservation of energy and momentum; - coefficient of restitution.

### Learning Outcomes

On completion of this module, students should be able to: - solve games related problems in 3D geometry; - apply Newton's laws to modelling the motion of objects subject to a range of forces; - demonstrate and apply standard projections of 3D space onto a surface; - represent and manipulate objects with respect to different co-ordinate systems; - compare and contrast numerical algorithms

### Teaching / Learning Strategy

T he University 'Strategy for Learning' documentation has informed the learning and teaching strategy for this module. The module material will be introduced through lectures, while practical exercises, based on the lecture material, will be given to students for their laboratory sessions. Tutorials will be used to help explain and elaborate on both the lecture material and the laboratory exercises. The course material will be introduced through lectures, while practical exercises, based on the lecture material, will be given to students in tutorials. Tutorials will be used to help explain and elaborate on both the lecture material and the laboratory exercises.

Bourg, D. (2002) Physics for Games Developers, O'Reilly, (ISBN 0-596-00006-5) Lengyel, E., (2003), Mathematics for 3D Game Programming and Computer Graphics, 2Rev Ed, Charles River Media, (ISBN 1584502770) Plastock, R. et al (2000) Schaum's outline of Computer graphics, 2Rev Ed , Schaum Outline Series; (ISBN: 0071357815) Van Verth, J.M & Bishop L.M. (2004) Essential Mathematics for Games and Interactive Applications: A Programmer's Guide, The Morgan Kaufmann Series in Interactive 3d Technology (ISBN: 155860863X)

### Transferrable Skills

D1 Critical thinking and problem solving. D2 Cognitive/intellectual skills. D3 Knowledge and understanding in the context of the subject. D6 Independent working. D12 IT skills D13 Communication skills, written, oral and listening. D14 Numeracy skills.

### Module Structure

Activity Total Hours
Assessment (FT) 20.00
Lectures (FT) 36.00
Tutorials (FT) 24.00
Independent Learning (FT) 120.00

### Assessment Methods

Component Duration Weighting Threshold Description
Coursework 1 n/a 30.00 35% Practically Based Assignment
Exam (Exams Office) 2.00 70.00 35% Written Examination