Mathematics

Sustainable Development Goals (SDG)

• 4. Quality education

Objectives

The general objective of this subject is to offer students basic knowledge of mathematics that is useful for professionals in the multimedia world.

Based on the usual needs in the design and creation of multimedia resources, some of the basic mathematical tools in this profession are worked on. For example, in animations objects move, rotate, move away and look smaller and smaller, etc. How is this achieved? The proportions of the object must be preserved, which must always look the same. This is where one of the most important mathematical tools appears: matrix algebra. On the other hand, if we see objects moving in an animation, how can we create the instructions that allow them to move appropriately? Here functions and their properties are crucial. When creating attractive geometric objects or natural landscapes, symmetries and fractals are involved at all times.

Although there are design or animation programs that allow us to do many of these operations automatically, the multimedia engineer often has to program and create new processes, so if he does not know the basic tools to do so, he is limited to the standard tools offered by the market. Therefore, it is essential to know the mathematical bases that have allowed the creation of the multimedia world.

Finally, whenever necessary, programming (Python) is used to clarify numerical concepts.

Learning outcomes

• RA1. Analyzes and solves problems analytically or numerically.
• LO2. Identify and correctly use mathematical terminology, notation and methods.
• LO3. Demonstrates skills for critical reflection in processes linked to the exercise of the profession.
• LO4. Apply scientific and technological knowledge to solve problems in the professional field.

Competencies

General skills

• Combine scientific knowledge with technical skill and technological resources to deal with difficulties in professional practice.

Specific skills

• Solve problems at the appropriate level of abstraction for each situation, using scientific and technological skill and knowledge.

Basic skills

• Students have demonstrated knowledge and understanding in a field of study that builds on general secondary education with the support of advanced textbooks and knowledge of the latest advances in this field of study.

Core skills

• Be a critical thinker before knowledge in all its dimensions. Show intellectual, cultural and scientific curiosity and a commitment to professional rigor and quality.

Content

Part A. Mathematics and movement

1. Coordinate systems and vector calculus
2. Matrix calculus. Movements and transformations of objects in space
   • Matrices. Operations and properties
   • Determinants
   • Application of matrix calculus to geometry: affine transformations
3. Geometry and systems of linear equations
   • Geometric interpretation of a system of linear equations
   • Systems resolution and discussion
   • Distances and relative positions of points, lines and planes
4. Rotations in space and quaternions
5. Physical functions and simulations
   • Function concept
   • Continuous and differentiable functions
   • Parameterization of multidimensional functions and motion
   • Basic interpolation and environment construction

Part B. Mathematics and design

6. Design and proportion
   • What is proportion. Rational and irrational proportions. Golden ratio. Thales' theorems. Fibonacci sequence and golden ratio
   • Recursion
   • Homothecia
   • Successions and progressions
7. Symmetry
   • What is symmetry. Types of symmetries
   • Symmetry group of a figure
   • Tessellations
8. Fractal geometry and natural structures
   • Algorithms and recursion
   • Scale symmetries and fractals
   • Fractals in structural modeling

Evaluation

The evaluation is based on different activities:

• Subject monitoring (exercises): 30% (non-recoverable)
• Two specific tests: 35% + 35% (only one of the two is recoverable)

The description of the activities is given in the subject presentation. The retake exam consists of only one of the two specific tests, which the student can choose. An average cannot be made if any of the three activities has not passed the grade of 3 points.

Methodology

• Whole class work sessions with the teacher
• Using Python programming to clarify specific concepts
• There is no basic text to follow. Class notes are essential. However, you are encouraged to explore the recommended bibliography.

Bibliography

Key references

• Burgos, Juan de. (2006). Álgebra lineal y geometría cartesiana. McGraw–Hill.
• Calle, M.L. i Vendrell, R. (1992). Problemes d'àlgebra lineal i càlcul infinitessimal. Eumo editorial.
• Estadella, Robert; Anglada, Guillem; Vílchez, Rosendo; López, Rosario; Sala, Ferran (1995). Álgebra lineal i geometria lineal. Problemes. Publicacions Universitat de Barcelona.
• Marschner, N. & Shirley, P. (2021). Fundamentals of Computer Graphics: International Student Edition (5 ed.). CRC Press.
• Van Verth, J., i Bishop, L. (2008). Essential Mathematics for Games and Interactive Applications: A Programmer's Guide (2 ed.). Morgan Kaufmann.

Further reading

Teachers will provide complementary bibliography and compulsory reading throughout the course via the Virtual Campus.