106131 - MATHEMATICAL FUNDAMENTALS OF VIDEO GAMES

Subject type: Mandatory

Coordinator: Adso Fernández Baena

Trimester: Third term

Credits: 4

The classes of the subject will be done in Catalan. Bibliography and supporting material may also be in Spanish and English.

The subject of Mathematical Fundamentals of Video Games is within the framework of the subject of Development, largely includes the simulation of real physical phenomena such as the movement of characters and objects, shocks, translations, rotations, camera movements, scaling d image and other phenomena that require the use of fundamental mathematical tools and concepts, such as geometry, algebra or trigonometry. The subject consists of theoretical sessions. In order to achieve the knowledge of the subject, individual exercises are evaluated on the one hand and exercises to be carried out in groups on the other.

This subject has methodological and digital resources to make possible its continuity in non-contact mode in the case of being necessary for reasons related to the Covid-19. In this way, the achievement of the same knowledge and skills that are specified in this teaching plan will be ensured.

The Tecnocampus will make available to teachers and students the digital tools needed to carry out the course, as well as guides and recommendations that facilitate adaptation to the non-contact mode.

At the end of the course students must be able to:

E6.1. Design the software architecture of a video game according to specifications

The subject uses the following work methodologies:

Master class, problem solving and collaborative learning.

Topic 0. Basic mathematical tools

0.1. Solving first degree equations.

0.2. Solving quadratic equations.

0.3. Systems of equations.

Topic 1. Algebra and Geometry in plane I (2D)

1.1. Coordinate systems.

1.2. Games: Applied coordinate systems (world, screen, camera, objects).

1.3. Vectors. Concepts. Coordinates and module. Free vectors and fixed vectors. Unit vectors.

1.4. Basic operations with vectors: addition, subtraction, product for a scalar.

1.5. Games: Positions, distances and routes.

1.6. Scalar product and vector product. Angles and relative position between vectors. Parallelism and perpendicularity.  

1.7. Games: Vector properties of game objects.

1.8. Games: Images and coordinate systems, vector images.

1.9. Vectors in space

Topic 2. Trigonometry

2.1. Measurement of angles. Units.

2.2. Trigonometric ratios.

2.3. Equivalent triangles. Symmetry. Complementary angles. 

2.4. Vectors and trigonometry: Cartesian coordinates and polar coordinates. 

2.5. Unit vectors and trigonometric ratios. 

2.6. Games: Projection of shadows, angles between objects, decomposition of vector quantities.

Topic 3. Rectilinear trajectories in the plane (2D)

3.1. Equation of the line. Sloping and ordered at the origin.

3.2. Rectilinear routes.

3.3. Linear interpolation.

3.4. Relative position of two straight lines. Angle, intersection, parallelism, perpendicularity.

3.5. Specular reflection.  

3.6. Games: rectilinear trajectories, projectile simulation, surface reflection.

3.7. Games: Intersection of trajectories, interpolation of motion.

Item 4. Physics of motion

4.1. Uniform rectilinear motion.

4.2. Uniformly accelerated rectilinear motion.

4.3. Circular movement.

4.4 Particular cases: free fall and parabolic shooting.

4.5. Games: gravity creation, free fall, parabolic jump, friction, wind, projectile launch.

4.6. Elastic and inelastic shocks. Refund coefficient.

4.7. Physics of motion in space

Item 5. Algebra and Geometry in Plan II (2D)

5.1. Matrix. Concept, representation and basic operations.

5.2. Identity matrix. Diagonal matrix. Inverse matrix.

5.3. Vector spaces and bases. Matrix representation. 

5.4. Reference system. Base change matrices.

5.5. Games: changes in reference systems. 

5.6. Transformation matrices: translation, rotation, scaling, deformation.

5.7. Games: translation of objects and characters, rotation, scaling. Camera movements. 

In order to gather evidence of the achievement of the expected learning outcomes, the following evaluative activities will be carried out (related to all the common competences):

A1. Exercises at home (Evidence of all learning outcomes)

Exercises to solve at home and present in class. This activity cannot be recovered.

A2. Exercises in class: Mathaton (Evidence of all learning outcomes)

It will consist of one or more resolution sessions of a specific practical case, where it will be necessary to apply the theoretical concepts and practical procedures of the contents of the subjects of the subject that are established. Class activities can be carried out collectively and proactively with classmates (groups of up to 4 students), always respecting the time indicated and delivery times. This activity cannot be recovered.

A3. Individual work: Exercises and problems (Evidence of all learning outcomes)

Individual work of theoretical-practical application (resolution of exercises and problems, questions) of the theoretical concepts and practical procedures of the contents of the subjects of the asignatura that establish. This activity cannot be recovered.

A4. Final exam (Evidence of all learning outcomes)

Individual examination of theoretical-practical application (resolution of exercises and problems, questions) of the theoretical concepts and practical procedures of the contents of the subjects of the subject that establish. It is essential to bring a calculator.

General criteria of the activities:

• The teacher will present a statement for each activity and the evaluation and / or rubric criteria.
• The teacher will inform of the dates and format of the delivery of the activity.

The grade of each student will be calculated following the following percentages:

A1. Exercises at home 10%

A2. Exercises in class: Mathaton 10%

A3. Individual work: Exercises and problems 30%

A4. Final exam 50%

Final grade = A1 0,1 + A2 0,1 + A3 0,3 + A4 0,5

Considerations:

• It is necessary to obtain a mark higher than 4 in the final exam to pass the subject.
• An activity not delivered or delivered late and without justification (court summons or medical matter) counts as a 0.
• It is the responsibility of the student to prevent plagiarism in all its forms. In the case of detecting a plagiarism, regardless of its scope, in some activity it will correspond to have a note of 0. In addition, the professor will communicate to the Head of Studies the situation so that measures applicable in matter of sanctioning regime are taken. .
• Students who have failed the subject, may take a resit exam on the dates pre-established by the same university in the official academic calendar. 

Recovery:

• You must obtain a grade higher than 5 in the resit exam to pass the course.
• In case of passing the recovery, the maximum final mark of the subject will be 5.

LENGYEL, E. (2012). "Mathematics for 3D Game Programming and Computer Graphics" (Third edition). Boston, MA (USA): Course Technology PTR (Cengage Learning)

"Discover Math with GeoGebra." GeoGebra - Dynamic Mathematics, www.geogebra.org

DUNN, F .; PARBERRY, I. (2002). "3D Math Primer for Graphics and Game Development". Plano, Texas (USA): Wordware Publishing, Inc.