Math Review

Intro to Computer Graphics

© 2020 Enrique Rosales

Learning goals

© 2020 Enrique Rosales

Intro to Computer Graphics

• Vectors

• Dot product

• Cross product

• Difference between vectors and points

• Bivectors

• Trivectors

• Matrices

• Coding example

Notation

© 2020 Enrique Rosales

Intro to Computer Graphics

Real-time rendering by Tomas Akenine-Moller [et. al.] page 5

What is a Vector?

© 2020 Enrique Rosales

Intro to Computer Graphics

It depends of who is asking

© 2020 Enrique Rosales

Intro to Computer Graphics

What is a Vector?It depends of who is asking

© 2020 Enrique Rosales

Intro to Computer Graphics

• Programming

• Mathematical

• Geometric

Programming perspective

© 2020 Enrique Rosales

Intro to Computer Graphics

Data

It is a n-tuple of real numbers

𝑎 = 𝑎1, 𝑎2, … , 𝑎𝑛

Mathematical perspective

© 2020 Enrique Rosales

Intro to Computer Graphics

This data is a matrix

𝑎𝑇 = 𝑎𝑥 , 𝑎𝑦 , 𝑎𝑧𝑎 =

𝑎𝑥𝑎𝑦𝑎𝑧

Geometric perspective

© 2020 Enrique Rosales

Intro to Computer Graphics

Geometric interpretationLinear Algebra

© 2020 Enrique Rosales

Intro to Computer Graphics

• Position

• Vector (a direction with a magnitude)

Geometric interpretationGeometric Algebra

© 2020 Enrique Rosales

Intro to Computer Graphics

• Position

• Vector (Directed length)

• Bivector (Directed area)

• Trivector (Directed volume)

Dot Product

© 2020 Enrique Rosales

Intro to Computer Graphics

Measures the similarity of two vectors

𝑎𝑏 =

𝑖=1

𝑛

𝑎𝑖𝑏𝑖

Dot Product

© 2020 Enrique Rosales

Intro to Computer Graphics

Measures the similarity of two vectors

𝑎𝑏 =

𝑖=1

𝑛

𝑎𝑖𝑏𝑖

𝑎𝑏 = 𝑎𝑥𝑏𝑥 + 𝑎𝑦𝑏𝑦 + 𝑎𝑧𝑏𝑧

Cross Product

© 2020 Enrique Rosales

Intro to Computer Graphics

Measures the difference of two vectors

𝑎 × 𝑏 = 𝑎𝑦𝑏𝑧 − 𝑎𝑧𝑏𝑦 , 𝑎𝑧𝑏𝑥 − 𝑎𝑥𝑏𝑧, 𝑎𝑥𝑏𝑦 − 𝑎𝑦𝑏𝑥

𝑎

𝑏𝑑

Cross Product

© 2020 Enrique Rosales

Intro to Computer Graphics

Cross Product

© 2020 Enrique Rosales

Intro to Computer Graphics

Cross Product

© 2020 Enrique Rosales

Intro to Computer Graphics

© 2020 Enrique Rosales

Intro to Computer Graphics

Wedge Product

Bivector

(directed area)

(Geometric Algebra)

𝑎𝑏 = 𝑎𝑦𝑏𝑧 − 𝑎𝑧𝑏𝑦 , 𝑎𝑧𝑏𝑥 − 𝑎𝑥𝑏𝑧, 𝑎𝑥𝑏𝑦 − 𝑎𝑦𝑏𝑥

Cross Product / Wedge Product

© 2020 Enrique Rosales

Intro to Computer Graphics

Bivector

Wedge Product

© 2020 Enrique Rosales

Intro to Computer Graphics

𝑎

𝑏𝑐

𝑎𝑏𝑐

(Geometric Algebra)

Wedge Product

© 2020 Enrique Rosales

Intro to Computer Graphics

Trivector

(directed volume)

𝑎

𝑏

𝑐𝑎𝑏𝑐

(Geometric Algebra)

Wedge Product

© 2020 Enrique Rosales

Intro to Computer Graphics

Trivector

(directed volume)

𝑎𝑏𝑐

(Geometric Algebra)

Colum vectors or Row vectors?

© 2020 Enrique Rosales

Intro to Computer Graphics

𝑎𝑇 = 𝑎𝑥 , 𝑎𝑦 , 𝑎𝑧𝑎 =

𝑎𝑥𝑎𝑦𝑎𝑧

Chapters 4 and 5 (Vectors)

Chapter 7 (Matrices)

3D math primer for graphics and game development by Fletcher Dunn, Ian Parberry

Thanks

Enrique Rosales