Math Review - University of British Columbia3D math primer for graphics and game development by Fletcher Dunn, Ian Parberry Thanks Enrique Rosales Title Shading Models Author enrique

Math Review

Intro to Computer Graphics

© 2020 Enrique Rosales

Learning goals



• Vectors

• Dot product

• Cross product

• Difference between vectors and points

• Bivectors

• Trivectors

• Matrices

• Coding example

Notation



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

What is a Vector?



It depends of who is asking



What is a Vector?It depends of who is asking



• Programming

• Mathematical

• Geometric

Programming perspective



Data

It is a n-tuple of real numbers

𝑎 = 𝑎1, 𝑎2, … , 𝑎𝑛

Mathematical perspective



This data is a matrix

𝑎𝑇 = 𝑎𝑥 , 𝑎𝑦 , 𝑎𝑧𝑎 =

𝑎𝑥𝑎𝑦𝑎𝑧

Geometric perspective



Geometric interpretationLinear Algebra



• Position

• Vector (a direction with a magnitude)

Geometric interpretationGeometric Algebra



• Position

• Vector (Directed length)

• Bivector (Directed area)

• Trivector (Directed volume)

Dot Product



Measures the similarity of two vectors

𝑎𝑏 =

𝑖=1

𝑛

𝑎𝑖𝑏𝑖

Dot Product



Measures the similarity of two vectors

𝑎𝑏 =

𝑖=1

𝑛

𝑎𝑖𝑏𝑖

𝑎𝑏 = 𝑎𝑥𝑏𝑥 + 𝑎𝑦𝑏𝑦 + 𝑎𝑧𝑏𝑧

Cross Product



Measures the difference of two vectors

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

𝑎

𝑏𝑑

Cross Product



Cross Product



Cross Product





Wedge Product

Bivector

(directed area)

(Geometric Algebra)

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

Cross Product / Wedge Product



Bivector

Wedge Product



𝑎

𝑏𝑐

𝑎𝑏𝑐

(Geometric Algebra)

Wedge Product



Trivector

(directed volume)

𝑎

𝑏

𝑐𝑎𝑏𝑐

(Geometric Algebra)

Wedge Product



Trivector

(directed volume)

𝑎𝑏𝑐

(Geometric Algebra)

Colum vectors or Row vectors?



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

Math Review - University of British Columbia3D math primer for graphics and game development by Fletcher Dunn, Ian Parberry Thanks Enrique Rosales Title Shading Models Author enrique

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