05. vectors

Video Game Development: Vectors A. Babadi 1 of 16

In The Name Of God

Video Game Development

Amin Babadi

Department of Electrical and Computer Engineering

Isfahan University of Technology

Spring 2015

Vectors


Outline

What is a game engine?

Vector definition

The dot product

Vector projection

The cross product


Game Engine

A game engine is a software designed for the creation and development of video games.

The core functionality typically provided by a game engine includes: o A rendering engine,

o A physics engine,

o Scripting support,

o Multi-threading support,

o Animation support,

o Etc.

More about game engines comes in next lectures.


Game Engine

Examples of well-known game engines: o CryEngine,

o Unreal Engine,

o Gamebyro,

o Unity,

o Wintermute Engine,

o XNA Game Studio,

o Game Maker.


Vectors

Vectors are of fundamental importance in any game engine.

They mainly are used for representing points in space and spatial directions. o But for now, we make no distinction between vectors representing

points and vectors representing directions, nor do we concern ourselves with how vectors are transformed from one coordinate system to another.


Vector Properties

We usually restrict ourselves to vectors defined by n-tuples of real numbers, where n is typically 2, 3, or 4.

𝑉 = 𝑣1, 𝑣2, … , 𝑣𝑛

The numbers 𝑣𝑖 are called the components of the vector 𝑉.

𝑉 may also be represented by a 𝑛 × 1 matrix:

𝑉 =

𝑣1𝑣2:𝑣𝑛

= 𝑣1 𝑣2 … 𝑣𝑛𝑇

The components can usually be labeled with the name of the axis to which they correspond.

𝑃 = 𝑝𝑥 , 𝑝𝑦, 𝑝𝑧


Vector Properties

𝛼𝑉 = 𝑉𝛼 = 𝛼𝑣1, 𝛼𝑣2, … , 𝛼𝑣𝑛 𝑃 + 𝑄 = 𝑝1 + 𝑞1, 𝑝2 + 𝑞2, … , 𝑝𝑛 + 𝑞𝑛

𝑉 = 𝑣𝑖2

𝑛

𝑖=1

If 𝑉 = 1, 𝑉 is called a unit vector. If 𝑉 > 0, 𝑉 can be converted to a unit vector. This operation is called normalization.

𝑉′ =1

𝑉𝑉


The Dot Product

The dot product of two vectors supplies a measure of the similarity (or difference) between the directions in which the two vectors point.

𝑃. 𝑄 = 𝑄. 𝑃 = 𝑝𝑖𝑞𝑖

𝑛

𝑖=1

It is also known as the scalar product or inner product, and is one of the most heavily used operations in 3D graphics.

𝑃. 𝑄 may also be expressed as the matrix product:

𝑃𝑇𝑄 = 𝑝1 𝑝2 … 𝑝𝑛

𝑞1𝑞2:𝑞𝑛


The Dot Product

𝑃. 𝑄 = 𝑃 𝑄 cos 𝛼

The dot product is related to the angle between two vectors.


The Dot Product

The sign of the dot product tells us whether two vectors lie on the same side or on opposite sides of a plane.


The Dot Product

Vectors whose dot product yields zero are called orthogonal (or perpendicular) vectors.

We define the zero vector 𝟎 = 0,0,… , 0 to be orthogonal to every vector 𝑃.


cos 𝛼 =𝑥

𝑃⇒ 𝑥 = 𝑃 cos 𝛼 =

𝑃. 𝑄

𝑄

We now have this formula for the projection of 𝑃 onto 𝑄:

𝑝𝑟𝑜𝑗𝑄𝑃 =𝑃.𝑄

𝑄 2𝑄

Projecting Vectors

𝛼

𝑃

𝑄

𝑥


The Cross Product

The cross product (vector product) of two 3D vectors, returns a new vector that is perpendicular to both of the vectors being multiplied together.

𝑃 × 𝑄 =

𝑖 𝑗 𝑘𝑝𝑥 𝑝𝑦 𝑝𝑧𝑞𝑥 𝑞𝑦 𝑞𝑧

= 𝑝𝑦𝑞𝑧 − 𝑝𝑧𝑞𝑦, 𝑝𝑧𝑞𝑥 − 𝑝𝑥𝑞𝑧, 𝑝𝑥𝑞𝑦 − 𝑝𝑦𝑞𝑥


The Cross Product

Theorem: 𝑃 × 𝑄 . 𝑃 = 𝑃 × 𝑄 .𝑄 = 0

The right hand rule provides an easy way for determining in which direction the cross product points.


The Cross Product

Theorem: 𝑃 × 𝑄 = 𝑃 𝑄 sin 𝛼

The right hand rule provides an easy way for determining in which direction the cross product points.


References

Lengyel’s textbook,

Wikipedia, and

Some other sources on the Internet.