1. Basic Math in 2D Game Development
I have to use math? I am out of here…
Unity 2D Game Development
Telerik Academy Plus

2. Table of Contents Introduction to Linear Algebra Vectors Quaternions*
Table of Contents
Introduction to Linear Algebra
Vectors
What is a Vector?
Adding, Subtracting
Scalar Multiplication
Length, Distance
Dot Product
Quaternions
Other helpful functions
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

3. But… I thought math is for kids!*
Linear Algebra
But… I thought math is for kids!
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

4. Linear Algebra Why should I care?*
Linear Algebra
Why should I care?
Linear Algebra is the study of vectors
Vectors are really helpful in game development
2D or 3D
Used for positions, directions and velocity in Cartesian coordinate system
The better you understand linear algebra, the better your games! Deal with it! 
*NOTE: images and information taken from:
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

5. What is a Vector What is a Vector? A vector is just a set of numbers*
What is a Vector
What is a Vector?
A vector is just a set of numbers
(2, -3) or (2,5,-1) or (1, 0)
Doesn't give much information without context
In games – position, direction or velocity
It is important to keep track of your units!
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

6. What is a Vector How to create vectors in Unity with C#?*
What is a Vector
How to create vectors in Unity with C#?
using UnityEngine; …
Vector2 position2D = new Vector2(2.5f, 3.8f); // 2D vector
Vector2 direction2D = new Vector2(0, 1); // another one
Vector2 zero = Vector2.zero; // (0, 0)
Vector2 one = Vector2.one; // (1, 1)
Vector2 right = Vector2.right; // (1, 0)
Vector2 left = -Vector2.right; // (-1, 0)
Vector2 up = Vector2.up; // (0, 1)
Vector2 down = -Vector2.up; // (0, -1)
Vector3 position3D = new Vector3(1, 2, 3); // 3D vector
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

7. Vector Addition How to add vectors? Just add each component*
Vector Addition
How to add vectors?
Just add each component
(1, 2, 3) + (3, 4, 5) = (1+3, 2 + 4, 3 + 5) = (4, 6, 8)
Commonly used in game's physics
Example:
Velocity – (1, 3)
Gravity – (0, -1)
var one = new Vector2(2.5f, 3.8f);
var two = new Vector2(-2, 5.2f);
var result = one + two; // (0.5, 9)
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

8. Vector Subtraction How to subtract vectors?*
Vector Subtraction
How to subtract vectors?
Just subtract each component
(4, 3, 1) - (5, 8, 0) = (4 - 5, 3 - 8, 1 - 0) = (-1, 6, 8)
Commonly used when you need to get a vector that points from one position to another
Example:
Player – (1, 2)
Enemy – (4, 3)
var one = new Vector2(2.5f, 3.8f);
var two = new Vector2(-2, 5.2f);
var result = one - two; // (4.5, -1.4)
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

9. Scalar-vector Multiplication*
Scalar-vector Multiplication
How to multiply a vector by scalar?
Scalars are just individual numbers: 5, 6, -3
Just multiply each component by the scalar
0.7 * (10, 20) = (0.7 * 10, 0.7 * 20) = (7, 14)
Example:
Player's velocity on ground – (5, 0)
Player's velocity in water – 0.7 * (5, 0)
var velocity = new Vector2(5, 0);
var waterFactor = 0.7f;
var waterVelocity = velocity * waterFactor; // (3.5, 0)
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

10. Length of a Vector How to find the length (magnitude) of a vector?*
Length of a Vector
How to find the length (magnitude) of a vector?
Written with |V| - length of vector V
Calculated with Pythagorean theorem
Example – (4, 3)
|V| = sqrt(42, 32) = 5
var velocity = new Vector2(4, 3);
var length = velocity.magnitude; // 5
// use velocity.sqrMagnitude for comparison
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

11. Distance Between Two Vectors*
Distance Between Two Vectors
How to find the distance between two vectors?
Just subtract the vectors and find the length of the result vector
Distance = |(4, 3) – (3,1)| = |(3,1) – (4,3)| = |(1,2)| = sqrt(12+22) = sqrt(5) =
var first = new Vector2(1, 2);
var second = new Vector2(3, 3);
var distance = Vector2.Distance(first,
second); // 2.23
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

12. Normalization What is normalization? How to normalize a vector?*
Normalization
What is normalization?
Making a vector's to have length of 1
Commonly used with directions
How to normalize a vector?
Just divide each component by the vector's length
(3, 4) => (3/5, 4/5) = (0.6, 0.8)
var position = new Vector2(3, 4);
var normalizedVector = position.normalized;
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

13. Dot Product How to calculate dot product of two vectors?*
Dot Product
How to calculate dot product of two vectors?
First multiply the components and then add the results
(a1,a2)•(b1,b2) = a1b1 + a2b2
Example:
(3,2)•(1,4) = 3*1 + 2*4 = 11
var position = new Vector2(3, 4);
var direction = new Vector2(1, 0);
var dotProduct = Vector2.Dot(position, direction);
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

14. Dot Product Ok, but why dot product is useful? Example: Solution:*
Dot Product
Ok, but why dot product is useful?
Example:
Hero at (3, 0)
Guard at (1, 1) facing (1, 1) direction and has 180º field of view
Is the guard seeing our hero?
Solution:
Find the vector between the two:
V = H – G = (2, -1)
Find the dot product between V and D
D•V = (1, 1)•(2, -1) = 2 – 1 = 1
1 is positive => the guard sees our hero!
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

15. *
Dot Product
Dot product shows the extend to which two vectors are pointing in the same direction
But to what extend? What is the angle?
The exact rotation is:
A•B = |A||B|cos(θ)
We can solve that for θ
θ = acos((A•B)/(|A||B|))
If we normalize A and B:
θ = acos(A•B)
var position = new Vector2(3, 4);
var direction = new Vector2(1, 0);
var angle = Vector2.Angle(position, direction);
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

16. Quaternion Here comes the bad guy! *
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

17. Quaternion What is Quaternion? Used to represent rotations*
Quaternion
What is Quaternion?
Used to represent rotations
They use complex numbers and are quite hard to understand right away
Luckily most often we would not use them (Unity does all the heavy lifting)
Quaternion.Euler – rotate by Euler rotation
Quaternion.Slerp – interpolate rotations
Quaternion.identity – rotation origin
Quaternion.Angle – angle between two rotations
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

18. Other Helpful Math Functions*
Other Helpful Math Functions
Math? Again? Come on!
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

19. Mathf Mathf is helper class provided by Unity*
Mathf
Mathf is helper class provided by Unity
Like System.Math in .NET but extended and with full floating point numbers support
Besides the normal Sin, Abs, etc. members:
Deg2Rad and Rad2Deg
Clamp and Clamp01
Lerp and LerpAngle
ClosestPowerOfTwo
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

20. Random The random generator is also enhanced*
Random
The random generator is also enhanced
It is the static class Random
Range - random floating point number between min and max (max is exclusive)
value - random number between 0.0 and 1.0
rotation - random rotation
There are more but these are the most useful
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

21. Summary Quaternions – used for rotation Mathf – helper class for math*
Summary
Linear Algebra
Vectors – used for game math
Properties of vectors – length, distance
Operations with vectors – adding, subtracting
Quaternions – used for rotation
Mathf – helper class for math
Random – helper class for random numbers
(c) 2007 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

22. Basic Math in Unity 2D

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