Lecture notes for Chapter 2 (Part 1) of Ian Parberry's "Introduction to Game Physics with Box2D", AK Peters, 2013.

1. Introduction to Game Physics with Box2D
2. Mathematics for Game Physics
Lecture 2.1: Basic Math
Ian Parberry
Dept. of Computer Science & Engineering
University of North Texas

2. Contents of This Lecture
1. Geometry
2. Vectors
3. Orientation
Chapter 2 Introduction to Game Physics with Box2D 2

3. René Descartes
• 1596 – 1650, French
philosopher, physicist,
physiologist, & mathematician.
• Famous for (among other
things) recognizing that linear
algebra and geometry are the
same thing.
• Particularly useful for us,
because the CPU does linear
algebra and what we see on
the screen is geometry.
Chapter 2 Introduction to Game Physics with Box2D 3

4. Geometry
Chapter 2 Introduction to Game Physics with Box2D 4

5. 2.1.3 2.2.2 2.2.1
Vector Ball to Line Ball to Wall
3-7 3,4
Magnitude Collision Collision
Chapters Chapters
2.1.3 2.2.2 2.1.4
Theorem of Vector Dot Vector
4, 5, 8
Pythagoras Product Orientation
Chapters
2.1.3 2.1.4
Law of
6,8
Cosines
Chapters
2.1.3 2.2.3
Pythagorean Ball to Ball Quadratic
Identity Collision Equations
3,4
Chapters
Chapter 2 Introduction to Game Physics with Box2D 5

6. Things You Might Remember From School
Chapter 2 Introduction to Game Physics with Box2D 6

7. Trig Functions
Chapter 2 Introduction to Game Physics with Box2D 7

8. Mnemonics
1. Sohcahtoa
2. Some Old Horse Caught Another Horse
Taking Oats Away.
3. Some Old Hippy Caught Another Hippy
Toking On Acid.
Chapter 2 Introduction to Game Physics with Box2D 8

9. Theorem of Pythagoras
Chapter 2 Introduction to Game Physics with Box2D 9

10. Proof of the Theorem of Pythagoras
Chapter 2 Introduction to Game Physics with Box2D 10

11. Pythagorean Identity
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12. Law of Cosines
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13. Proof of the Law of Cosines
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14. More Useful Trig Identities
Chapter 2 Introduction to Game Physics with Box2D 14

15. Vectors
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16. What’s Our Vector, Victor?
Chapter 2 Introduction to Game Physics with Box2D 16

17. Programming Vectors
• Vectors correspond very naturally to an array
in most programming languages.
• D3DX has a structure D3DXVECTOR2 that we
will use to implement 2D vectors in code.
• A D3DXVECTOR2 v has two floating point
fields v.x and v.y.
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18. Chapter 2 Introduction to Game Physics with Box2D 18

19. Vector Multiplication by a Scalar
Chapter 2 Notes 3D Math Primer for Graphics & Game Dev 19

20. Vector Addition: Algebra
Chapter 2 Notes 3D Math Primer for Graphics & Game Dev 20

21. Vector Addition: Geometry
=
=
Chapter 2 Introduction to Game Physics with Box2D 21

22. Vector Addition in Code
D3DXVECTOR2 u, v, w;
v = D3DXVECTOR2(3.1415f, 7.0f);
u = 42.0f * v;
w = u + D3DXVECTOR2(v.x, 9.0f);
u += w;
Chapter 2 Introduction to Game Physics with Box2D 22

23. Vector Magnitude
Chapter 2 Introduction to Game Physics with Box2D 23

24. Vector Normalization
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25. Vector Magnitude in Code
• D3DXVec2Normalize normalizes a
D3DXVECTOR2, that is, makes its length 1.
• D3DXVec2Length computes the length of a
D3DXVECTOR2.
– Square roots are expensive.
– Often it is just as easy to work with squares of
vector lengths as with lengths.
– If so, use the faster D3DXVec2LengthSq
function instead of D3DXVec2Length.
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26. Orientation
Chapter 2 Introduction to Game Physics with Box2D 26

27. Relative Orientation
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28. Orientation
Chapter 2 Introduction to Game Physics with Box2D 28

29. Length & Orientation to Vector
Step 1.
Step 2. Step 3.
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30. Vector to Orientation
Step 2.
Step 1.
Step 3.
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31. A Gotcha
Chapter 2 Introduction to Game Physics with Box2D 31

32. Rotating a Vector
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33. Rotating a Vector
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34. Rotating a Vector
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35. Vector Rotation Code
void Rotate(const D3DXVECTOR2& u,
D3DXVECTOR2& v, float beta){
float alpha = atan2(u.y, u.x);
v.x = cos(alpha + beta);
v.y = sin(alpha + beta);
} //Rotate
This works but we can do better. The atan2
can be optimized out.
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36. Optimizing the Rotation Code
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37. Optimized Vector Rotation Code
void Rotate(const D3DXVECTOR2& u,
D3DXVECTOR2& v, float beta){
v.x = u.x * cos(beta) - u.y * sin(beta);
v.y = u.x * sin(beta) + u.y * cos(beta);
} //Rotate
We’ve replaced an arctangent with four floating
point multiplications, which is faster in practice.
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38. Conclusion
Chapter 2 Introduction to Game Physics with Box2D 38

39. Suggested Reading
Section 2.1
Suggested Activities
Problems 1-4 from Section 2.5.
Chapter 2 Introduction to Game Physics with Box2D 39