Basic Mathematical Concepts

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Fac. of Comp., Eng. Tech. Staffordshire
University
Programming Physics Engines for Games
Basic Mathematical Concepts
Dr. Claude C. Chibelushi
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Outline

• Introduction
• Cartesian coordinates
• Vectors
• Vector notation
• Basic vector operations
• negation, addition, scaling
• Matrices
• Addition, multiplication
• Summary

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Introduction

• Many mathematical techniques / tools used in
  computer games, e.g.
• 2D or 3D coordinates
• typically used in representing spatial
  relationships and shapes e.g. trajectory
  modelling, scene configuration, shape modelling
• Vector operations
• typically used in processes which depend on
  direction or orientation, e.g.
• 2D or 3D motion under control of force
• 3D sound production

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Cartesian Coordinates

• Coordinate systems

Right-handed coordinate system
Left-handed coordinate system
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Cartesian Coordinates

• Position in 2D or 3D space can be specified using
  2 or 3 Cartesian coordinates, respectively

Point p at position (x, y, z)
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Vectors
Example

• Flight simulation what is shortest path from
  Stafford to Stoke-on-Trent?

Vector s distance and direction
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Vectors

• Vector directed line segment
• has length (magnitude) and direction
• unit vector has length equal to 1
• e.g. basis vectors special unit vectors along
  Cartesian coordinate axes

Orthonormal basis
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Vectors

• Some common vector notations
• Boldface notation
• Arrow notation
• Underscore notation
• 3D vector

ax, ay, az vector components
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Vectors

• Vector a from p1 (x1, y1, z1) to p2 (x2, y2,
  z2)
• ax x2 - x1
• ay y2 - y1
• az z2 - z1
• Example displacement of ball is
• s 7 i 13 j 2 k

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Vectors

• Vector length (magnitude)

Vector magnitude
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Vectors

• If magnitude and orientation of 2D vector are
  known, vector components can be calculated

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Vectors

• Some basic vector operations

Vector addition
Vector negation
Vector scaling by s
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Matrices
Matrix addition
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Matrices
Matrix multiplication row-by-column sum of
products
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Matrices
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Suggested Reading

• Relevant parts of Ch. 1 Appendix A, D.M. Bourg,
  Physics for Game Developers, OReilly
  Associates, 2002.

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Summary

• Coordinates specify position
• right-handed or left-handed Cartesian coordinate
  system
• Vector
• mathematical representation for entity that
  possesses length (magnitude) and orientation
• some basic vector operations addition, negation,
  scaling
• Matrices
• some basic matrix operations addition,
  multiplication