Mathematics for Game Development

Mathematics for Game Development is a field of applied mathematics focused on the principles required to build interactive virtual worlds and their underlying systems. It heavily utilizes linear algebra, including vectors and matrices, to manipulate the position, rotation, and scale of objects in 2D and 3D space, forming the backbone of graphics rendering and object transformation. Trigonometry is essential for calculating angles and trajectories, while calculus underpins the physics engines that simulate realistic motion, gravity, and collisions. Furthermore, concepts from probability and statistics are crucial for designing game mechanics such as procedural content generation, AI behavior, and balanced loot systems, making math an indispensable tool for nearly every aspect of creating a game.

1. Foundational Mathematical Concepts
   1. Number Systems and Properties
      1. Real Numbers
      2. Integer Operations
      3. Rational and Irrational Numbers
      4. Properties of Operations
   2. The Cartesian Coordinate System
      1. 2D Coordinates
         1. X and Y Axes
         2. Quadrants
         3. Plotting Points in 2D Space
         4. Distance Between Points in 2D
         5. Midpoint Formula
      2. 3D Coordinates
         1. X, Y, and Z Axes
         2. Octants
         3. Plotting Points in 3D Space
         4. Distance Between Points in 3D
         5. Midpoint Formula in 3D
      3. Coordinate System Conventions
         1. Left-Handed Systems
         2. Right-Handed Systems
         3. Converting Between Systems
         4. Game Engine Conventions
   3. Basic Algebra Review
      1. Variables and Expressions
         1. Algebraic Notation
         2. Evaluating Expressions
         3. Combining Like Terms
      2. Linear Equations
         1. One-Variable Equations
         2. Two-Variable Equations
         3. Systems of Linear Equations
         4. Graphical Solutions
      3. Polynomials
         1. Polynomial Expressions
         2. Degree and Leading Coefficient
         3. Adding and Subtracting Polynomials
         4. Multiplying Polynomials
         5. Factoring Polynomials
         6. Roots and Zeros
      4. Exponents and Radicals
         1. Laws of Exponents
         2. Radical Expressions
         3. Rational Exponents
      5. Order of Operations
      6. Substitution and Rearrangement
   4. Fundamental Geometry
      1. Basic Geometric Elements
         1. Points
         2. Lines
         3. Rays
         4. Line Segments
         5. Planes
      2. Relationships Between Elements
         1. Parallel Lines
         2. Perpendicular Lines
         3. Intersecting Lines
         4. Skew Lines
      3. Angles
         1. Types of Angles
         2. Measuring Angles
         3. Angle Relationships
         4. Degrees and Radians
         5. Converting Between Degrees and Radians
      4. Triangles
         1. Types of Triangles
         2. Triangle Properties
         3. Pythagorean Theorem
         4. Triangle Inequality
      5. Polygons
         1. Regular Polygons
         2. Irregular Polygons
         3. Interior and Exterior Angles
      6. Circles
         1. Circle Properties
         2. Circumference and Area
         3. Arcs and Sectors
      7. 2D Shape Properties
         1. Perimeter Calculations
         2. Area Calculations
         3. Composite Shapes
      8. 3D Solid Properties
         1. Volume Calculations
         2. Surface Area Calculations
         3. Common 3D Shapes