Game Development Game Mathematics Syllabus

Game Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Game Mathematics in Game Development, not a summary of it.

1. 1

   Fundamentals of Mathematics

   5 topics

   • Arithmetic
   • Algebra
   • Geometry
   • Trigonometry
   • Calculus
2. 2

   Linear Algebra

   5 topics

   • Vectors
   • Matrices
   • Transformations
   • Dot Product
   • Cross Product
3. 3

   Analytic Geometry

   4 topics

   • Coordinate Systems
   • Lines and Planes
   • Curves and Surfaces
   • Distance and Angles
4. 4

   Calculus for Game Development

   4 topics

   • Differentiation
   • Integration
   • Differential Equations
   • Multivariable Calculus
5. 5

   Discrete Mathematics

   4 topics

   • Graph Theory
   • Combinatorics
   • Boolean Algebra
   • Set Theory
6. 6

   Probability and Statistics

   5 topics

   • Basic Probability
   • Random Variables
   • Distributions
   • Hypothesis Testing
   • Regression Analysis
7. 7

   Numerical Methods

   5 topics

   • Root Finding Algorithms
   • Interpolation
   • Numerical Integration
   • Numerical Differentiation
   • Solving Differential Equations Numerically
8. 8

   Physics for Games

   5 topics

   • Kinematics
   • Dynamics
   • Collision Detection
   • Rigid Body Dynamics
   • Fluid Dynamics
9. 9

   Transformations and Projections

   4 topics

   • 2D Transformations
   • 3D Transformations
   • Projection Techniques
   • Homogeneous Coordinates
10. 10

    Shaders and Graphics Programming

    5 topics

    • Basic Shader Mathematics
    • Lighting Models
    • Texture Mapping
    • Normal Mapping
    • Shadow Mapping

Game Mathematics flashcards for Game Development

18 of 50 cards from the Game Mathematics deck — real questions with worked answers.

1. What is the distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ in 2D?

   $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

2. What is the distance formula between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in 3D?

   $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

3. How do you compute the magnitude (length) of a vector $\vec{v} = (x, y, z)$?

   $$\|\vec{v}\| = \sqrt{x^2 + y^2 + z^2}$$

4. How do you normalize a vector $\vec{v}$ to get a unit vector $\hat{v}$?

   Divide the vector by its magnitude: $$\hat{v} = \frac{\vec{v}}{\|\vec{v}\|}$$ The result has length $1$ and points in the same direction.

5. What is the formula for the dot product of $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$?

   $$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$$

6. How is the dot product related to the angle $\theta$ between two vectors?

   $$\vec{a} \cdot \vec{b} = \|\vec{a}\|\,\|\vec{b}\|\cos\theta$$ so $\cos\theta = \dfrac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|\,\|\vec{b}\|}$.

7. In game math, what does the sign of the dot product tell you about two vectors' directions?

   If $\vec{a} \cdot \vec{b} > 0$ they point roughly the same way ($\theta < 90^\circ$); if $= 0$ they are perpendicular ($\theta = 90^\circ$); if $< 0$ they point roughly opposite ways ($\theta > 90^\circ$).

8. What is the formula for the cross product $\vec{a} \times \vec{b}$ of $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$?

   $$\vec{a} \times \vec{b} = (a_2 b_3 - a_3 b_2,\; a_3 b_1 - a_1 b_3,\; a_1 b_2 - a_2 b_1)$$

9. What geometric quantity does the magnitude of the cross product equal, and what direction does it point?

   $\|\vec{a} \times \vec{b}\| = \|\vec{a}\|\,\|\vec{b}\|\sin\theta$, the area of the parallelogram spanned by the vectors. It points perpendicular to both $\vec{a}$ and $\vec{b}$ (direction given by the right-hand rule).

10. Contrast the dot product and cross product: what do they return and when are they used?

    The dot product returns a scalar and measures alignment/projection (used for angles, lighting). The cross product returns a vector perpendicular to both inputs and measures area/orientation (used for surface normals, torque).

11. How do you compute the scalar projection of $\vec{a}$ onto $\vec{b}$?

    $$\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|}$$

12. How do you compute the vector projection of $\vec{a}$ onto $\vec{b}$?

    $$\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{b}}\,\vec{b}$$

13. State the Pythagorean theorem for a right triangle with legs $a$, $b$ and hypotenuse $c$.

    $$a^2 + b^2 = c^2$$

14. Give the fundamental identities for $\sin\theta$, $\cos\theta$, and $\tan\theta$ in a right triangle.

    $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{\sin\theta}{\cos\theta}$.

15. State the Pythagorean trigonometric identity.

    $$\sin^2\theta + \cos^2\theta = 1$$

16. How do you convert an angle from degrees to radians, and radians to degrees?

    $$\text{radians} = \text{degrees} \times \frac{\pi}{180}, \qquad \text{degrees} = \text{radians} \times \frac{180}{\pi}$$

17. What are the sine and cosine addition formulas?

    $$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$$ $$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$$

18. How do you use $\text{atan2}(y, x)$ in games and why is it preferred over $\arctan(y/x)$?

    $\text{atan2}(y, x)$ returns the angle of the vector $(x, y)$ measured from the positive $x$-axis in $(-\pi, \pi]$. It is preferred because it uses both signs to give the correct quadrant and avoids division by zero when $x = 0$.

See more Game Mathematics flashcards →

Planning Game Mathematics for Game Development

Game Mathematics (Game Development) FAQ