Game Development Game Mathematics Flashcards

24 sample cards from the Game Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. What distinguishes a left-handed from a right-handed 3D coordinate system?

   In a right-handed system, curling the right hand's fingers from $+x$ to $+y$ points the thumb toward $+z$ ($\vec{x}\times\vec{y}=\vec{z}$). A left-handed system has $+z$ flipped (into the screen). Convention affects cross products, winding order, and depth.

2. How do polar coordinates $(r, \theta)$ convert to Cartesian coordinates $(x, y)$?

   $$x = r\cos\theta, \qquad y = r\sin\theta$$ Inverse: $r = \sqrt{x^2 + y^2}$, $\theta = \text{atan2}(y, x)$.

3. What is a $2\times 2$ rotation matrix that rotates a 2D vector counterclockwise by angle $\theta$?

   $$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$

4. What is the identity matrix and what does multiplying by it do?

   The identity $I$ has $1$s on the diagonal and $0$s elsewhere, e.g. $$I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ For any matrix $M$, $MI = IM = M$; it represents no transformation.

5. How is matrix multiplication defined for element $(i,j)$ of the product $C = AB$?

   $$C_{ij} = \sum_{k} A_{ik} B_{kj}$$ i.e. row $i$ of $A$ dotted with column $j$ of $B$. It requires $A$'s column count to equal $B$'s row count and is not commutative ($AB \neq BA$ in general).

6. Why do 3D game engines use $4\times 4$ matrices and homogeneous coordinates instead of $3\times 3$?

   A $4\times 4$ matrix can encode translation as well as rotation, scale, and shear in one matrix. Points become $(x, y, z, 1)$ so translation appears in the fourth column, letting transforms compose by multiplication.

7. What does the transpose $A^T$ of a matrix do, and what is special about the transpose of a pure rotation matrix?

   The transpose swaps rows and columns: $(A^T)_{ij} = A_{ji}$. For an orthonormal rotation matrix $R$, the transpose equals the inverse: $R^{-1} = R^T$.

8. What is the inverse $A^{-1}$ of a matrix, and what happens when a matrix is not invertible?

   $A^{-1}$ satisfies $A A^{-1} = A^{-1} A = I$; it undoes the transformation. A matrix is not invertible (singular) when its determinant is $0$, meaning the transform collapses space to a lower dimension.

9. What is the determinant of a $2\times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, and what does it represent geometrically?

   $$\det = ad - bc$$ Its absolute value is the area scale factor of the transform; a negative sign means orientation (handedness) is flipped.

10. In what order do translation ($T$), rotation ($R$), and scale ($S$) usually compose into a model matrix, and why?

    $$M = T \cdot R \cdot S$$ Applied to a column vector, scale acts first, then rotation, then translation. This scales the object in local space, orients it, then places it — avoiding skew from scaling after translating.

11. Define the derivative of a function $f(x)$ using the limit definition.

    $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$ It gives the instantaneous rate of change / slope of the tangent line.

12. State the power rule and the constant rule for differentiation.

    Power rule: $\dfrac{d}{dx} x^n = n\,x^{n-1}$. Constant rule: the derivative of a constant is $0$.

13. State the product rule and the quotient rule for derivatives.

    Product: $(fg)' = f'g + fg'$. Quotient: $$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$$

14. State the chain rule for differentiating a composite function $f(g(x))$.

    $$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$$

15. In game physics, how are position, velocity, and acceleration related through differentiation?

    Velocity is the derivative of position, and acceleration is the derivative of velocity: $$\vec{v} = \frac{d\vec{x}}{dt}, \qquad \vec{a} = \frac{d\vec{v}}{dt} = \frac{d^2\vec{x}}{dt^2}$$

16. State the power rule for integration (the reverse of differentiation).

    $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$

17. State the Fundamental Theorem of Calculus (evaluation form).

    If $F$ is an antiderivative of $f$, then $$\int_a^b f(x)\, dx = F(b) - F(a)$$ The definite integral gives the signed area under $f$ from $a$ to $b$.

18. What is semi-implicit (symplectic) Euler integration for updating velocity and position with timestep $\Delta t$?

    $$\vec{v}_{n+1} = \vec{v}_n + \vec{a}\,\Delta t, \qquad \vec{x}_{n+1} = \vec{x}_n + \vec{v}_{n+1}\,\Delta t$$ Using the updated velocity for position makes it more stable than explicit Euler for games.

19. What is a differential equation, and give the ODE governing simple harmonic motion (e.g. a spring).

    A differential equation relates a function to its derivatives. Simple harmonic motion: $$\frac{d^2 x}{dt^2} = -\omega^2 x$$ with solution $x(t) = A\cos(\omega t + \phi)$.

20. What is a partial derivative, and what notation is used?

    A partial derivative measures the rate of change of a multivariable function with respect to one variable while holding the others constant, written $\dfrac{\partial f}{\partial x}$ or $f_x$.

21. What is the gradient $\nabla f$ of a scalar field $f(x, y, z)$, and what does it represent?

    $$\nabla f = \left(\frac{\partial f}{\partial x},\; \frac{\partial f}{\partial y},\; \frac{\partial f}{\partial z}\right)$$ It points in the direction of steepest increase of $f$, with magnitude equal to that rate.

22. What is the formula for a quadratic Bézier curve with control points $P_0$, $P_1$, $P_2$?

    $$B(t) = (1-t)^2 P_0 + 2(1-t)t\, P_1 + t^2 P_2, \quad t \in [0, 1]$$

23. What is the surface normal of a triangle with vertices $A$, $B$, $C$, and how do you compute it?

    It is the unit vector perpendicular to the triangle's plane: $$\vec{n} = \frac{(B - A) \times (C - A)}{\|(B - A) \times (C - A)\|}$$ Winding order determines which way it faces.

24. How do you reflect an incoming direction $\vec{d}$ off a surface with unit normal $\vec{n}$?

    $$\vec{r} = \vec{d} - 2(\vec{d} \cdot \vec{n})\,\vec{n}$$ Used for bouncing projectiles and mirror-like reflections.

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