Game Development Game Mathematics Syllabus

What is the slope-intercept form of a line and what do the parameters mean?

$$y = mx + b$$ where $m$ is the slope (rise over run) and $b$ is the $y$-intercept (value of $y$ when $x = 0$).

What is the parametric (ray) form of a line through point $\vec{p_0}$ with direction $\vec{d}$?

$$\vec{p}(t) = \vec{p_0} + t\,\vec{d}$$ For a ray, $t \geq 0$; for a segment, $t \in [0, 1]$.

What is the equation of a plane with normal $\vec{n} = (a, b, c)$ passing through point $(x_0, y_0, z_0)$?

$$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$$ or equivalently $ax + by + cz + d = 0$ where $d = -(ax_0 + by_0 + cz_0)$.

How do you compute the signed distance from a point $\vec{p}$ to a plane with unit normal $\vec{n}$ and offset $d$ (plane: $\vec{n}\cdot\vec{x} + d = 0$)?

$$\text{dist} = \vec{n} \cdot \vec{p} + d$$ The sign tells you which side of the plane the point is on.

What is the standard equation of a circle of radius $r$ centered at $(h, k)$, and a sphere of radius $r$ centered at $(a, b, c)$?

Circle: $$(x - h)^2 + (y - k)^2 = r^2$$ Sphere: $$(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2$$

How do you find the midpoint of a segment between $(x_1, y_1)$ and $(x_2, y_2)$?

$$\left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right)$$

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.

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

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

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

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

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.

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).

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.

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: $R^{-1} = R^T$.

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.

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.

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.

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.

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 $0$.

State the product rule and the quotient rule for derivatives.

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

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

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

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: $$\vec{v} = \frac{d\vec{x}}{dt}, \qquad \vec{a} = \frac{d\vec{v}}{dt} = \frac{d^2\vec{x}}{dt^2}$$

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

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

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 $b$.

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.

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 $x(t) = A\cos(\omega t + \phi)$.

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 $f_x$.

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.

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

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

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.

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.