Waterloo CS488 - Introduction to Computer Graphics Flashcards

Question 1

3D Scaling Matrix

Accepted Answer

Scales geometry along X, Y, and Z axes relative to the origin:
$$\begin{bmatrix} s_x & 0 & 0 & 0 \ 0 & s_y & 0 & 0 \ 0 & 0 & s_z & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$$

Question 2

Affine Transformation

Accepted Answer

A transformation that preserves points, straight lines, and planes, as well as parallelism. It is a combination of a linear transformation (rotation/scaling/shear) followed by a translation ($T(x) = Ax + b$).

Question 3

Homogeneous Coordinates

Accepted Answer

A coordinate system using $N+1$ dimensions to represent $N$-dimensional space (e.g., $[x, y, z, w]^T$). It allows translation to be expressed as matrix multiplication alongside scaling and rotation. Points have $w = 1$, while directional vectors have $w = 0$.

Question 4

2D Translation Matrix

Accepted Answer

Represented in homogeneous coordinates as:
$$\begin{bmatrix} 1 & 0 & t_x \ 0 & 1 & t_y \ 0 & 0 & 1 \end{bmatrix}$$
Translates a point $(x, y)$ by vector $(t_x, t_y)$ via matrix multiplication.

Question 5

3D Rotation Matrix (Z-axis)

Accepted Answer

Rotates a point around the Z-axis by angle $	heta$ counterclockwise:
$$\begin{bmatrix} \cos	heta & -\sin	heta & 0 & 0 \ \sin	heta & \cos	heta & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$$

Question 6

Graphics Viewing Pipeline

Accepted Answer

The sequence of coordinate transformations to display 3D scenes on a 2D screen:
1. **Object/Model Space** $\rightarrow$ World Space (Model Matrix)
2. **World Space** $\rightarrow$ View/Camera Space (View Matrix)
3. **View Space** $\rightarrow$ Clip Space (Projection Matrix)
4. **Clip Space** $\rightarrow$ Screen Space (Viewport Transform)

Question 7

Orthographic vs. Perspective Projection

Accepted Answer

- **Orthographic**: Parallel projection where parallel lines remain parallel; no foreshortening (distance does not affect object size).
- **Perspective**: Models human vision/camera lenses; objects further from the center of projection appear smaller due to foreshortening.

Question 8

Perspective Divide

Accepted Answer

The step following perspective projection where homogeneous clip coordinates $[x_c, y_c, z_c, w_c]^T$ are converted to Normalized Device Coordinates (NDC) by dividing $x_c, y_c, z_c$ by $w_c$. This division creates perspective foreshortening.

Question 9

Bresenham's Line Algorithm

Accepted Answer

An efficient 2D rasterization algorithm that uses integer-only arithmetic (addition and subtraction) to determine which pixels on a discrete grid should be filled to approximate a straight line.

Question 10

Barycentric Coordinates

Accepted Answer

Expresses a point $\mathbf{P}$ inside a triangle with vertices $\mathbf{A}, \mathbf{B}, \mathbf{C}$ as $\mathbf{P} = \alpha\mathbf{A} + \beta\mathbf{B} + \gamma\mathbf{C}$, where $\alpha + \beta + \gamma = 1$.
Used for inside/outside triangle testing and interpolating attributes (color, normals, UVs) across rasterized primitives.

Question 11

Z-Buffering (Depth Buffering)

Accepted Answer

A hidden-surface removal algorithm that tracks the minimum depth ($Z$-value) of the closest drawn pixel in a secondary 2D buffer (`Z-buffer`). If a new fragment's depth is closer than the stored value, the color buffer and Z-buffer are updated.

Question 12

Z-Fighting

Accepted Answer

A rendering artifact occurring when two primitives have nearly identical depth values in clip space. Limited floating-point precision in the Z-buffer causes the primitives to repeatedly overwrite each other, producing flickering.

Question 13

Phong Reflection Model

Accepted Answer

Calculates local surface illumination using three components:
$$I = I_{ambient} + I_{diffuse} + I_{specular}$$
$$I = k_a I_a + k_d I_d (\mathbf{N} \cdot \mathbf{L}) + k_s I_s (\mathbf{R} \cdot \mathbf{V})^n$$
Where $\mathbf{N}$ is normal, $\mathbf{L}$ is light vector, $\mathbf{R}$ is reflection vector, $\mathbf{V}$ is view vector, and $n$ is specular shininess.

Question 14

Blinn-Phong Shading Model

Accepted Answer

A modification of the Phong model that uses the halfway vector $\mathbf{H} = \frac{\mathbf{L} + \mathbf{V}}{\|\mathbf{L} + \mathbf{V}\|}$ instead of computing the reflection vector $\mathbf{R}$.
The specular term becomes $k_s I_s (\mathbf{N} \cdot \mathbf{H})^n$, which is computationally faster than computing $\mathbf{R}$ per pixel.

Question 15

Flat Shading vs. Gouraud Shading vs. Phong Shading

Accepted Answer

- **Flat**: Evaluates the lighting model *once per polygon face*; uniform color across the primitive.
- **Gouraud**: Evaluates lighting *per vertex* and linearly interpolates the resulting *colors* across the face.
- **Phong**: Interpolates surface *normals* across the face and evaluates the lighting model *per pixel/fragment*.

Question 16

Vertex Shader

Accepted Answer

A programmable GPU pipeline stage that runs per vertex. Its primary responsibility is transforming vertex positions from model space to clip space (via MVP matrix) and computing per-vertex data like normals or texture coordinates.

Question 17

Fragment Shader

Accepted Answer

A programmable GPU pipeline stage that runs per fragment (potential pixel). It evaluates texture sampling, lighting algorithms, and material properties to produce the final output color and depth value of a pixel.

Question 18

Whitted-Style Recursive Ray Tracing

Accepted Answer

A global illumination algorithm that shoots primary rays from the camera through pixels. Upon intersecting geometry, it recursively casts:
1. **Shadow rays** to light sources
2. **Reflection rays** in the mirror direction
3. **Refraction rays** according to Snell's Law

Question 19

Bounding Volume Hierarchy (BVH)

Accepted Answer

A tree structure used to accelerate spatial queries (like ray-triangle intersections) by partitioning geometric objects into hierarchical bounding volumes (e.g., AABBs). Reduces ray intersection complexity from $O(N)$ to $O(\log N)$.

Question 20

UV Mapping

Accepted Answer

The 2D image mapping process of projecting a 2D texture onto a 3D model surface using normalized coordinates $(u, v) \in [0, 1]^2$, where $u$ represents the horizontal axis and $v$ the vertical axis.

Question 21

Supersampling (SSAA) vs. Multisampling (MSAA)

Accepted Answer

- **SSAA**: Renders the scene at a higher resolution and downsamples, executing both vertex and fragment shaders at multiple sub-pixel samples.
- **MSAA**: Evaluates the fragment shader *once per pixel*, but evaluates depth and coverage tests at multiple sub-pixel locations to anti-alias geometry edges efficiently.

Waterloo CS488 - Introduction to Computer Graphics

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