TOPIC 2

Computer Graphics

Introduction

In computer graphics, we work with objects defined in a three-dimensional world.
- 2D objects and worlds are special cases of the general three-dimensional context.
All objects to be drawn, along with their respective cameras, possess:
  - Shape
  - Position
  - Orientation
We write computer programs that describe these objects and their lighting.
- The aim is to compute the final pixel values displayed on the screen.

Fundamental Tools in Graphics

Two fundamental sets of tools assist in graphics:
- Vector Analysis
- Transformations
Methods are developed to describe various geometric objects.
We learn to convert geometric ideas into numerical representations.
- This results in crucial algorithms applicable in graphics programming.

Easy Problems for Vectors

Key problems can be simplified using vectors:
  - Calculate the center of a circle defined by three points.
  - Determine the shape and position of images on the viewplane.
  - Identify the reflection of a cube on a shiny cone and the shape of that reflection.

Vectors

Vectors are advantageous for solving challenging problems.
- Defined by: Length and Direction.
- They have no fixed position in space (relative to a coordinate system) and can be moved freely.
Points have a fixed position but lack length and direction.
Scalars represent size only (a numerical value).

Basics of Points and Vectors

All points and vectors are defined in relation to a coordinate system.
- Example depicted with a 2D coordinate system and right-handed and left-handed 3D coordinate systems.

Left and Right Handedness

Right-Handed System:
- Curl your fingers from the x-axis towards the y-axis; your thumb points along the z-axis.
Left-Handed System:
- Your thumb points opposite the direction of the z-axis.

Review of Vectors

Vectors visual representation:
- Illustrated as arrows of a specific length pointing in a specific direction.
Example: Displacements of stars in the Big Dipper over the next 50,000 years.

Vectors and Coordinate Systems

To calculate a vector $v$ from points $P = (1, 3)$ to $Q = (4, 1)$:
  - $v$ has components $(3, -2)$, derived from the calculation $Q - P$.
  - Movement from $P$ to $Q$ includes:
    - Down by 2 units.
    - Right by 3 units.
  - Vectors can be represented without position; hence vectors with the same components are equivalent.

Vector Operations

Difference between points forms a vector:
- $v = Q - P$.
Adding a vector to a point results in another point:
- $P + v = Q$.
An n-dimensional vector is typically represented as an n-tuple:
- For 2 or 3 dimensions: $v = (3, -2)$.

Vector Representations

Row vector representation:
- $v = (33, 142.7, 89.1)$.
Column vector representation:
- $v = \begin{pmatrix} 33 \ 142.7 \ 89.1 \end{pmatrix}$ .

Fundamental Vector Operations

Two basic operations:
- Addition of two vectors.
- Multiplication by a scalar.
If $a$ and $b$ are vectors, then:
- $a + b$ is also a vector.
- For a scalar $s$, $sa$ is a vector.

Vector Operations (Continued)

Subtracting vector $c$ from $a$ is equivalent to adding $a$ to $-c$:
- $-c = (-1)c$.

Linear Combinations of Vectors

Defined as:
- $v_1 ext{ ± } v_2 = (v_{1x} ext{ ± } v_{2x}, v_{1y} ext{ ± } v_{2y}, v_{1z} ext{ ± } v_{2z})$
- $sv = (sv_x, sv_y, sv_z)$.
A linear combination of vectors $v_1, v_2, ext{…}, v_m$ is given by:
- $w = a_1v_1 + a_2v_2 + ext{…} + a_mv_m$.
Example:
- $2(3, 4, -1) + 6(-1, 0, 2) = (0, 8, 10)$.

Affine and Convex Combinations

An affine combination exists if:
- $a_1 + a_2 + … + a_m = 1$.
Example:
- $3a + 2b - 4c$ is affine, but $3a + b - 4c$ is not.
Convex combinations require:
- $a_i ext{ ≥ } 0$ for all $1 ext{ ≤ } i ext{ ≤ } m$.
Example:
- $0.3a + 0.7b$ is convex, but $1.8a - 0.8b$ is not.

Set of All Convex Combinations of Two or Three Vectors

Convex combination represented as:
- $v = (1 - a)v_1 + av_2$, where $a$ varies from 0 to 1.
A corresponding example illustrated.

Vector Magnitude and Unit Vectors

Magnitude of an n-vector $w$ is denoted as $|w|$.
Examples:
- For $w = (4, -2)$, find $|w| = ext{??}$.
- For $w = (1, -3, 2)$, find $|w| = ext{??}$.
A unit vector has a magnitude of:
- $|v| = 1$.
The unit vector aligning in the same direction as vector $a$ is denoted (if $|a|
eq 0$).
- The process of converting $a$ into a unit vector is termed normalizing.

Properties of Unit Vectors

A unit vector can refer to a direction.
Any vector can be expressed as the product of its magnitude and direction:
- $a = |a| ext{ } ext{ direction}$.

Vector Dot Product

The dot product for n-vectors $v$ and $w$:
- $v ullet w = v_1w_1 + v_2w_2 + … + v_nw_n$.
Properties of the dot product:
  - Commutative: $v ullet w = w ullet v$.
  - Distributive: $(a ext{ ± } b) ullet c = a ullet c ext{ ± } b ullet c$.
  - Associative over scalar multiplication: $(sa) ullet b = s(a ullet b)$.
  - Self dot product yields the magnitude squared: $b ullet b = |b|^2$.

Applications: Angle Between Two Vectors

Vectors represented:
- $b = (|b| ext{ cos } heta_b, |b| ext{ sin } heta_b)$.
- $c = (|c| ext{ cos } heta_c, |c| ext{ sin } heta_c)$.
The connection:
  - $b ullet c = |b| |c| ext{ cos }( heta_c - heta_b)$ where $ heta = heta_c - heta_b$ is the smaller angle between $b$ and $c$.
  - The relationship is given by:
  - $cos( heta) = rac{ ext{ }{ullet ext{ }} ullet ext{ }}$.

Angle Conditions

Conditions determined by the angle $ heta$:
- Positive: if $| heta| < 90^{ ext{o}}$. - Zero: if $| heta| = 90^{ ext{o}}$. - Negative: if $ heta > 90^{ ext{o}}$.
Vectors $b$ and $c$ are orthogonal if:
- $b ullet c = 0$.

Standard Unit Vectors

Standard unit vectors in 3D:
- $i = (1,0,0)$, $j = (0, 1, 0)$, and $k = (0, 0, 1)$ (pointing in the positive z direction).
In 2D:
- $i = (1,0)$ and $j = (0, 1)$.
Noteworthy property: the standard unit vectors are orthogonal.

Finding a 2D Perpendicular Vector

Given a vector $a = (a_x, a_y)$,
  - The vector perpendicular in a counterclockwise sense is:
    - $a^{ot} = (-a_y, a_x)$.
  - In the clockwise sense, it's $-a^{ot}$.
In 3D, any vector in the plane perpendicular to $a$ defines a "perp" vector.

Properties of Perpendicular Vectors

Key properties:
  - $(a ext{ ± } b)^{ot} = a^{ot} ext{ ± } b^{ot}$.
  - $(sa)^{ot} = s(a^{ot})$.
  - $(a^{ot})^{ot} = -a$.
  - Inner product relationships:
    - $a^{ot} ullet b = -b^{ot} ullet a = -a_yb_x + a_xb_y$.
  - Orthogonality condition:
    - $a^{ot} ullet a = a ullet a^{ot} = 0$.
  - Magnitude property:
    - $|a^{ot}| = |a|$.

Orthogonal Projections and Distances from a Line

Given points $A$ and $C$ and a vector $v$:
  - Address the following queries:
    - How far is $C$ from line $L$ (passing through $A$ in direction $v$)?
    - If we drop a perpendicular from $C$ to $L$, where does it intersect?
    - How can we decompose vector $c = C - A$ into components along $L$ and perpendicular to $L$?

Answering the Projection Questions

Define $c = C - A$:
- We can express $c$ as:
- $c = Kv + Mv^{ot}$.
Analyzing through dot products:
- $c ullet v = Kv ullet v + Mv^{ot} ullet v = K|v|^2$ gives:
- $K = rac{c ullet v}{|v|^2}$ (projection of $c$ onto $v$).
Dot product with $v^{ot}$ provides:
- $M = rac{c ullet v^{ot}}{|v|^2}$.
Answers to original questions include:
- Distance: $|Mv^{ot}|$.
- Point of intersection: $A + Kv$ or $C - Mv^{ot}$.

Application of Projection: Reflections

A reflection scenario occurs when light strikes a shiny surface or when a billiard ball collides with the edge of a wall.

Reflections Law

When light reflects from a mirror, the angle of reflection equals the angle of incidence:
- $ heta_1 = heta_2$.
Vectors and projections permit calculation of the new direction $r$ in 2D or 3D scenarios.

Reflection Mathematical Representation

Illustrated relationships:
- $e = a - m$ and $r = e - m = a - 2m$.
For $m$:
- $m = rac{(a ullet n)}{|n|^2}n = (a ullet n)n$; returns projection of $a$ onto vector $n$.

Vector Cross Product (3D Vectors Only)

Cross product defined as:
- $a imes b = (a_yb_z - a_zb_y)i - (a_xb_z - a_zb_x)j + (a_xb_y - a_yb_x)k$.
Determinant representation:
- $a imes b = \begin{vmatrix} i & j & k \ a_x & a_y & a_z \ b_x & b_y & b_z \end{vmatrix}$ .

Properties of Cross Product

Key properties include:
  - $i imes j = k$; $j imes k = i$; $k imes i = j$.
  - Antisymmetry: $a imes b = -b imes a$.
  - Distributivity: $a imes (b ext{ ± } c) = a imes b ext{ ± } a imes c$.
  - Scalar multiplication: $(sa) imes b = s(a imes b)$.
  - Non-valid associativity: $a imes (b imes c)
eq (a imes b) imes c$.
Resultant vector $c = a imes b$ remains perpendicular to both $a$ and $b$; direction follows right/left-hand rules based on coordinate systems.

Additional Cross Product Properties

Self dot product:
- $a ullet (a imes b) = 0$.
Magnitude representation:
- $|a imes b| = |a| |b| ext{ sin } heta$, where $ heta$ is the smaller angle between $a$ and $b$.
Cross product also provides the area of the parallelogram formed by vectors $a$ and $b$.
Equals zero when $a$ and $b$ are collinear or one has length zero.

Geometric Interpretation of the Cross Product

The area of the parallelogram spanned by vectors $a$ and $b$ is determined by:
- $Area = |a imes b|$.

Application: Finding the Normal to a Plane

Given three non-collinear points $A$, $B$, and $C$, we can determine a normal vector to the plane defined by these points:
- Let $a = B - A$, $b = C - A$; the normal vector $n = a imes b$.
- The opposite normal on the other side of the plane is $-n$.

Representations of Key Geometric Objects

Representation of lines and planes is crucial in graphics for distinguishing points:
- This process is simplified by using 4 coordinates instead of 3 (homogeneous representation).

Coordinate Systems and Frames

Points and vectors have coordinates in a coordinate system.
Multiple coordinate systems can coexist, defined by origins at various spatial locations.
A coordinate frame comprises a single origin point ($O$) along with 3 mutually perpendicular unit vectors: $a$, $b$, and $c$.

Coordinate Frames (Continued)

A vector $v$ can be represented as:
- $v = v_1a + v_2b + v_3c$.
A point $P$ from the origin can be expressed as:
- $P - O = p_1a + p_2b + p_3c$.

Homogeneous Coordinates

Both points and vectors are treated uniformly as entities in homogeneous coordinates:
  - A vector is represented as $\begin{pmatrix} v_1 \ v_2 \ v_3 \ 0 \end{pmatrix}$ .
  - A point is expressed via:
  - $\begin{pmatrix} v_1 \ v_2 \ v_3 \ 1 \end{pmatrix}$ .

Changing to and from Homogeneous Coordinates

Conversion to homogeneous coordinates:
- If representing a vector, append a 0 as the 4th coordinate.
- If representing a point, append a 1.
To revert from homogeneous coordinates, simply remove the 4th coordinate.
  - OpenGL utilizes 4D homogeneous coordinates for processing all vertices:
    - A 3-tuple $(x, y, z)$ is converted to $(x, y, z, 1)$.
    - A 2D point $(x, y)$ is converted to $(x, y, 0, 1)$.
All calculations in OpenGL are managed in 4D homogeneous coordinates.

Example: Cross Product Calculation

Let:
  - $A = (1, 2, 0)$
  - $B = (-2, 1, 3)$,
  - Compute:
  - $a = (1, 2, 0)$
  - $b = (-2, 1, 3)$
  - Compute $a imes b$:
  - Evaluate determinant:

   $ext{det} = \begin{pmatrix} i & j & k \ 1 & 2 & 0 \ -2 & 1 & 3 \end{pmatrix}$
  - Calculating individual components:
  - $i(6 - 0) - j(3 - 0) + k(1 + 4)$ provides $6i - 3j + 5k$.

Example: Normal to Plane Calculation

Given:
  - $A = (1,2,0)$,
  - $B = (-2, 1, 3)$,
  - $C = (-1, 1, 1)$,
  - Compute:
  - Let $a = A - C = (2, 1, -1)$
  - Let $b = B - A = (-3, -1, 3)$,
  - Calculate $a imes b$:
  - $= \begin{pmatrix} 2 & 1 & -1 \ -3 & -1 & 3 \end{pmatrix}$ :

$a imes b = -2i + 3j + k$
- Result: $c ullet a = 2(2) -3 -1 = 0$ (indicates orthogonality).

Here are 20 multiple choice questions (MCQs) with their respective choices and answers:

What does GPU stand for?
- A) Graphic Processing Unit
- B) General Processing Unit
- C) Graphical Product Unit
- D) Geometry Processing Unit
- Answer: A
Which of the following is a common graphics API?
- A) OpenGL
- B) HTML
- C) CSS
- D) PHP
- Answer: A
What is the primary purpose of the rasterization process?
- A) To compute pixel colors
- B) To create 3D models
- C) To shade objects
- D) To perform collision detection
- Answer: A
Which of the following describes a vertex in computer graphics?
- A) A pixel on the screen
- B) A point in 3D space
- C) A type of shader
- D) A mathematical function
- Answer: B
What is the term for a series of transformations applied to an object in a 3D space?
- A) Projection
- B) Animation
- C) Transformation
- D) Simulation
- Answer: C
What do shaders primarily do in graphics programming?
- A) Store textures
- B) Define how objects interact with light
- C) Create 3D models
- D) Handle user input
- Answer: B
In a right-handed coordinate system, what direction does the thumb point?
- A) Positive x-axis
- B) Positive y-axis
- C) Positive z-axis
- D) Negative z-axis
- Answer: C
What polygon is most commonly used in 3D graphics?
- A) Triangle
- B) Square
- C) Rectangle
- D) Hexagon
- Answer: A
Which term describes the visual distance between two points in 3D space?
- A) Angle
- B) Length
- C) Depth
- D) Width
- Answer: B
Which of the following algorithms is commonly used for surface rendering?
- A) Ray tracing
- B) Depth sorting
- C) Clipping
- D) Z-buffering
- Answer: A
What is the main function of the z-buffer in computer graphics?
- A) To store texture data
- B) To handle colors
- C) To manage depth information
- D) To define vertices
- Answer: C
Which of the following methods is used to achieve anti-aliasing in graphics?
- A) Smoothing
- B) Sampling
- C) Vectorization
- D) Clipping
- Answer: A
What is the primary advantage of using a vertex buffer?
- A) Reduces memory usage
- B) Increases performance
- C) Simplifies coding
- D) Enhances image quality
- Answer: B
In computer graphics, what does texture mapping refer to?
- A) Adjusting the brightness of an object
- B) Applying an image to a surface
- C) Rendering shadows
- D) Creating outlines
- Answer: B
What is the typical frame rate for smooth animation in computer graphics?
- A) 15 frames per second
- B) 24 frames per second
- C) 30 frames per second
- D) 60 frames per second
- Answer: D
What does the term 'normal vector' refer to in 3D graphics?
- A) The vector pointing to the camera
- B) A vector perpendicular to a surface
- C) A vector defining an object's movement
- D) A vector connecting two points
- Answer: B
Which geometry does not represent a shape in 3D graphics?
- A) Point
- B) Line
- C) Plane
- D) Surface
- Answer: A
What is the main purpose of transformations in computer graphics?
- A) To change memory allocation
- B) To simulate physics
- C) To manipulate objects in space
- D) To add textures
- Answer: C
Which of the following is true about ray tracing?
- A) It is faster than rasterization
- B) It simulates lighting effects accurately
- C) It is a real-time rendering technique
- D) It requires less computational power
- Answer: B
What type of graphics uses lines and shapes instead of pixels?
- A) Bitmap graphics
- B) Vector graphics
- C) Raster graphics
- D) Photorealistic graphics
- Answer: B

These questions can be utilized for quizzing individuals on their knowledge of computer graphics.