2D and 3D Transformations

Unit Structure

            
Computer Graphics: 2D and 3D Transformations
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├── 1. 2D Transformation
│   ├── Introduction to 2D Transformation
│   ├── Basic Transformations
│   │   ├── Translation
│   │   ├── Rotation
│   │   └── Scaling
│   ├── Composite Transformations
│   │   ├── Reflection
│   │   └── Shearing
│   └── Transformations Between Coordinate Systems
│
└── 2. 3D Transformation
    ├── Introduction to 3D Transformation
    ├── Parallel Projection
    ├── Perspective Projection
    ├── Visible Lines and Surfaces Identification
    └── Hidden Surface Removal Algorithms

When you're working with computer graphics, you're not just drawing shapes — you're controlling how those shapes move, rotate, resize, and appear in space. That’s where 2D and 3D transformations come in. These transformations help bring flat designs to life and add depth and realism to 3D scenes.
We’ll start with 2D transformations, where you’ll learn the basics like translation, rotation, and scaling — the building blocks of motion and shape manipulation on a flat plane. Then we’ll move into composite transformations, which let you combine effects like reflection and shearing for more complex visual effects. You’ll also explore how these transformations work between different coordinate systems.
Then we jump into the exciting world of 3D transformations. This is where we add a third dimension to everything — including parallel and perspective projections (which help mimic how our eyes see the world). You'll also learn how to identify which parts of 3D objects should be visible and how to hide the rest using hidden surface removal algorithms.

2D Transformation

2D transformation refers to operations that change the position, size, orientation, or shape of 2D objects in a two-dimensional plane (X-Y plane).
These transformations are fundamental in computer graphics, animation, and geometric modeling.

In practice, transformations are often represented using matrices, which allow multiple transformations to be combined into a single matrix operation using matrix multiplication.

Translation

Translation is the process of moving an object along a straight path from one coordinate location to another.
It is a type of rigid body transformation, meaning the shape and size of the object remain unchanged (no deformation or rotation occurs).

(tx, ty) is called the translation vector or shift vector. It indicates how far the object is moved along the x and y axes.
The translation equations are:
\( x' = x + t_x \)
\( y' = y + t_y \)
The matrix representation of the translation is:

\( \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} t_x \\ t_y \end{bmatrix} \)

Q1: Translate a point \( P = (2, 4) \) using the translation vector \( T = (-1, 14) \). Find the new coordinates \( P' \).

Given:
\( P = (2, 4) \)
\( T = (t_x, t_y) = (-1, 14) \)

Apply translation:
\( P' = P + T \)
\( \Rightarrow P' = (x + t_x, \; y + t_y) \)
\( \Rightarrow P' = (2 + (-1), \; 4 + 14) \)
\( \Rightarrow P' = (1, \; 18) \)

Q2: Translate a polygon with vertices \( A(2, 7) \), \( B(7, 10) \), and \( C(10, 2) \) by 3 units in the X direction and 4 units in the Y direction.

Translation vector: \( T = (3, 4) \)

Apply translation to each point:
\( A' = (2 + 3, \; 7 + 4) = (5, \; 11) \)
\( B' = (7 + 3, \; 10 + 4) = (10, \; 14) \)
\( C' = (10 + 3, \; 2 + 4) = (13, \; 6) \)

Scaling

Scaling is a transformation that changes the size of an object either by enlarging or shrinking it.
It is applied by multiplying the original coordinates with scaling factors along the x and y axes.
The shape of the object is preserved, but its size changes proportionally or non-proportionally depending on the scaling factors.

(sx, sy) are the scaling factors along the x-axis and y-axis respectively.
The scaling equations are:
\( x' = x \cdot s_x \)
\( y' = y \cdot s_y \)
The matrix representation of scaling is:
\( \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \)

Q1: Scale a point \( P = (3, 5) \) using the scaling factors \( s_x = 2 \), \( s_y = 3 \). Find the new coordinates \( P' \).

Given:
\( P = (3, 5) \)
\( s_x = 2, \; s_y = 3 \)

Apply scaling:
\( x' = x \cdot s_x = 3 \cdot 2 = 6 \)
\( y' = y \cdot s_y = 5 \cdot 3 = 15 \)
\( \Rightarrow P' = (6, \; 15) \)

Q2: Scale a triangle with vertices \( A(1, 2) \), \( B(4, 2) \), and \( C(1, 6) \) using scaling factors \( s_x = 0.5 \), \( s_y = 2 \).

Scaling vector: \( s_x = 0.5, \; s_y = 2 \)

Apply scaling to each point:
\( A' = (1 \cdot 0.5, \; 2 \cdot 2) = (0.5, \; 4) \)
\( B' = (4 \cdot 0.5, \; 2 \cdot 2) = (2, \; 4) \)
\( C' = (1 \cdot 0.5, \; 6 \cdot 2) = (0.5, \; 12) \)

Reflection

Reflection is a transformation that produces a mirror image of a shape or point across a specified axis (X-axis, Y-axis, or line \( y = x \)).
The size and shape of the object remain unchanged, but the orientation may change depending on the axis of reflection.
It is a rigid body transformation, meaning no distortion occurs—only position changes.

Reflection axes:
- X-axis → mirror across horizontal axis
- Y-axis → mirror across vertical axis
- Line \( y = x \) or \( y = -x \) → diagonal mirror
The reflection equations are:
- Across X-axis: \( x' = x, \; y' = -y \)
- Across Y-axis: \( x' = -x, \; y' = y \)
- Across line \( y = x \): \( x' = y, \; y' = x \)
Matrix representations:

Across X-axis:
\( \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \)

Across Y-axis:
\( \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \)

Across line \( y = x \):
\( \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \)

Q1: Reflect a point \( P = (3, 7) \) across the X-axis. Find the coordinates of \( P' \).

Given:
\( P = (3, 7) \)
Axis: X-axis

Apply reflection:
\( x' = x = 3 \)
\( y' = -y = -7 \)
\( \Rightarrow P' = (3, -7) \)

Q2: Reflect a triangle with vertices \( A(1, 4) \), \( B(5, 2) \), and \( C(3, 6) \) across the Y-axis.

Axis: Y-axis

Apply reflection to each point:
\( A' = (-1, 4) \)
\( B' = (-5, 2) \)
\( C' = (-3, 6) \)

Introduction to 3D Transformation

Okay, imagine you're playing a 3D video game. You're moving around the game world — going forward, rotating to see behind, or zooming in and out. What's actually happening behind the scenes is 3D Transformation.
Just like we had 2D transformations (moving, rotating, resizing in a flat plane), we now want to move and manipulate objects in a 3D space. This means we work with three axes:
- X-axis → left–right
- Y-axis → up–down
- Z-axis → forward–backward (depth)

Why do we need 3D transformations?

To move objects around in a 3D world.
To rotate or scale them realistically.
To simulate how things look in the real world when seen from different angles.
So instead of dealing with points like (x, y) in 2D, we now deal with (x, y, z) in 3D.

Transformation Type	Description	Example
Translation	Move an object in 3D space	Moving a cube 3 units up and 2 right
Rotation	Rotate around X, Y, or Z axis	Turning a box 90° to the right
Scaling	Resize in 3D (wider/taller/etc.)	Making a ball 2× bigger in all axes

Let's now see how 3D gets converted into 2D for our screens!

Parallel Projection

Let's say you want to draw a 3D object (like a cube) on a 2D screen (like your laptop). But our screen is flat — so how do we show depth?
One way is parallel projection.

What is it?

Parallel projection is like tracing a shadow of a 3D object using light rays that are parallel to each other. No matter how far the object is, its size doesn't change on the screen.
Think of it like architectural blueprints:
- A wall that is far away looks the same size as a closer wall.

Property	Parallel Projection
Depth perception	No (objects look same size)
Rays	Parallel
Use Case	Engineering, CAD drawings

Types of Parallel Projections:

Type	Description
Orthographic	Projectors are perpendicular to the screen.
Oblique	Projectors are at an angle.

Orthographic is like a front-view or top-view drawing. Oblique adds a bit of angle to show more dimensions.

Perspective Projection

Now, let's make things look more realistic.
In real life, when you look at a road, it seems to narrow down as it goes farther, right? That's perspective.

What is Perspective Projection?

It uses rays that converge at a point (like your eye or a camera). So, objects closer to the viewer look bigger, and those far away look smaller — just like in real life!

Property	Perspective Projection
Depth perception	Yes (looks realistic)
Rays	Converging
Use Case	Games, 3D animations, VR

There are different types of perspective projections:

Type	Description
One-point	One vanishing point (like a hallway view)
Two-point	Two vanishing points (like a building corner)
Three-point	Adds a third vanishing point (for tall scenes)

Visible Lines and Surfaces Identification

Alright, now we know how to project a 3D object onto 2D.
But there's a problem...
If you're looking at a cube, you don't want to see the back side of it, right?
So, we need to identify which lines and surfaces should be visible to the viewer.

How does it work?

The system checks:
- Which faces are facing towards the camera/viewer.
- Which ones are turned away (and therefore invisible).
Imagine you're holding a cardboard box. You can see the front, top, and one side. The system calculates that and only shows those parts.

Concept	Example
Back-face culling	Hide faces that point away from viewer
Front-face rendering	Show only visible surfaces

This step cleans up the image and reduces the number of calculations needed in rendering.

Hidden Surface Removal Algorithms

Okay, final piece of the puzzle!
Now, imagine a 3D model of a city. There are tons of buildings, and some are in front of others.
We don't want to draw things that are blocked from view, right? That wastes time and resources.
So we use Hidden Surface Removal Algorithms.

What do they do?

These algorithms figure out which surfaces or parts of objects are not visible from the viewer's position — and remove or skip them during rendering.

Algorithm	Description
Z-buffer	Keeps track of depth (z-value) of each pixel and draws only the closest.
Painter's Algorithm	Draws far objects first, then closer ones on top (like painting layers).
BSP Trees	Breaks the scene into front/back using planes.

Real-world Example:

In video games, when you're walking through a building, the system uses these algorithms so you only see the visible walls — not all the ones behind or hidden.

Summary

Here's a quick table to wrap up what we just learned:

Topic	Key Idea
3D Transformation	Moving/rotating/scaling 3D objects in space
Parallel Projection	Flat, non-realistic view (good for diagrams)
Perspective Projection	Realistic view with depth
Visible Lines/Surfaces	Identify what parts of 3D shapes should be shown
Hidden Surface Removal Algorithms	Hide or skip parts that are blocked or not visible to the viewer