Unit 4: Clipping & Projection

4.1 Clipping Algorithms

Analogy: Taking a photo. Anything outside the viewfinder doesn't get recorded.

1. Cohen-Sutherland Line Clipping

Uses 4-bit Region Codes (TBRL - Top, Bottom, Right, Left) to quickly accept or reject lines.

The Logic:

Center (Visible): 0000
Corners: e.g., Top-Left = 1001 (Top=1, Left=1).
Rule 1 (Trivial Accept): If both points are 0000, keep it.
Rule 2 (Trivial Reject): If (Code1 AND Code2) $\neq$ 0000, throw it away. (Both are outside the same edge).
Rule 3 (Clip): If neither, calculate intersection and chop.

2. Mid-Point Subdivision Clipping

Concept: A "Binary Search" approach. Instead of calculating complex intersections (slow math), it simply cuts the line in half.

Is the whole line visible? No.
Divides into two halves.
Is the first half visible? Yes -> Keep. No -> Divide again.
Repeat until pixel size is reached.

3. Sutherland-Hodgman Polygon Clipping

Problem: Clipping a shape is harder than a line because cutting off a corner creates new edges.

The Pipeline Approach:

Clip the entire polygon against one boundary at a time.

Input Polygon $\rightarrow$ Left Clipper $\rightarrow$ Output
Input $\rightarrow$ Right Clipper $\rightarrow$ Output
Input $\rightarrow$ Top Clipper $\rightarrow$ Output
Input $\rightarrow$ Bottom Clipper $\rightarrow$ Final Polygon

Exam Warning (Numericals):

Ensure you can solve a Liang-Barsky clipping problem. It is more efficient than Cohen-Sutherland and frequently carries 10+ marks.

mindmap
  root((Clipping Logic))
    Code Checks
      Cohen-Sutherland
      TBRL Codes
      Accept/Reject
    Divide Checks
      Mid-Point Subdiv
      Binary Search
      Good for Hardware
    Shape Checks
      Sutherland-Hodgman
      Pipeline Cutters
      L -> R -> T -> B

4.2 Projections

Analogy: Casting a shadow.

Parallel: Sun is far away. Shadows are same size as object.
Perspective: Flashlight is close. Shadows get bigger; objects look smaller.

Feature	Parallel Projection	Perspective Projection
Visual	Parallel lines stay parallel. No vanishing point.	Parallel lines meet at a "Vanishing Point".
Size	True to size (Measurements are accurate).	Distant objects look smaller (Realistic).
Use Case	Architects, Engineers, CAD.	Games, Movies, Human Vision.

Sub-Types

Parallel Values:
- Orthographic: Top, Front, Side views.
- Oblique: Slanted view (Cavalier, Cabinet).
Perspective Values: 1-point, 2-point, 3-point.

mindmap
  root((View Types))
    Parallel
      CAD / Architect
      True Size
      No Vanishing Point
    Perspective
      Human Eye / Game
      Shrink with Dist
      Vanishing Points

4.3 The Math (Deep Dive)

Formulas often asked in "Hard" sections.

1. Perspective Formula (Similar Triangles)

Concept: Divide by depth ($z$).

If simple camera at Origin and Screen at distance $d$: $$x' = x \cdot \frac{d}{z}$$ $$y' = y \cdot \frac{d}{z}$$
(As $z$ increases, $x'$ gets smaller).

2. Oblique Projection Math

The front face is flat ($z=0$), but sides go back at angle $\phi$.

$x' = x + L \cdot z \cdot \cos\phi$
$y' = y + L \cdot z \cdot \sin\phi$

Variations:

Cavalier: Angle $45^\circ$, Length $L=1$. (Looks long).
Cabinet: Angle $63.4^\circ$, Length $L=0.5$. (Looks realistic).

3. Vanishing Points Logic

Tricky Question: What makes it 1, 2, or 3 point perspective?

Answer: It depends on how many axes ($x,y,z$) intersect/cut the Projection Plane.

1-Point: Plane cuts 1 axis ($z$). (e.g., Tunnel view).
2-Point: Plane cuts 2 axes ($x, z$). (e.g., Street corner).
3-Point: Plane cuts 3 axes ($x,y,z$). (e.g., Bird's eye view from skyscraper).

mindmap
  root((Projection Math))
    Perspective
      Divide by Z
      x' = x (d/z)
    Oblique
      Add Offset
      x' = x + LzCos
      Cavalier vs Cabinet
    Vanishing Pts
      Axes Intersected
      1-Pt (Z only)
      2-Pt (X, Z)
      3-Pt (X, Y, Z)