UNIT 2
PART A |
homogeneous coordinates - represent all transformations as a uniform matrix multiplication; easy composition |
viewing pipeline - mapping world coordinate system to viewport (display device) |
modeling coordinates are local to the object and world coordinates are the common reference system |
normalization transformation - maps the world coordinate window to a standard, device-independent coordinate system (simplify clipping and viewport mapping) |
Sutherland-Hodgman:
P1 inside, P2 inside → Output P2
P1 inside, P2 outside → Output Intersection
P1 outside, P2 inside → Output I, then P2
P1 outside, P2 outside → Output nothing |
Line clipping algorithms: Cohen-Sutherland, Liang-Barsky (or Nicholl–Lee–Nicholl (NLN), Cyrus-Beck |
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Homogeneous Coordinates
Extension of 2D Cartesian coordinates (x, y) to 3D (x, y, w). |
Point represented as a column vector [x, y, 1]^T. |
Without homogeneous coordinates, translation is vector addition (T + v), while others are matrix multiplications (M * v). |
1. Unified Representation: 3x3 matrix multiplication |
2. Efficient Concatenation: multiple transformations |
3. Elegant Handling of Translation: translation requires addition |
4. Ability to Represent Points at Infinity: advanced graphical concepts |
5. Simplified Inverse Transform: inverse of the single composite matrix, |
Trade-off with Cartesian system:
1. Conceptual Complexity - understanding an extended coordinate system
2. Computational Complexity - slight increase (3x3 instead of 2x2) per operation; though there is a computational gain through matrix concatenation (composite matrix)
⇒ composite matrix transformations happen right to left
Polygon tables
Vertex + Polygon Tables |
Single Integrated Polygon Table |
Three-Table Model |
Vertex table (8 coords) + Polygon table (6 faces) |
Each face stores full vertex coordinates (6 faces, 3 coordinates) |
Vertex table + Edge table + Polygon table |
24 floats + 24 ints |
72 floats |
24 floats + 48 ints |
efficient, no duplication |
simple |
full topology, adjacency queries |
no explicit edge info |
high memory |
more memory + complex |
Data Efficiency |
best |
worst |
moderate |
Integrated = simple but wasteful | Three-table = powerful but heavy | Vertex+Polygon = best balance
2D viewing pipeline process
1. World Window Definition |
a rectangular region in the World Coordinate System (clipping boundary) |
2. Viewport Definition |
a rectangular region on the Device Coordinate System |
3. Window-to-Viewport Mapping Process |
linear transformation (mapping) from window to viewport |
Steps |
1. Translate window → origin |
2. Scale to viewport size |
3. Translate to viewport position |
(Additionally) Pscreen=Mscreen×Mviewport×Mzoom×Mpan×Pworld |
Pan: Mpan_i=Translate by (-Txi, -Tyi)
Zoom: M_zoom_i = Scale by (Si, Si) |
Flexibility: Separating pan & zoom matrices allows independent control. |
Efficiency: GPU-friendly single matrix multiplication |
Combined Importance
1. Separation of Concerns (decouples scene design from display specs)
2. Flexibility & Reuse
3. Device Independence
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Sutherland–Hodgman
Inside test |
xmin≤x≤xmax, ymin≤y≤ymax |
computationally cheap |
Intersection Checks for Lines |
point where a polygon edge crosses a clip boundary. |
Use line equations to calculate intersection coordinates |
intersection becomes a new vertex; closed and correctly shaped |
Iterative Vertex Generation |
processes the polygon against each clip boundary (left, right, bottom, top) successively. |
Modular and systematic - 4 simple clipping steps
Builds final result incrementally
Cohen–Sutherland
Assign Region Codes |
For both end points P1 and P2 |
Perform Trivial Tests |
Trivial Acceptance: (code1∣code2)=0000 |
Trivial Rejection: (code1&code2)≠0000 |
Select Outside Point |
Choose endpoint with non-zero region code |
Find Intersection |
Left/Right boundary (x = constant): y=y1+m(x−x1) Top/Bottom boundary (y = constant): x=x1+ (y−y1)/m |
Repeat |
Continue until: Accepted (both codes = 0000), or Rejected (AND ≠ 0) |
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