UNIT 4
PART A |
modeling transformations |
Viewing transformations |
world coordinate system. |
re-position the scene in front of a camera |
view orientation transformation |
DC to WC; camera at origin looking along -z axis |
parallel projection |
perspective projection |
preserves true measurements; less realistic |
realistic (depth); does not preserve true measurements |
Center of Projection (COP) or Projection Reference Point (PRP) is the single point where all projectors converge |
Viewport Clipping - after projection in standard viewing pipeline (before the perspective projection division; avoids division by zero) |
Back-Face Detection condition |
V = N • D |
Scan-Line Method |
determines the closest object along the line of sight (depth) and its corresponding color |
Painter's Algorithm. - Depth-Sorting Method |
Primary criterion: sorted by minimum z-depth
Ambiguity tests: Bounding rectangle overlap test, depth overlap test, surface intersection test, and final "complete behind" test. |
composite transformation matrix is not commutative
limitation of Back-Face
Detection : multiple front faces overlap or obscure each other (hard to detect)
|
|
Parallel vs. Perspective projection
Parallel projection |
Perspective projection |
Orthographic: perpendicular rays
Oblique: angled rays |
converge at a point (foreshortening) |
No depth cues, looks artificial, parallel lines stay parallel |
Realistic, depth visible, lines converge (vanishing points) |
Technical Drawings |
Realistic Visualization |
Parallel → accuracy & measurement
Perspective → realism & depth
Back-Face Detection algorithm
Removes polygons whose normal faces away from viewer |
N⋅V≤0⇒discard |
Sufficient for: |
Single, convex, closed objects (e.g., cube) |
Limitations |
1. Concave Objects: |
May remove faces that are actually visible |
2. Multiple Objects: |
Cannot decide visibility between different objects; No depth comparison |
3. Transparent Surfaces: |
Incorrectly removes visible back faces |
4. Open Surfaces: |
Back-face concept may not apply |
5. Does not resolve cyclic overlap cases |
6. Object-space only; doesn’t compare between objects |
does not work for scene-level visibility |
It is a pre-processing culling step that reduces the polygon count sent to the rendering pipeline (by ~50%).
followed by a true Hidden Surface Removal (HSR) algorithm (e.g., Depth-Buffer/Z-Buffer, Painter's, BSP Trees)
3D clipping
Steps |
1. Transform vertices → Normalized Device Coordinates (NDC) using projection |
2. Clip against 6 planes of canonical cube: |
x=±1, y=±1, z=0/1 |
3. Test vertices (inside/outside) |
4. Compute intersection points if edges cross planes |
5. Discard outside parts, pass clipped primitives for viewport mapping |
Standardized Cube |
Uniform planes (inside/out and intersection math faster); hardware optimized; simplified depth (0 to 1) |
Clipping in normalized space: consistency & simplicity (axis-aligned with plane equations than skewed, perspective dependent frustum); device independence; efficiency
|
|
3D projection
Perspective Projection |
1. find parameter t |
t=zp-zCoP/z-z~CoP |
2. projected x' and y' |
x′=xCoP+t(x−xCoP)
y′=yCoP+t(y−yCoP) |
3. projected z' |
z′=zp |
Orthographic Projection |
1. Keep x and y unchanged |
x′=x, y′=y |
2. Set z to projection plane |
z′=zp |
Multi-Viewport Viewing Pipeline for CAD
1. Divergence of Single Modeling Stream |
One modeling transformation pipeline (world/model matrix) feeds all viewports. |
2. Viewing & Projection Management |
Each viewport has its own:
View Matrix (camera orientation)
Projection Matrix
Orthographic (Top, Front, Side)
Perspective (3D view) |
3. Clipping per Viewport |
Clipping done independently for each viewport (frustum) |
4. Overall architecture |
Model Transform → Stream Duplication → Parallel Per-Viewport (View+Proj Transform → Clipping → Viewport Transform). |
Multi-Viewport Viewing Pipeline for CAD
1. Divergence of Single Modeling Stream |
One modeling transformation pipeline (world/model matrix) feeds all viewports. |
2. Viewing & Projection Management |
Each viewport has its own:
View Matrix (camera orientation)
Projection Matrix
Orthographic (Top, Front, Side)
Perspective (3D view) |
|
