Proseminar Web3D 3D Computer Fundamentals Part I

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Proseminar Web3D3-D Computer
Fundamentals (Part I)
Naoufel Boulila

• Christian Trübswetter

TUM 2002
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Content

• 2-D/3-D Transformations
• Rasterization of 2-D Primitives
• 2-D/3-D Clipping

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2-D/3-D Transformations

• What is a transformation?
• What kind of transformations are there?
• How can I compute them?

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Rasterization of 2-D Primitives

• What is rasterization?
• What are the difficulties?

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2-D/3-D Clipping

• Clipping sounds simple
• But usually it isnt
• Problems with convexe polygons
• 3-D clipping

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Transformations
 

• 2-D Translation
• Scaling
• Rotation
• 3-D Basics
• 3-D Rotation
• Characteristics of the 3-D Space
• Optimizations
• Projection Transformation
• Perspective Projection
• Fixed Point Arithmetics

 
2-D/3-D Transformations
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Translation in 2-D
 
Figure Translation of a set of points,
translation of space.
Point-Vector /- Translation-Vector new
Point-Vector
2-D/3-D Transformations
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2-D Scaling
 
 Figure Scaling transformation.

• Computation
• Scalar Vector Vector

2-D/3-D Transformations
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Rotation in the Plane
 
With 2-D Rotation Matrix Rotation-Matrix
Point-Vector Point-Vector
 
Figure 2.8 Rotating the coordinate system.
2-D/3-D Transformations
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3-D Basics
 

• Spaces and Angles
• Roll (alpha, Z-axis), pitch (beta, X) and yaw
  (gamma, Y)

 
Figure 3-D Coordinates
2-D/3-D Transformations
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3-D Rotations
 

• Mathematical representation by a special kind of
  3-by-3 matrix
• A point (array or vector) is multiplied by a
  rotation matrix in order to get the new
  coordinates
• Multiplication of matrixes equals combined
  rotations
• Associative but not commutative

 
2-D/3-D Transformations
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Characteristics of the 3-D Space
 

• Different mathematical representation
• Representation of translation with addition of a
  vector
• Scaling by multiplying scalar
• Rotate with multiplication of matrixes
• What we want uniform behaviour of all those
  transformations
• Problem 3 by 3 matrix not possible, because
  translation requires affine, not linear
  transformation
• Solution 4 by 4 matrix

 
2-D/3-D Transformations
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Optimizations
 

• Sums instead of multiplications
• Frequent computation of sin, cos
• Matrix elements contain lots of trig. Functions
• Taylor Power Series, first two summands
• Precomputation of values
• Iterative algorithms

2-D/3-D Transformations
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Projection Transformation

• Mapping 3-D coordinates into 2-D screen
• Parallel projection

 
Figure Parallel projection.
2-D/3-D Transformations
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Perspective Projection
 

• Perspective Projection
• - xxfocus/z

  Figure Perspective projection.
  Figure Geometry of perspective projection.  

• Problems
• Singularity at z0
• Division is costly
• Visibility changes after perspective
  transformation

 
2-D/3-D Transformations
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Fixed Point Arithmetics
 

• Good fast additions and subtractions esp. when
  no floating point unit
• Bad overflow more likely

 
2-D/3-D Transformations
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Rasterization
 

• Rasterizing Points
• Rasterizing Line Segments
• Rasterizing Polygons
• Rendering Interpolatively Shaded Polygons
• Rendering Textured Polygons
• Anti-Aliasing

 
Rasterization of 2-D Primitives
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Rasterizing Points
 

• Points and pixels
• Discrete nature of pixels
• Address of pixel (x,y) is ysize(X)x

 
Figure Image bitmap layout.
Rasterization of 2-D Primitives
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Rasterizing Line Segments
 

• Finding points on a line
• Computation of the y-coordinate for every
  discrete x-value
• Problems
• - Which pixel(s) should be drawn
• - Degeneration of vertical lines
• - Computing every pixel necessary?
• Iterative computing of lines

 
Figure 3.3 Calculating Y function of X verses X
function of Y for some line.
Rasterization of 2-D Primitives
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Rasterizing Polygons
 

• Different kinds of polygons
• Simple, convex, concave, semi-simple
• Scanlines

 
  Figure Concave and convex polygons.  
Rasterization of 2-D Primitives
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Rasterizing convex Polygons
 

• One scanline
• Horizontal lines

 
Figure Array to store polygons pixel lines.
Rasterization of 2-D Primitives
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Rasterizing concave Polygons

• Division in convex polygons
• Disadvantage two many objects, lines are doubled
• Special algorithms for concave P.
• Ordered, active/passive edges

 
Figure Steps in finding convex polygons pixel
lines.
Rasterization of 2-D Primitives
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Rendering Interpolatively Shaded Polygons
 

• Pixels with additional values
• Linear interpolation

 
  Figure Interpolatively shaded polygons.
Figure 3.14 Interpolating color intensities.
Rasterization of 2-D Primitives
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Rendering Textured Polygons
 

• Warping when using linear mapping
• In case of shading normally its not noticed
• Solution Scanline subdivision, approximations

  Figure Linear texture mapping. Note unnatural
warping.
Figure Linear texture mapping for a
perspectively projected polygon.
 
  Figure Perspective texture mapping.
Figure Scanline subdivision and linear
approximation.
Rasterization of 2-D Primitives
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Anti-Aliasing
 

• Discrete nature of pixels and frames causes
  undesirable effects
• Stairs in lines, patterns in surfaces, optical
  illusions in videos

 
Figure Filtering.
Rasterization of 2-D Primitives
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Clipping

• 2-D Clipping Strategies
• Clipping Points
• Clipping Line Segments
• Clipping Polygons
• 3-D Clipping Strategies

2-D/3-D Clipping
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2-D Clipping Strategies
 

• Preclipping for simple primitives and rectangle
  clip. area
• Clipping during Rasterization for more complex
  primitives
• Postclipping for complex clipping areas

 
  Figure Different types of clipping.  
2-D/3-D Clipping
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Clipping Points
 

• Easy for rectangle clipping area
• - check if point is in the x- and y-range
• For others clipping areas
• - go through with scanline
• - check range(s)

2-D/3-D Clipping
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Clipping Line Segments
 

• Trivial rejection, based on region outcodes
• In case of acceptance clipping of he line against
  an edge
• Intersection computed by binary search

 
Figure Outcodes.
Figure Clipping a segment against a vertical
edge.
2-D/3-D Clipping
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Clipping Polygons
 

• Clip polygons iteratively against all edges
• Determine intersection points, new vertices, new
  edges
• Problem convex polygons
• - More than one part possible
• - How to treat them

 
Figure 4.8 Iterating polygons edges against a
single clipping edge.
Figure A scan-line coinciding with the
horizontal edge.
2-D/3-D Clipping
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3-D Clipping Strategies
 

• Clipping in case of parallel projection
• - Rectangular clipping areas
• Perspective clipping very complex and time
  consuming
• Singularities/overflow for near objects
• Limited number of primitives
• No 3-D effects for far objects
• Simplify the clipped volume perspective angles
  of 90
• Bounding boxes for detaild objects

 
Figure Bounding box and sphere for an object of
complex geometry.    
2-D/3-D Clipping
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Summary

• Transformation techniques allow to describe
  location, orientation and motion of virtual
  objects and also their visibility to the viewer
• The rasterization routines are used to picture
  geometric primitives. Shading and texture mapping
  can be added to improve visual realism
• Clipping discards the objects in the world space
  which would not contribute to the generation of
  the image

2-D/3-D Clipping
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Thats it

• Thank you for listening
• Questions please