3D Computer Fundamentals Part 1

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3-D Computer Fundamentals- Part 1 -

• Supervisor
• Naoufel Boulila

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What are we going to talk about?

• 3-D Transformations
• Translation, Scaling, Rotation
• Transformations in a matrix form, Projection
• Fixed point arithmetic
• Rasterization of 2-D primitives
• Points, Line segments, Polygons
• Textured polygons
• Anti-aliasing
• 2-D and 3-D Clipping
• 2-D Clipping strategies
• 3-D Clipping strategies

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Basics

• Euclidean spaces
• Operations on vectors

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Translation

• Moving the points of an object
• Moving the coordinate system (opposite direction)
• Moving (translating a set of points in 2D 3D)
• In 2D along X und Y axes
• In 3D along X, Y und Z axes

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Scaling

• Expansion or contraction
• of an object
• Optional Scaling of the space itself
• 0 lt factor lt 1 ? Contraction
• factor gt 1 ? Expansion

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Rotation

• Rotation in 2D using
• sin cos functions
• Optional Rotating space
• 3D Rotation with the help of 2D Rotations.
• Important
• Reference system?
• Positive directions?
• Sequence of rotations?

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Coordinate systems

• Right- and left-handed notation
• We are using the left-handed notation coordinate
  system (see picture)
• a angle roll
• ß angle pitch
• ? angle yaw

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3D Rotation
x' z sin() x cos() x'' x
x''' y'' sin() x''
cos() y' y y''
y' cos() z' sin() y''' y''
cos() x'' sin() z' z cos() x sin()
z'' y' sin() z' cos()
z''' z''

• x''' xmx1 ymy1 zmz1
• y''' xmx2 ymy2 zmz2
• z''' xmx3 ymy3 zmz3

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Matrix form

• Representing the transformation coefficients in a
  matrix form
• Scaling and Rotating with 3x3 matrix (we use also
  4x4 matrices to assimilate with translating)
• Translating with 4x4 matrix
• Example 3D Rotation matrix

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Projection transformations

• How can we map the 3D space onto a 2D screen?
• Parallel projection
  Perspective projection

Viewing
direction.
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Parallel projection

• All the points belonging to parallel lines in 3D
  space would come together in one point in 2D
  plane
• Angle between projection lines and projection
  plane
• 90 than is orthographic projection
• Otherwise oblique projection
• When camera position is not parallel to XY plane
  than we do first affine transformation and after
  that we use the parallel projection

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Perspective projection

• Similar triangles
• Problem with the z coordinate
• If zz then can come to distortions
• Solution is z-C/z where C is some constant
• If z0 than we get error because we divide by
  zero
• If z negative we get points (pixels) that are
  behind the viewer

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Fixed point arithmetic

• sin cos functions with real values
• Instead expensive floating-point operations we
  use fixed-point arithmetic

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Rasterization

• Raster graphics
• Image on the screen composed from a mosaic of
  pixels
• Rasterizing points
• Rasterizing line segments
• Rasterizing polygons
• Convex polygons
• Concave polygons

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Rasterizing points

• Each pixel is represented by one unit of memory
  (byte or word)
• Address of pixel (x,y) ysizex x with sizex
  and sizey as a dimensions of the picture
• Problem
• Point with a real coordinates
• Rounding to the nearest integer

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Rasterizing line segments

• Finding all pixels intersected by a given line
  (Xstart ,Ystart) and (X end ,Y end)
• ?X gt ?Y ?
• ?Y gt ?X ?
• Performance?
• Alternatives?
• Forward differences J.E.Bresenham Method

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Rasterizing polygons

• Different types of polygons convex, concave
• Rasterization method scan-line
• Concave (left) convex (right)
  polygon with pixel line

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Rasterizing convex polygons

• Finding and rasterizing horizontal pixel-lines
• We are using arrays to store the polygons pixel
  lines
• Edge scanning process (just like rasterization of
  a line segment)

Start
End
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Rasterizing concave polygons

• Splitting into convex polygons
• Problem
• Finding intersections of scan-line and edges is
  expensive
• Solution
• Rasterizing with pre-sorted list and active edge
  list

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Rendering shaded polygons

• Technique
• include lighting during rasterization
• apply a material texture onto the polygon
• Shading
• We integrate rasterization by interpolating light
  values for every vertex at the time of pixel line
  computations
• Ix color intensity value (see pic.)

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Rendering textured polygons

• Texturing a texture associates color value to
  two parameters (u,v) of polygons area
• procedural textures
• store a texture as a bitmap
• Combining interpolation technique for calculating
  color intensity and texturing a polygons area

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Texturing perspective projection
Screen space
Texture space

• Problem
• The effect of perspective destortion in linear
  texture mapping
• Solution
• Subdivision of the polygon before per.pr.
• Approximation instead of real texture mapping

A
B
A
B
C
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C
D
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Anti-aliasing

• Problem
• staircase effect of jagged lines and edges when
  using straightforward rasterization algorithms
• Solution
• The techniques
• using filtering on
• the rendered
• image and making
• it smoother

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Clipping

• Restriction of the coordinates of primitives to
  the boundaries of the screen
• Problem
• Point with z0 cannot be transformed into the
  screen
• Objects whith negative z are behind the viewer,
  and, thus, invisible
• Two sort of clippings
• 2D screen boundaries clipping
• 3D volume clipping

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2D Clipping strategies

• Three different approaches to 2D Clipping
• Clipping before the rasterization phase (for
  polygons) - Preclipping
• Clipping during the rasterization phase (for
  complex primitives)
• Clipping after the rasterization phase -
  Postclipping

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Clipping points line segments

• Points
• we are checking if the point coordinates
• are within the clipping area
• Line segments divide conquer
• We are finding the part of the line segment
• by clipping against some boundary and
• proceeding to other boundaries whith the
• sub-problem (usefully for any clipping area)

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Clipping line segments

• Can we speed up the proces?
• Region outcodes trivial
• acceptance or rejection
• Both endpoints are outside the
• clipp. area ? the line segment
• is rejected (acceptance analog)
• Bitwise and and or to decide
• whether the line segment would
• be accepted, rejected or taken
• through the clipping process

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Clipping line segments

• Finding intersections with
• or
• binary search

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Clipping polygons

• Problem
• Polygon can change its shape as a
• result of clipping
• Clipping divide conquer
• - at each iteration we
  obtain a progressively smaller
  problem

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Clipping polygons

• Proceeding
• We are considering pol. edges in order of
  appearance and use clipping (for AB and CD edges,
  picture below) ABCD ? AI1I2D
• The same proceeding for concave polygons

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3D Clipping strategies

• Two kind of points
• Projected inside the screen, belonging to the
  view volume
• Projected outside the screen (overflow problem)
• Parallel projection
• 2D boundaries clipping
• Perspective projection
• Volume clipping
• No effects for far objects
• Bounding box and sphere

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Conclusion

• Transformation
• Used to describe location, orientation and motion
  of virtual objects and their visibility to the
  viewer
• Rasterization
• Used to picture geometric primitives in raster
  display device. More complex objects are usually
  presented as a collection of simpler primitives
• Clipping
• Used to discard the objects in the world space
  which cant be rasterized or transformed with the
  perspective projection. It insures consistency of
  these two algorithms.