Graphics by ewghwehws


									    159.235 Graphics & Graphical
          Lecture 19 - Introduction to 3D

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           3D Intro - Outline
•   2D Viewing
•   Parametric Equations
•   3D Models
•   Projective Geometry
•   3d Graphics Rendering and Voxels
•   Movies, games, and Virtual Reality
•   Polygons and scenegraphs
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                   2D Viewing
• Our “model” so far has been entirely 2-dimensional
• Model uses x,y Cartesian coordinates
• Simple mapping to pixel space coordinates for our
• Cropping (what to do if model is outside our
  renderable window)
• Not trivial but details hidden from us by the graphics
• Scaling and other Affine Transforms possible
• Colours, text, drawing primitives such as line,
  rectangle, filled shapes …
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  Details of Drawing Shapes and
           Clipping Them
• Graphics library (in our case the java Swing
  Library) does all this.
• Some interesting algorithms in fact needed
  for this
• Shapes drawn parametrically
• Clipping regions calculated rather than just
  implemented naively
• Important for speed and Performance
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     Parametric Equation for a Circle
• x = R cos(theta)
• y = R sin(theta)
• R = sqrt( x^2 + y^2 )
• R is radius (a constant for a particular circle)
• theta is a parameter of the equation (radians angle - 2Pi
  -> 1 full rotation or 360 degrees)
• Implement as:
    – for( theta = 0; theta < 2Pi; theta += increment )

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      Circle Equation is Basis for
           Polar Coordinates
• (r,theta) <--> (x,y)
• given a particular origin (x_0,y_0)

• We have affine transform utility:
     – g2.translate( x-amount, y-amount);

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           What about 3D Models?
• What do we do if we want to render something
  in 3D?
• Our model has (x,y,z) cartesian coordinates
• How do we translate this into a rendered view?
• We use some projective geometry:
• Project what you would see as a flat image or
  picture if you looked at a real world 3D object
  from a particular viewpoint
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            Projective Geometry
• We need transformations to project the model
  coordinates to a view plane coordinate space (what we
  or a camera “sees”)
• But we see a distorted view - things further away
  look different from things that are closer
• So also apply a perspective transformation - to distort
  what a viewer would see from a particular point in
  model space
• So a lot more computation needed to make all this
  work - luckily modern libraries will do a lot of the
  work for us
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          Geometry Implementation
• If we needed to build a library we would
  need to go into details of the Mathematical
  formulations of the projection transforms
• Typically write a a matrix that operates on
  a world coordinate space to a viewing one
• World might be (x,y,z)
• View might be (u,v) - like normal (x,y)
  but in the plane of the view
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      3D Graphics Implementation
• Generally your application maintains a
  world model
• This is mapped to the 3D internal
• A projection computes what would be seen
  in a 2D view space
• A rendering turns the 2D view into pixels

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             Rendering Pipeline
• This sequence of getting from model world to
  pixels is often called the rendering pipeline
• Sometimes have special hardware to help the
  various stages - eg polygon calculations
• Our world model might be based on a wire-frame
  model built from polygonal shapes and with surface
  textures or images that we paste on top of them
  – Together this information constitutes the “scene-graph”

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• We sometimes imagine a voxelated model
• Voxel = volume element
• Pixel = picture element
• But until recently we did not have
  hardware that could “render” voxels
• Polymer prototyping machines come close
• The scene-graph model is more useful
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3D Graphics is the basis of VR
• Virtual Reality (VR) is when we try to fool
  our human senses into thinking we are
  interacting with a model world that need
  never actually exist in reality
• It is entirely simulated in our program
• We render it into displays that we can
  surround ourselves with and total immersion
  VR goes further and simulates other sensory
  information such as sounds or movements
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    3D Graphics used in Movies &
• Recent examples - Shrek and Gollum
• Wire-frame models that are updated according to
  some approximate physics calculations (not yet in
• Superpose texture maps (colours etc) onto the
  surfaces defined by the wire-frames
• Render and image - becomes still frame in a movie or
• Around 25 frames-per-second fools the human vision
  system into thinking we see continuous motion
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          3D Practicalities
• How do we represent the world model?
• Wireframe model is set of polygons
• Possibly a very large set if the world
  object has smooth rounded surfaces
• A cube could be modeled with just 6
  polygons (6 squares in fact)
• Each might have a separate texture and
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• A polygon - many sided shape will have a set of
  vertices and edges
• A Graphics library may provide a Polygon class
• There are various file formats for storing polygon
• A growing set of proprietary and some free tools that
  let us design things - create the polygon information
• We can then store/exchange/combine these files
• Often our own applications generate the polygons
  from some “physics” (rules about our model world)
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          Generating World “Objects”
• We can develop and combine algorithms to
  generate shapes
• A cube is 6 squares
• A sphere could be generated parametrically from
  parametric spherical trigonometrical equation
• We can approximate a sphere by a surface made
  up of lots of polygons, with an opaque colour

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• 3D Graphics allows us to render a real world
  model into a projected view
• The view can be rendered as per normal
• All the information in a “scene-graph” together
  with the viewpoint and lighting model etc tells
  us what we would see if the model were “real”
• Uses in Movies, Games and VR systems
• Computationally very expensive -
  supercomputers needed

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