159'235 Graphics

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159.235 Graphics Graphical Programming

• Lecture 19 - Introduction to 3D Graphics

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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 rendering
• Cropping (what to do if model is outside our
  renderable window)
• Not trivial but details hidden from us by the
  graphics library
• 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( x2 y2 )
• R is radius (a constant for a particular
  circle)
• theta is a parameter of the equation (radians
  angle - 2Pi -gt 1 full rotation or 360 degrees)
• Implement as
• for( theta 0 theta lt 2Pi theta increment )

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Circle Equation is Basis for Polar Coordinates

• (r,theta) lt--gt (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 representation
• 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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Voxels

• We sometimes imagine a voxelated model
• Voxel volume element
• Pixel picture element
• But until recently we did not have hardware
  that could render voxels directly
• 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 Games

• Recent examples - Shrek and Gollum
• Wire-frame models that are updated according to
  some approximate physics calculations (not yet
  in real-time!)
• Superpose texture maps (colours etc) onto the
  surfaces defined by the wire-frames
• Render and image - becomes still frame in a
  movie or game.
• 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 colour

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Polygons

• 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 information
• 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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Summary

• 3D Graphics allows us to render a real world
  model into a projected view
• The view can be rendered as per normal
  graphically
• 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