4ICT10 Computer Graphics and Virtual Reality

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4ICT10 Computer Graphics and Virtual Reality

• 9. World Windows, Viewports Clipping
• Dr Ann McNamara

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Definitions

• World Coordinate System (Object Space)
• This space in which the application model is
  defined. The representation of an object is
  measured in some physical or abstract units
• Screen Coordinate Space (Image Space)
• The space in which the image is displayed.
  Ususally measured in pixels but could use any
  units.

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Definitions

• Window, the rectangle defining the part of the
  world we wish to display
• Interface Window the visual represntation of the
  screen coordinate system for windowed displays
  (coordinate system moves with the interface
  window)

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Definitions

• Viewport the rectangle on the raster graphics
  screen (or interface window for window
  displays) defining where the image will appear,
  usually the entire screen or interface window
• Viewing Transformations the process of mapping
  from a window in world cordintates to a viewport
  in screen coordinates

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Windows and Viewports
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Window-Viewport Transformation
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Viewing Transformation Example
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Window-Viewport Transformation
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Window-Viewport Transformation

• Proportionality in mapping x to sx

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Window to Viewport Mapping

• sx Ax C sy By D
• A C V.l AW.l
• B D V.b BW.b

V.r-V.l
W.r-W.l
V.t-V.b
W.t-W.b
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Properties of Mapping

• If x is at the windows left edge x W.l the
  xs is at the viewports left edge sx V.l
• If x is at the windows right edge then sx is at
  the viewports right edge
• If x is a fraction of f of the way across the
  window then sx is a fraction of f the way across
  the viewport
• If x is outside the window to the left (xltW.l)
  theh xs is outside the viewport to the left
  (sxltV.l) and analogously if x is outside to the
  right
• One should also check these properties for y to sy

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Example

• Window (W.l, W.r, W.b, W.t)(0, 2.0,0,1.0)
• Viewport(V.l, V.r, V.b, V.t)(40,400,60,300)

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Example

• A 180, C 40, B 240, D 60
• sx 180x 40
• sy 240x 60
• Check!
• Each corner is indeed mapped
• The center of the window maps to the center of
  the viewport

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Example 2

• Find the values of A,B,C,D for the case of a
  world window (-10.0, 10.0, -6.0, 6.0) and a
  viewport of (0, 600, 0, 400)

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Window-Viewport Transformation in OpenGL

• gluOrtho2d does it for us!
• If my window is 5 meters by 4 meters and has a
  lower left corner at (2, 0) in world coordinates
• gluOrtho2D (2, 7, 0, 4)
• Will automatically transform the vertices
• If you do the transform manually, you normally
  dont include the last step (translating to
  screen coordinates) because you probably set up
  your interface window so that the origin was at
  the lower-left corner
• Generally, also need glViewport(l, b, r, t) but
  the default viewport uses the whole interface
  window

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Sequences of OpenGL Transformations
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2D in OpenGL
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Notes on Window-Viewport Transformation

• Panning
• Moving the window about the world
• Zooming
• Reducing/increasing the window size
• As the window increases in size the image in the
  viewport decreases in size and vice versa
• See Thursdays Tutorial for Examples!

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Setting the Window Automatically
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Setting the Viewport Automatically

• Largest undistorted version that will fit in
  screen?
• R Aspect Ratio of World
• Two situations

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Setting the Viewport Automatically

• RgtW/H
• World window is short and stout compared to
  screen window viewport with a matching aspect
  ratio R will extend fully across but will be some
  space unused above/below
• At largest therefore the viewport will have width
  W and height W/R
• setViewport(0, W, 0, W/R)

• RltW/H
• World window is Tall and Narrow compared to
  screen window viewport with a matching aspect
  ratio R will extend fully from top to bottom but
  will be some space unused at left/right
• At largest therefore the viewport will have width
  HR and height H
• setViewport(0, HR, 0, H)

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Example, Tall Window

• If the world window has R1.6 and the screen has
  H 200 and W 360, W/H 1.8. Therefore we have
  case B and the viewport is set to 200 pixels high
  and 320 wide
• (Height H 200, Width HR 2001.6 320!)

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Example, Short Window

• If the world window has R2.0 and the screen has
  H 200 and W 360, W/H 1.8. Therefore we have
  case B and the viewport is set to 180 pixels high
  and 360 wide
• (Width 360, Height W/R 360/2 180!)

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Clipping Lines
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Cohen-Sutherland Algorithm

• Region Checks
• Trivially accept/reject lines and points
• Fast for large windows and small windows
• Everything is either inside or outside!
• Each vertex is assigned a 4-bit outcode

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Cohen-Sutherland Algorithm
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Cohen-Sutherland Clipping
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Cohen-Sutherland Clipping

• Trivially Accepted
• Both endpoints have outcode FFFF
• Trivially Rejected
• Any correspoding bits in the two outcodes are T.
  (this means both endpoints are to the right, to
  the left, above or below the window)

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Clipping Lines not Accepted or Rejected

• In case where a line is neither trivially
  accepted or rejected
• divide and conquer approach

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Cohen-Sutherland Algorithm

• A must be computed
• x W.right
• y needs adjusting p1.y by the amount d
• By similar triangles
• e is p1.x-W.right
• delx p2.x p1.x
• dely p2.y p1.y
• p1.y-(p1.x-W.right)dely/delx

d/dely e/delx
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Polygon Clipping

• Polygons can be clipped against each edge of the
  window one edge at a time. Window/Edge
  intersections (if any) are easy to find since the
  x and y coordinates are known.
• Vertices which are kept after clipping against
  one window edge are saved for clipping against
  remaining edges. Note that the number of
  vertices ususally changes and will often increase.

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Polygon Clipping Algorithm

• The window boundary determines a visible or
  invisible region
• The edge from vertex i to vertex i1 can be one
  of 4 types
• Exit visible region save the intersection
• Wholly outside visible region save nothing
• Enter visible region save intersection and end
  point
• Wholly inside visible region save endpoint

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Polygon Clipping Issues

• The final output, if any, is always considered a
  single polygon
• The spurious edge may not be a problem since it
  always occurs on a window boundary, but can be
  eliminated if necessary

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Pipelined Polygon Clipping

• Because polygon clipping does not depend on any
  other polygons it is possible to arrange the
  clipping stages in a pipeline. The input polygon
  is clipped against one edge and any points that
  are kept are passed on as input to the next stage
  of the pipeline.
• This way four polygons can be at different stages
  of the clipping process simultaneously. This is
  often implemented in hardware

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Clipping in the OpenGL Pipeline