Window Viewport and Clipping

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Window Viewport and Clipping

• Pradondet Nilagupta
• Dept. of Computer Engineering
• Kasetsart University

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Terminology (1/4)

• World Coordinate System (Object Space) the
  coordinate system used to describe the objects
  being displayed
• Window - The rectangle defining the part of the
  world we wish to display.

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Terminology (2/4)

• Screen Coordinate System (Image Space) - The
  space within the image is displayed
• Interface Window - The visual representation of
  the screen coordinate system for window
  displays (coordinate system moves with interface
  window).

Interface Window
Viewport
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Terminology (3/4)
Maximum Values Of Screen Co-ordinates

• Viewport - a rectangular region in the image
  space
• Viewing Transformation - The process of going
  from a window in world coordinates to a viewport
  in screen coordinates.

Viewport 2
Viewport 1
Screen Co-ordinates
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Terminology (4/4)

• Clipping

y
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Device Coordinates

• The integer (x,y) pixel coordinates are called
  device coordinates.

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Normalized device coordinates

• Normalised device coordinates (NDC) space is a
  square region with 0 ? x ? 1 and 0 ? y ? 1.
  The origin (0,0) is the lower left hand corner of
  the square.

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Mapping NDC to Device Coordinate (1/3)

• If a pixel is square, we can map NDC to the
  largest square region of the screen.
• For example, if the screen has device coordinates
  (X,Y) with 0 ? X ? 1023 and 0 ? Y ? 767, then we
  could map the NDC (x,y) using
• Xmax 1023
• Ymax 767
• X0 (Xmax - Ymax)/2
• Y0 0
• and
• X Ymax x X0
• Y Ymax y Y0

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Mapping NDC to Device Coordinate(2/3)

• If pixels are not square, define the aspect ratio
  R as the ratio of the height to width of a pixel.
• Assuming the horizontal width of a pixel is 1,
  the horizontal size of the screen is (Xmax1) and
  the vertical size is (Ymax1).

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Mapping NDC to Device Coordinate(3/3)

• Suppose tbe vertical size is less than the
  horizontal size, the corresponding mapping from
  NDC (x,y) to DC (X,Y) is
• Xmax 1023
  Ymax 767 X0
  (Xmax - RYmax)/2 Y0 0
  
• X RYmax x X0
  Y Ymax y Y0
• Can map a non-square region of NDC to DC so that
  all of the device coordinates are used.

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Viewing Transformation
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Mathematic (1/4)
y
First translate the window
World Co-ordinates
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Mathematic (2/4)
Mwv T(?) . S(?) . T(??)
T(?) T(umin, vmin)
T(??) T(-xmin, -ymin)
S(?) S
Putting it together.
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Mathematic (3/4)
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Mathematic (4/4)
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Line clipping

• A line clipping algorithm needs to determine if a
  line is inside, outside, or across the window.
• The whole line is accepted, If it is inside.
• The whole line is rejected, If it is outside.
• If it is across the window, the intersections of
  the line and the window boundaries need to be
  calculated and the line clipped to the
  boundaries.

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Cohen Sutherland Algorithm (1/2)

• The region code has four bits with meanings
• bit 0 1 if endpoint to on outside of left
  boundary.
• bit 1 1 if endpoint on outside of right
  boundary.
• bit 2 1 if endpoint on outside of bottom
  boundary.
• bit 3 1 if endpoint on outside of top boundary.

Region Outcodes
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Cohen Sutherland Algorithm (2/2)

• A line can be trivially accepted if both
  endpoints have an outcode of 0000.
• A line can be trivially rejected if any
  corresponding bits in the two outcodes are both
  equal to 1. (This means that both endpoints are
  to the right, to the left, above, or below the
  window.)
• if (outcode 1 outcode 2) ! 0000, trivially
  reject!

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Clipping Lines Not Accepted or Rejected

• In the case where a line can be neither trivially
  accepted nor rejected, the algorithm uses a
  divide and conquer method.

Line AD 1) Test outcodes of A and D --gt cant
accept or reject. 2) Calculate intersection point
B, which is on the dividing line between the
window and the above region. Form new line
segment AB and discard BD because above the
window. 3) Test outcodes of A and B.
Reject. Line EH ??
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Polygon Clipping
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Pipelined Polygon Clipping
Clip Top
Clip Right
Clip Bottom
Clip Left
Clipping against one edge is independent of all
others, 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.
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Sutherland Hodgman Algorithm (1/4)

• clips polygons by clipping the polygon to each
  window boundary in turn.
• The output of the algorithm is a set of vertices
  defining an area that is to be shaded.

The polygon is defined as an ordered sequence of
vertices
p4
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Sutherland Hodgman Algorithm (2/4)

• Each vertex is tested in turn against an extended
  window boundary.
• Vertices inside the window boundary are saved
  for clipping against the next boundary
• Vertices outside are discarded.

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Sutherland Hodgman Algorithm (3/4)
q1
P6
q2
New vertices q1,q2 are added
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Sutherland Hodgman Algorithm (4/4)

• If proceeding from inside the window to outside
• keep the inside vertex,
• add a new vertex where the edge crosses the
  window boundary,
• discard the outside vertex.
• If proceeding from outside to inside
• add a new vertex where the edge intersects the
  boundary,
• discard the outside vertex,
• keep the inside vertex.

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Canonical View Volume

• 3-D Extension of 2-D Cohen-Sutherland Algorithm,
  Outcode of six bits. A bit is true (1) when the
  appropriate condition is satisfied
• Bit 1 - Point is above view volume y gt -z
• Bit 2 - Point is below view volume y lt z
• Bit 3 - Point is right of view volume x gt -z
• Bit 4 - Point is left of view volume x lt z
• Bit 5 - Point is behind view volume z lt -1
• Bit 6 - Point is in front of view volume z gt zmin

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3D Line Clipping

• A line is trivially accepted if both endpoints
  have a code of all zeros.
• A line is trivially rejected if the bit-by-bit
  logical AND of the codes is not all zeros.
• Otherwise Calculate intersections.
• On the y z plane from parametric equation of
  the line
• y0 t( y1 - y0) z0 t( z1 z0)
• Solve for t and calculate x and y. Already know
  z y