TwoDimensional Viewing

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Two-Dimensional Viewing
Chapter 6
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The Viewing Pipeline

• Window
• A world-coordinate area selected for display.
• defines what is to be viewed
• Viewport
• An area on a display device to which a window is
  mapped.
• defines where it is to be displayed
• Viewing transformation
• The mapping of a part of a world-coordinate scene
  to device coordinates.
• A window could be a rectangle to have any
  orientation.

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Two-Dimensional Viewing
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The Viewing Pipeline
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The Viewing Pipeline
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Viewing Effects

• Zooming effects
• Successively mapping different-sized windows on a
  fixed-sized viewports.
• Panning effects
• Moving a fixed-sized window across the various
  objects in a scene.
• Device independent
• Viewports are typically defined within the unit
  square (normalized coordinates)

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Viewing Coordinate Reference Frame

• The reference frame for specifying the
  world-coordinate window.
• Viewing-coordinate origin P0 (x0, y0)
• View up vector V Define the viewing yv direction

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Window-to-Viewport Coordinate Transformation
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Window-to-Viewport Coordinate Transformation
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Workstation transformtion
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Clipping Operations

• Clipping
• Identify those portions of a picture that are
  either inside or outside of a specified region of
  space.
• Clip window
• The region against which an object is to be
  clipped.
• The shape of clip window
• Applications of clipping
• World-coordinate clipping

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Clipping Operations

• Viewport clipping
• It can reduce calculations by allowing
  concatenation of viewing and geometric
  transformation matrices.
• Types of clipping
• Point clipping
• Line clipping
• Area (Polygon) clipping
• Curve clipping
• Text clipping
• Point clipping (Rectangular clip window)

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Line Clipping

• Possible relationships between line positions and
  a standard rectangular clipping region

Before clipping
after clipping
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Line Clipping

• Possible relationships
• Completely inside the clipping window
• Completely outside the window
• Partially inside the window
• Parametric representation of a line
• x x1 u(x2 - x1)
• y y1 u(y2 - y1)
• The value of u for an intersection with a
  rectangle boundary edge
• Outside the range 0 to 1
• Within the range from 0 to 1

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Cohen-Sutherland Line Clipping

• Region code
• A four-digit binary code assigned to every line
  endpoint in a picture.
• Numbering the bit positions in the region code as
  1 through 4 from right to left.

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Cohen-Sutherland Line Clipping

• Bit values in the region code
• Determined by comparing endpoint coordinates to
  the clip boundaries
• A value of 1 in any bit position The point is in
  that relative position.
• Determined by the following steps
• Calculate differences between endpoint
  coordinates and clipping boundaries.
• Use the resultant sign bit of each difference
  calculation to set the corresponding bit value.

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Cohen-Sutherland Line Clipping

• The possible relationships
• Completely contained within the window
• 0000 for both endpoints.
• Completely outside the window
• Logical and the region codes of both endpoints,
  its result is not 0000.
• Partially

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Liang-Barsky Line Clipping

• Rewrite the line parametric equation as follows
• Point-clipping conditions
• Each of these four inequalities can be expressed
  as

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Liang-Barsky Line Clipping

• Parameters p and q are defined as

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Liang-Barsky Line Clipping

• pk 0, parallel to one of the clipping boundary
• qk lt 0, outside the boundary
• qk gt 0, inside the parallel clipping boundary
• pk lt 0, the line proceeds from outside to the
  inside
• pk gt 0, the line proceeds from inside to outside

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Liang-Barsky Line Clipping
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Liang-Barsky Line Clipping

• Parameters u1 and u2 that define that part of the
  line lies within the clip rectangle
• The value of u1 From outside to inside (pk lt 0)
• The value of u2 From inside to outside (pk gt 0)
• If u1 gt u2, the line is completely outside the
  clip window.

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Liang-Barsky Line Clipping
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Nicholl-Lee-Nicholl Line Clipping

• Compared to C-S and L-B algorithms
• NLN algorithm performs fewer comparisons and
  divisions.
• NLN can only be applied to 2D clipping.
• The NLN algorithm
• Clip a line with endpoints P1 and P2
• First determine the position of P1 for the nine
  possible regions.
• Only three regions need be considered
• The other regions using a symmetry transformation
  
• Next determine the position of P2 relative to
  P1.

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Nicholl-Lee-Nicholl Line Clipping
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Nicholl-Lee-Nicholl Line Clipping
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Nicholl-Lee-Nicholl Line Clipping
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Nicholl-Lee-Nicholl Line Clipping

• To determine the region in which P2 is located
• Compare the slope of the line to the slopes of
  the boundaries of the clip region.
• Example P1 is left of the clipping rectangle, P2
  is in region LT.

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Nonrectangular Clip Windows

• Algorithms based on parametric line equations can
  be extended
• Liang-Barsky method
• Modify the algorithm to include the parametric
  equations for the boundaries of the clip region
• Concave polygon-clipping regions
• Split into a set of convex polygons
• Apply parametric clipping procedures
• Circle or other curved-boundary clipping regions
• Lines can be clipped against the bounding
  rectangle

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Splitting Concave Polygons

• Identify a concave polygon
• Calculating the cross product of successive edge
  vectors.
• If the z component of some cross product is
  positive while others have a negative, it is
  concave.

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Splitting Concave Polygons

• Vector method
• Calculate the edge-vector cross product in a
  counterclockwise order.
• If any z component turns out to be negative
• The polygon is concave.
• Split it along the line of the first edge vector
  in the cross-product pair.

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Splitting Concave Polygons

• Rotational method
• Proceeding counterclockwise around the polygon
  edges.
• Translate each polygon vertex Vk to the
  coordinate origin.
• Rotate in a clockwise direction so that the next
  vertex Vk1 is on the x axis.
• If the next vertex, Vk2, is below the x axis,
  the polygon is concave.
• Split the polygon along the x axis.

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Splitting Concave Polygons
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Polygon Clipping
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Sutherland-Hodgeman Polygon Clipping

• Processing the polygon boundary as a whole
  against each window edge
• Processing all polygon vertices against each clip
  rectangle boundary in turn

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Sutherland-Hodgeman Polygon Clipping

• Pass each pair of adjacent polygon vertices to a
  window boundary clipper
• There are four cases

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Sutherland-Hodgeman Polygon Clipping

• Intermediate output vertex list
• Once all vertices have been processed for one
  clip window boundary, it is generated.
• The output list of vertices is clipped against
  the next window boundary.
• It can be eliminated by a pipeline of clipping
  routine.
• Convex polygons are correctly clipped.
• If the clipped polygon is concave
• Split the concave polygon

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Sutherland-Hodgeman Polygon Clipping
v3
v2
v1
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Weiler-Atherton Polygon Clipping

• Developed as a method for identifying visible
  surfaces
• It can be applied with arbitrary polygon-clipping
  region.
• Not always proceeding around polygon edges
• Sometimes follows the window boundaries
• For clockwise processing of polygon vertices
• For an outside-to-inside pair of vertices, follow
  the polygon boundary.
• For an inside-to-outside pair of vertices, follow
  the window boundary in clockwise direction.

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Weiler-Atherton Polygon Clipping
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Other Clipping

• Curve clipping
• Use bounding rectangle to test for overlap with a
  rectangular clip window.
• Text clipping
• All-or-none string-clipping
• All-or-none character-clipping
• Clip the components of individual characters

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Exterior Clipping

• Save the outside region
• Applications
• Multiple window systems
• The design of page layouts in advertising or
  publishing
• Adding labels or design patterns to a picture
• Procedures for clipping objects to the interior
  of concave polygon windows

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Exterior Clipping