Clipping

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Clipping

• Clipping is a process of extracting a portion of
  the database (having many polygons)
• Clipping can be used to select a specific
  information to display
• Clipping can be 2D or 3D and implemented in
  hardware or software. For real time applications,
  clipping is typically implemented in hardware or
  firmware
• A regular clipping window is rectangular with its
  edges aligned with those of the object space or
  display device

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Clipping

• Points are interior if
• xl lt x lt xr and yb lt y lt yr
• (points on the boundary are included)
• A simple but inefficient method of clipping would
  be to have SetPixel(x,y,color) and ignore any
  requests to set a pixel outside the drawing
  window
• A better method is to clip the primitives
  themselves so that only the visible portions are
  further processed

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Cases for clipping lines
B
H
C
E
J
A
G
I
4
Cases for Clipping Lines
B
C
A
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Clipping Targets

• Points
• Lines
• Polygons
• Text
• Other Objects

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Clipping

• We will study problem of Clip Line against an
  axis-oriented rectangular window
• Expensive part of clipping is intersection
  calculations, so main goal of a clipping
  algorithm is to minimize this.

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Intersection calculation

• We can find where the line segment crosses each
  clipping-window edge by
• assigning the coordinate value for that edge to
  either x or y and solving for parameter t.
• ii) For instance, substitute xl for x, get t,
  calculate corresponding y.
• iii) If this value of t is outside the range from
  0 to 1, the line segment does not
• Intersect that window border line.
• iv) Otherwise, part of the line is inside. We
  process this inside portion against
• the other clipping boundaries until we have
  clipped the entire line.

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Clip to boundary
Clipping Edge
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Cohen-Sutherland Line clipping Algorithm

• Following algorithm considers efficiently
  clipping 2D lines to a rectangle .
• The motivating idea is to use inside/outside
  information about the line end-points. This
  generates the following possibilities
• Both end points are inside the rectangular
  window
• Line completely inside trivial accept
• Both end points are to the right of the right
  edge of the window
• Line completely outside trivial reject
• None of the above
• Clip against one edge and repeat

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Cohen-Sutherland Line clipping Algorithm
1001
1000
1010
y-top
0010
0001
0000
y-bottom
0110
0100
0101
X-right
X-left
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Look at Sign Bits (neg 1)

• Bit 1
• Bit 2
• Bit 3
• Bit 4

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Cohen-Sutherland Line clipping Algorithm

• The calculation of an outcode for each endpoint
  allows for an efficient implementation. An
  outcode consists of 4 bits TRBL, where
• T is set if y gt top
• B is set if y lt bottom
• R is set is x gt right , and
• L is set if x lt left

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Trivial accept cases

• The trivial accept case corresponds to ensuring
  that no outcode bits are set for both endpoints.
  This can be nicely accomplished with a bitwise OR
  operation.
• Out (A) 0101
• Out (B) 0100
• bitwise OR 0101 ? fails trivial accept
• Out(C) 0000
• Out (D) 0000 ? fails trivial accept

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Trivial reject cases

• The trivial reject case can be nicely
  accomplished with a bitwise AND operation. We can
  trivially reject a line segment which has a
  result not equal to 0000
• Out (A) 0101
• Out (B) 0100
• bitwise AND 0100 ? trivial reject
• Out (E) 1000
• Out (F) 0010
• bitwise AND 0000 ?fails trivial reject

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Example1
k
h
j
i
l
g
b
e
d
a
f
c
1000
1010
1001
T
0001
0010
0000
B
0101
0100
0110
L
R
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Clipping example1
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Cohen-Sutherland Line ClippingExample2
D
C
I
B
H
A
G
F
E
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Cohen-Sutherland Line ClippingExample2
D
C
1000
1001
1010
I
B
H
A
G
0000
0010
0001
F
E
0110
0101
0100
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Initial Outcode Calculations

• OC(D)1000 OC(A)0001
• OC(E)0100 OC(I)1010

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Clip and Continue

• Clip against the top boundary (and throw gt t)
• Calculate B. Keep AB
• Calculate H . Keep EH

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Cohen-Sutherland Line Clipping
B
H
A
G
F
E
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Clip and Continue

• Now test and reject AB because
• OC(A)0001 and OC(B)0001
• Reject AB on outcode basis
• Clip against the bottom boundary

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Outcode Calculations

• OC(H)0010 OC(E)0100
• Since product is 0, process HE to get FH

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Cohen-Sutherland Line Clipping
H
G
F
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Outcode Calculations

• OC(F)0000 OC(H)0010
• Since product is 0, process HF to get GF

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Clip and Continue

• Clip against the right boundary
• Get GF
• Done

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Cohen-Sutherland Line Clipping
G
F
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Process Ends

• Do not have to do
• Polygon clipping convex, concave?

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Cohen-Sutherland in 3D

• The Cohen-Sutherland algorithm extends easily to
  3D.
• 2D is extended to the 3D clipping window
  boundaries to define 27 regions. Assign a 6 bit
  code to each region, that is, for each point
• Left (first) bit set (1) point lies to left of
  window
• Right (second) bit set (1) point lies to right
  of window
• Bottom (third) bit set (1) point lies below
  window
• Top (fourth) bit set (1) point lies above
  window
• Near (fifth) bit set (1) point lies to in front
  of window (near)
• Far (sixth) bit set (1) point lies to behind of
  window (far)

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• The Left, Right, Bottom, Top, Near, Far (LRBTNF)
  outcode can be used to determine segments that
  are trivially visible, trivially invisible, or
  indeterminate. In the indeterminate case we
  intersect the line segment with faces of clipping
  cube determined by the outcode of an end-point
  that is outside of the clipping cube. More
  specifically, in the indeterminant case, use
  parametric form of the line.
• To clip against a face, say y1, compute
• and use it to evaluate the x and z intersections
  

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3D Clipping against a right parallelepiped
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Clipping for Orthographic Viewing
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Clipping for Oblique Viewing
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Distortion of VV by shear a) top view before
shear, b) top view after shear
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Equations

• Explicit equation y f(x)
• Dependent on choice of coordinate axes
• Not defined if more than one f(x) for x, sqrt(9)
• Implicit equation f(x, y) 0
• Quick to see if (x,y) is on curve
• Difficult to generate series of f(x) for x0g10
• Parametric x f(t), y f(t)

37
Plane Equations

• Implicit plane equation
• Ax By Cz D F(x,y,z)F(x,y,z)0 for points
  on the plane
• This can be rewritten asF(P) N.P D
• Parametric plane equation
• Plane(s,t) P0 s(P1-P0) t(P2-P0), provided
  P0, P1, and P2 are non-colinear.
• Plane(s,t) P0 s V1 t V2, where V1 and V2
  are two basis vectors.
• Explicit plane equation
• z -(A/C)x - (B/C)y - D/C , valid if C is
  non-zero.

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Line-plane Intersection
(Will also see in Ray-casting)

• L(t) Pa t(Pb - Pa)Substituting into the
  plane equation, and solving for t

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Graphics (rendering) pipeline
Viewing Geometry Processing World Coordinates
(floating point)
Rendering Pixel Processing Screen Coordinates
(integer)
Conservative VSD Selective traversal of object
database (or traverse scene graph to get CTM)
Transform Vertices To canonical view
volume
Light at Vertices Calculate light intensity at
vertices (lighting model of choice)
Conservative VSD Back-face Culling
Conservative VSD View Volume Clipping
Image- precision VSD Compare pixel
depth (Z-buffer)
Shading Interpolate color values (Gouraud) or
normals (Phong)
Projected!
per polygon
per pixel of polygon
avd November 9,
2004 VSD 6/48
avd November 5,
2002 VSD 5/45
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