Clipping
15. Clipping
Procedures for eliminating all parts of primitives outside
of the specified view volume are referred to as clipping
algorithms or simply clipping
This takes place as part of the Projection and
Perspective Division transformations
Eye Coordinates
Projection Matrix
Projection
Transformation
Clip Coordinates
Perspective
Division
Normalized Device Coordinates
Projection is actually a two step process that is not
completed until after the Perspective Division operation
Clipping is done following the perspective normalization
transformation
E.R. Bachmann & P.L. McDowell
MV 4202 Page 1 of 6
Clipping
• Normalized View Volumes
Clipping against a regular parallelepiped is simplified
because each clipping plane is perpendicular to one of
the coordinate axes
- Finding the intersection of a line with a plane at an
arbitrary spatial orientation requires finding the
equation for the plane and floating point division
- In a parallelepiped:
◊ bottom and top clipping planes have a constant
y coordinate
◊ right and left clipping planes have a constant x
coordinate
◊ near and far clipping planes have a constant z
coordinate
♦ Perspective Normalization Transformation
Prior to clipping, the frustum, used for perspective
projection, is reduced to a rectangular parallelepiped
using shearing and scaling
- Shearing in the x and y directions puts the center
of projection on a line perpendicular to the
projection plane
- Scaling converts the sides of the frustum into
rectangular sides of a parallelepiped
- All points in the frustum are moved into the
parallelepiped
E.R. Bachmann & P.L. McDowell
MV 4202 Page 2 of 6
Clipping
• Perspective Projection Revisited
The perspective projection matrix used in OpenGL
 2n
0
 r −1

2n
 0
P=
t −b
 0
0

 0
0

r +1
r −1
t +b
t −b
− ( f + n)
f −n
−1

0 

0 

− 2 fn 
f −n
0 
(15.1)
is the result of a multiplication by a shear matrix, H, a
scaling matrix, S, and finally a projection normalization
matrix, N.
P = NSH
(15.2)
Following multiplication by this matrix, the viewing
volume becomes a parallelepiped which is bounded by
the planes
E.R. Bachmann & P.L. McDowell
x = ±1
(15.3)
y = ±1
(15.4)
z=0
(15.5)
z =1
(15.6)
MV 4202 Page 3 of 6
Clipping
(x, y, z)
center
of
proj.
proj.
plane
(x’, y’, z’)
viewing
volume
center
of
proj.
near
far
proj.
plane near
far
• Clipping Lines Against a Normalized View Volume
Once primitives have been converted to normalized
coordinates each vertex is checked against each of the
six clipping planes
Each vertex is assigned a six bit region code that
identifies the relative position with respect to the viewing
volume
bit 1 = 1, if x < xvmin (left) - least significant bit
bit 2 = 1, if x > xvmax (right)
bit 3 = 1, if y < yvmin (below)
bit 4 = 1, if y > yvmax (top)
bit 5 = 1, if z < zvmin (front)
bit 6 = 1, if z > zvmax (back)
- A code of 000000 identifies a point within the
viewing volume
E.R. Bachmann & P.L. McDowell
MV 4202 Page 4 of 6
Clipping
- A code of 101000 identifies a point as above and
behind the viewing volume
- A line segment for which both end points have a
region code of 000000 is entirely within the
viewing volume
- If only one end point is within the viewing volume,
the line must be clipped
In order to clip a line, the intersection of the line with the
clipping plane must be found
Recall the parametric equation for a line
x = (x2 - x1) * t + x1
(15.7)
y = (y2 - y1) * t + y1
(15.8)
z = (z2 - z1) * t + z1
(15.9)
for t in the range [0,1]
To find the intersection, substitute the constant
coordinate of the clipping plane into the appropriate
parametric equation and solve for t
E.R. Bachmann & P.L. McDowell
MV 4202 Page 5 of 6
Clipping
For the near clipping plane
t=
min − z1
z 2 − z1
zv
(15.10)
The x and y coordinates of the intersection are given by
 zv min -z1 

xI = x1 + ( x2 − x1 )
 z 2 − z1 
(15.11)
 zv min − z1 

y I = y1 + ( y2 − y1 )
 z 2 − z1 
(15.12)
If either xI or yI is not in the range of the boundaries of
the viewing volume, the line intersects the front plane
outside the volume
• Polygon Clipping
Polygon clipping algorithms must deal with many
different cases
- New edges must be added
- Existing edges must be discarded, retained or
divided
- Multiple polygons may result from clipping a single
polygon
E.R. Bachmann & P.L. McDowell
MV 4202 Page 6 of 6