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The concept of homogeneous coordinates, its application in representing data and transforming objects in 3d computer graphics. It covers topics such as representing numbers, defining objects and transformations, perspective projection, and vertex transformation. The document also discusses techniques like culling, clipping, and projection matrices.
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Homogeneous Coordinates
A way of representing data
Representing n−d space by n+1 dimensions
for example,
16bit word for an integer between −32768 and 32767
How to represent a number > 32767?
a position [ 60000, y, z ]
homogeneous coordindates:
[ 30000, y/2, z/2, 1/2 ]
Homogeneous Coordinates (cont’d)
− distinguish between a vector and a point
− modify the position of the origin of the coordinate system
there is no room in the 3x3 matrix to specify translation!
3x
x 1
1x3 1x
linear transformation (^) translation
perspective transformation
overall scaling
(rotation, scaling, reflection, shearing, ...)
− there is no unique homegeneous coordinate representation! −
h
h = 0?
View Volume
back clip plane
front clip plane
view plane window
(View Frustum)
front clip plane back clip plane
view plane
COP (^) h
Perspective Projection
− foreshortening
− line width is not preserved, so are angles
− irreversible
What happens if we move COP to infinitely far away?
Expressed in Homogeneous Coordindates
X/w Y/w Z/w w/w
xs ys zs 1
xs =
d zv
xv ys^ =^
d zv
yv
X xv Y yv Z zv w 1
0 0 1/d 0
where M =
X x v Y yv Z zv w zv/d
that is, =
after perspective divide
xv/w yv/w d 1
0 0 1/d 0
For parallel projection:
For perspective projection:
x s = x v ys = y v zs = 0 (with projection plan at z=0)
P’ = M proj Mview P
An overall transformatin from World to Screen space can be expressed as:
model−view maxtrix
projection matrix
perspective division
viewport transformation
Vertices
Stages of Vertex Transformation
view coordinates
clip coordinates
normalized device coordinates
window coordinates
Clipping
Point Clipping
xmin < x < xmax ymin < y < ymax
(xmin, ymin)
(xmax,ymax)
Otherwise, do not draw the point
Line Clipping
Brute Force Line Clipping
Compute intersections of line with every window boundary −> expensive
clip window
A
C
B D
E
Code(P0)&Code(P1) = 0
Code(P0) = Code(P1) = 0
Cohen−Sutherland Algorithm (cont’d)
Algorithm:
− Test for trivial reject or accept (exit?)
− If ( code(P0)!=0 and code(P1)=0 )
One point is inside and the other is outside. The line must be shortened. One or two intersections must be computed. else if ( code(P0)&code(P1)=0 )
Both endpoints are outside. Intersect with one of the sides of the window, and check the code of the resulting point
p 1 = −dx, q 1 = x1 − xmin p 2 = dx, q 2 = x max − x 1 p 3 = −dy, q 3 = y1 − ymin p 4 = dy, q 4 = y max − y 1
Liang−Barsky Line Clipping (cont’d)
pk = 0
q k < 0 the line is completely outside the boundary
the line is parallel to one of the clipping boundaries
q1< p 1=
q 2> p 4>
pk > 0
q k > 0 the line is inside the parallel clipping boundary
the line proceeds from the inside to the outside
Sutherland−Hodgeman Polygon Clipping
− clip the polygon to each of the window boundaries in succession − edge and vertex defintions of the polygon are updated accordingly
Polygon Clipping
The output of a polygon clipper should be a sequence of vertices that defines the clipped polygon boundaries
Concave case:
clipped
tessellation
