Transcript 幻灯片 1 - Shandong University

Review on Graphics Basics
Outline
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•
•
•
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Polygon rendering pipeline
Affine transformations
Projective transformations
Lighting and shading
From vertices to fragments
Outline
•
•
•
•
•
Polygon rendering pipeline
Affine transformations
Projective transformations
Lighting and shading
From vertices to fragments
Polygon Rendering Pipeline
• Process objects one at a time in the order they
are generated by the application
• Pipeline architecture
application
program
display
• All steps can be implemented in hardware on
the graphics card
Vertex Processing
• Much of the work in the pipeline is in converting object
representations from one coordinate system to
another
– Object coordinates
– Camera (eye) coordinates
– Screen coordinates
• Every change of coordinates is equivalent to a matrix
transformation
• Vertex processor also computes vertex colors
Projection
• Projection is the process that combines the 3D
viewer with the 3D objects to produce the 2D
image
– Perspective projections: all projectors meet at the
center of projection
– Parallel projection: projectors are parallel, center
of projection is replaced by a direction of
projection
Primitive Assembly
Vertices must be collected into geometric
objects before clipping and rasterization can
take place
– Line segments
– Polygons
– Curves and surfaces
Clipping
Just as a real camera cannot “see” the whole
world, the virtual camera can only see part of
the world or object space
– Objects that are not within this volume are said to
be clipped out of the scene
Rasterization
• If an object is not clipped out, the appropriate pixels
in the frame buffer must be assigned colors
• Rasterizer produces a set of fragments for each
object
• Fragments are “potential pixels”
– Have a location in frame buffer
– Color and depth attributes
• Vertex attributes are interpolated over objects by the
rasterizer
Fragment Processing
• Fragments are processed to determine the
color of the corresponding pixel in the frame
buffer
• Colors can be determined by texture mapping
or interpolation of vertex colors
• Fragments may be blocked by other fragments
closer to the camera
– Hidden-surface removal
Outline
•
•
•
•
•
Polygon rendering pipeline
Affine transformations
Projective transformations
Lighting and shading
From vertices to fragments
Affine Transformations
• Line and parallelism preserving
• Affine transformations used in Graphics
– Rigid body transformations: rotation, translation
– Scaling, shear
• Importance in graphics is that we need only
transform endpoints of line segments and let
implementation draw line segment between
the transformed endpoints
Homogeneous Coordinates
The homogeneous coordinates form for a three dimensional
point [x y z] is given as
p =[x’ y’ z’ w] T =[wx wy wz w] T
We return to a three dimensional point (for w0) by
xx’/w
yy’/w
zz’/w
If w=0, the representation is that of a vector
Note that homogeneous coordinates replaces points in three
dimensions by lines through the origin in four dimensions
For w=1, the representation of a point is [x y z 1]
Homogeneous Coordinates and
Computer Graphics
• Homogeneous coordinates are key to all
computer graphics systems
– All standard transformations (rotation, translation,
scaling) can be implemented with matrix
multiplications using 4 x 4 matrices
– Hardware pipeline works with 4 dimensional
representations
Translation Matrix
We can also express translation using a
4 x 4 matrix T in homogeneous coordinates
p’=Tp where
T = T(dx, dy, dz) =
1
0

0

0
0
1
0
0
0 dx 
0 dy 

1 dz 

0 1
This form is better for implementation because all affine
transformations can be expressed this way and multiple
transformations can be concatenated together
Rotation (2D)
Consider rotation about the origin by q degrees
– radius stays the same, angle increases by q
x’ = r cos (f + q)
y’ = r sin (f + q)
x’=x cos q –y sin q
y’ = x sin q + y cos q
x = r cos f
y = r sin f
Rotation about the z axis
• Rotation about z axis in three dimensions leaves all
points with the same z
– Equivalent to rotation in two dimensions in planes
of constant z
x’=x cos q –y sin q
y’ = x sin q + y cos q
z’ =z
– or in homogeneous coordinates
p’=Rz(q)p
Rotation Matrix
cos q  sin q
 sin q cos q
R = Rz(q) = 
 0
0

0
 0
0
0
1
0
0
0
0

1
Scaling
Expand or contract along each axis (fixed point of origin)
x’=sxx
y’=syx
z’=szx
p’=Sp
S = S(sx, sy, sz) =
 sx
0

0

0
0
sy
0
0
0
0
sz
0
0
0

0

1
Reflection
corresponds to negative scale factors
sx = -1 sy = 1
original
sx = -1 sy = -1
sx = 1 sy = -1
Shear
• Helpful to add one more basic transformation
• Equivalent to pulling faces in opposite directions
Shear Matrix
Consider simple shear along x axis
x’ = x + y cot q
y’ = y
z’ = z
1 cot q
0
1

H(q) =
0
0

0
0
0
0
1
0
0
0
0

1
Outline
•
•
•
•
•
Polygon rendering pipeline
Affine transformations
Projective transformations
Lighting and shading
From vertices to fragments
Affine vs. Projective Transformation
a
e
Ma = 
i

0
b
f
j
0
c
g
k
0
d
h 
l

1
a
e
Mp = 
i

m
b
f
j
n
c
g
k
o
d
h 
l

1
• Difference is in the last line of the transformation
matrix.
– Projective transformation preserves collinearity but
not parallelism, length, and angle.
– Affine transformation is a special case of the
projective transformation. However, it preserves
parallelism.
Projections and Normalization
• The default projection in the eye (camera) frame is
orthogonal
• For points within the default view volume
xp = x
yp = y
zp = 0
• Most graphics systems use view normalization
– All other views are converted to the default view by
transformations that determine the projection matrix
– Allows use of the same pipeline for all views
Homogeneous Coordinate
Representation
default orthographic projection
xp = x
yp = y
zp = 0
wp = 1
pp = Mp
M=
1
0

0

0
0
1
0
0
0
0
0
0
0

0
0

1
In practice, we can let M = I and set
the z term to zero later
Simple Perspective
• Center of projection at the origin
• Projection plane z = d, d < 0
Perspective Equations
Consider top and side views
x
z/d
xp =
x
z/d
yp =
y
z/d
zp = d
Homogeneous Coordinate Form
consider q = Mp where
q=
x
 y
 
z
 
1 
M=

p=
1
0

0

0
0 0
1 0
0 1
0 1/ d
 x 
 y 


 z 


z / d 
0

0
0

0
Perspective Division
• However w  1, so we must divide by w to
return from homogeneous coordinates
• This perspective division yields
xp =
x
z/d
yp =
y
z/d
zp = d
the desired perspective equations
• We will consider the corresponding clipping
volume with the OpenGL functions
Outline
•
•
•
•
•
Polygon rendering pipeline
Affine transformations
Projective transformations
Lighting and shading
From vertices to fragments
Phong Model
For each light source and each color component,
the Phong model can be written as
I = (a+bd+cd2)-1[ kd Id l · n+ks Is (v · r )a +kaIa]
For each color component
we add contributions from
all sources
Example
Only differences in
these teapots are
the parameters
in the modified
Phong model
Outline
•
•
•
•
•
Polygon rendering pipeline
Affine transformations
Projective transformations
Lighting and shading
From vertices to fragments
Pipeline Clipping of Polygons
• Three dimensions: add front and back clippers
• Strategy used in SGI Geometry Engine
• Small increase in latency
Scan Conversion and Interpolation
C1 C2 C3 specified by glColor or by vertex shading
C4 determined by interpolating between C1 and C2
C5 determined by interpolating between C2 and C3
interpolate between C4 and C5 along span
C1
C4
C2
scan line
C5
span
C3
Hidden Surface Removal
• z-Buffer Algorithm
– Use a buffer called the z or depth buffer to store the depth of
the closest object at each pixel found so far
– As we render each polygon, compare the depth of each pixel
to depth in z buffer
– If less, place shade of pixel in color buffer and update z buffer