程序代写代做代考 algorithm cse3431-lecture9
cse3431-lecture9
Matrix Order
Normally projection has to apply to all
objects (i.e. the entire scene) thus it must
pre-multiply the modelview matrix
• M = MprojMmodelview or
• M = MprojMviewMmodel
However, with shaders you have absolute control of the matrices
and the way they are multiplied
Important
Projection parameters are given in CAMERA
Coordinate system (Viewing).
So if camera is at z = 50, is aligned with the
world CS, and you give near = 10 where is
the near plane with respect to the world?
Important
Projection parameters are given in CAMERA
Coordinate system (Viewing).
So if the camera is at z = 50, is aligned with
the world CS, and you give |near| = 10 where
is the near plane with respect to the world?
• Transformed by inverse(Mvcs)
• i.e. (0,0,40)
Nonlinearity of perspective
transformation
Tracks:
Left: x= -1, y = -1
Right: x = 1, y = -1
Z = -inf, inf
View volume:
Left = -1, right = 1
Bot = -1, top = 1
Near = 1, far = 4
Reminder: ZVCS:=-near –> ZNDCS = -1
-far –> 1
Notice that the curve flattens as
-ZVCS–> far
On systems with limited numerical precision for the z-buffer (e.g. 8 bits)
a large difference in near and far can result in multiple ZVCS values to
map on the same value in ZNDCS. As a result the graphics system cannot
resolve visibility correctly!
Rule of thumb: Limit the z-range as much as you can
Z in NDCS vs -Z in VCS
ZNDCS = 5/3 +8/(3 (-ZVCS))
near far
Viewport transformation
Modeling
transformation
Viewing
transformation
Projection
transformation
Perspective
division
Viewport
transformation
OCS WCS VCS CCS
NDCS
DCS
M-1cam
Viewport
(left,bottom)
(right,top)
(1,1,-1)
NDCS Screen Space
(1,1,-1) –> (top,bottom, 0)
(-1,-1,-1) –> (left,bottom, 1)
• Transforms the canonical coordinates to a viewport of
size W x H from (0,0) at lower left; thus,
viewport is [ 0, W ] x [ 0, H ]
• Scales and translates z to be in [0,1]
• How does a partial coverage matrix look like?
Example: Full window coverage
MFull
V P
=
2
66
4
1 0 0 W/2
0 1 0 H/2
0 0 1 0
0 0 0 1
3
77
5
2
66
4
W
2
0 0 0
0 H
2
0 0
0 0 1 0
0 0 0 1
3
77
5
2
66
4
1 0 0 0
0 1 0 0
0 0 0.5 0.5
0 0 0 1
3
77
5
So..Pixel Centers?
• Pixel size: 1×1
• Therefore pixel centers
at fractional locations
in screen space
pij = (i.5, j.5)
• In OpenGL the bottom left corner of the window is at (0,0)
• In some windowing systems the top left is at (0,0)
• When do you care about this?…When needing the location
of the mouse from the windowing system
(0,0)
(3,3)
(1.5,0)
Viewport in WebGL
• gl.viewport( x, y, width, height );
– (x,y): lower left corner of viewport rectangle in pixels.
– width, height: width and height of viewport in pixels.
– Generally put the code in a reshape callback.
– Example: the whole window
• gl.viewport(0,0,canvas.width, canvas.height);
Why viewports?
Undo the distortion of the projection
transformation
Stereo views
Render the scene twice from different points
of view
Example: Two viewports
void render()
{
gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT);
// Set the first viewport
gl.viewport(0,0,canvas.width/2,canvas.height/2);
// Set an orthrographic projection matrix
projectionMatrix = ortho(-3,3,-3,3,1,100) ;
modelViewMatrix = mat4() ;
var eye = vec3(0,0,10) ;
modelViewMatrix = mult(modelViewMatrix,lookAt(eye, at , up));
drawObjects() ;
// Set the second viewport
gl.viewport(canvas.width/2,canvas.height/2,canvas.width/2,canvas.height/2);
// Set an orthographicprojection matrix
projectionMatrix = ortho(-3,3,-3,3,1,100) ;
modelViewMatrix = mat4() ;
eye = vec3(10,10,0) ;
modelViewMatrix = mult(modelViewMatrix,lookAt(eye, at , up));
drawObj() ;
}
Example: Two viewports
• Viewport one: lower left quadrant
• Viewport one: top right quadrant Width: 500 pixels
H
ei
gh
t:
50
0
pi
xe
ls
Transformations in the pipeline
Modeling
transformation
Viewing
transformation
Projection
transformation
Perspective
division
Viewport
transformation
OCS WCS VCS CCS
NDCS
DCS
(e.g. pixels)
LookAt()
Ortho()
Frustrum()
Perspective
gl.viewport()
Translate()…
ModelView Matrix
Vertex Shader
Matrices in the Pipeline
Modeling
transformation
Viewing
transformation
Projection
transformation
Perspective
division
Viewport
transformation
OCS WCS VCS CCS
NDCS
DCS
M-1camMmod Mproj
Mvp
Vertex Shader
Vertex Shader
attribute vec4 vPosition;
attribute vec3 vNormal;
varying vec4 fColor;
void
main()
{
gl_Position = projectionMatrix * modelViewMatrix * vPosition;
fColor = vec4(1.0f, 0.0f, 0.0f, 1.0f) ;
}
// Notice that perspective division happens later.
// gl_Position is in homogeneous coordinates
Line Rendering Algorithm
Compute Mmod
Compute M-1cam
Compute Mmodelview = M-1cam Mmod
Compute MO
Compute MP // disregard MP here and below for orthographic-only case
Compute Mproj = MOMP
Compute MVP // Viewport transformation
Compute M = MVP Mproj Mmodelview
for each line segment i between vertices Pi and Qi do
P = MPi; Q = MQi
drawline(Px/hP, Py/hP, Qx/hQ, Qy/hQ) // hP,hQ are the 4th coordinates of P,Q
end for
