1 chapter 5 viewing. 2 opengl perspective matrix parallel perspective axonometric multiview...
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Chapter 5
Viewing
2
OpenGL Perspective Matrix
parallel perspective
axonometric multivieworthographic
oblique
isometric dimetric trimetric
2 point1 point 3 point
planar geometric projections
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Perspective Projection
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Parallel Projection
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Classical Viewing Viewing requires three basic elements
One or more objects A viewer with a projection surface Projectors that go from the object(s) to the projection surface
Classical views are based on the relationship among these elements The viewer picks up the object and orients it how she would like
to see it
Each object is assumed to constructed from flat principal faces Buildings, polyhedra, manufactured objects
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Planar Geometric Projections Standard projections project onto a plane Projectors are lines that either
converge at a center of projection are parallel
Such projections preserve lines but not necessarily angles
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Classical Projections
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Orthographic Projection Projectors are orthogonal to projection surface
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Multiview Orthographic Projection
Projection plane parallel to principal face Usually form front, top, side views
isometric (not multivieworthographic view)
in CAD and architecture, we often display three multiviews plus isometric
top
front
side
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Advantages and Disadvantages Preserves both distances and angles
Shapes preserved Can be used for measurements
Building plansManuals
Cannot see what object really looks like because many surfaces hidden from view Often we add the isometric
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Axonometric ProjectionsAllow projection plane to move relative to object
classify by how many angles ofa corner of a projected cube are the same
none: trimetrictwo: dimetricthree: isometric
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Types of Axonometric Projections
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Construction of Axonometric Projections
Projectors are still orthogonal to the projection plane, but Projection plane can have any orientation w.r.t. the object
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Advantages and Disadvantages Lines are scaled (foreshortened) but can find scaling factors Lines preserved but angles are not
Projection of a circle in a plane not parallel to the projection plane is an ellipse
Can see three principal faces of a box-like object Some optical illusions possible
Parallel lines appear to diverge
Does not look real because far objects are scaled the same as near objects
Used in CAD applications
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Oblique ProjectionArbitrary relationship between projectors and
projection plane
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Advantages and Disadvantages Can pick the angles to emphasize a particular face In physical world, cannot create with simple
camera; possible with bellows camera or special lens (architectural)
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Perspective Projection Projectors coverge at center of projection
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Vanishing Points Parallel lines (not parallel to the projection plan)
on the object converge at a single point in the projection (the vanishing point)
Drawing simple perspectives by hand uses these vanishing point(s)
vanishing point
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Three-Point Perspective No principal face parallel to projection plane Three vanishing points for cube
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Two-Point Perspective On principal direction parallel to projection plane Two vanishing points for cube
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One-Point Perspective One principal face parallel to projection plane One vanishing point for cube
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Advantages and Disadvantes Objects further from viewer are projected smaller than t
he same sized objects closer to the viewer (diminuition) Looks realistic
Equal distances along a line are not projected into equal distances (nonuniform foreshortening)
Angles preserved only in planes parallel to the projection plane
More difficult to construct by hand than parallel projections (but not more difficult by computer)
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Computer Viewing There are three aspects of the viewing process,
all of which are implemented in the pipeline, Positioning the camera
Setting the model-view matrix Selecting a lens
Setting the projection matrix Clipping
Setting the view volume
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The OpenGL Camera In OpenGL, initially the world and camera
frames are the same Default model-view matrix is an identity
The camera is located at origin and points in the negative z direction
OpenGL also specifies a default view volume that is a cube with sides of length 2 centered at the origin Default projection matrix is an identity
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Default ProjectionDefault projection is orthogonal
clipped out
z=0
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Moving the Camera Frame If we want to visualize object with both positive and negativ
e z values we can either Move the camera in the positive z direction
Translate the camera frame Move the objects in the negative z direction
Translate the world frame Both of these views are equivalent and are determin
ed by the model-view matrix Want a translation (glTranslatef(0.0,0.0,-d);) d > 0
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Moving Camera Back from Origin
default frames
frames after translation by –d d > 0
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Moving the Camera (Object) We can move the camera to any desired
position by a sequence of rotations and translations
Example: positive x (side view of an object) Rotate the camera Move it away from origin Model-view matrix C = TR
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OpenGL Code Remember that last transformation
specified is first to be applied
glMatrixMode(GL_MODELVIEW)glLoadIdentity();glTranslatef(0.0, 0.0, -d);glRotatef(90.0, 0.0, 1.0, 0.0);
We are changing the modelview matrix C. A vertex specified at qthrough glVertex, after the change is still at q in the world frame, but is at Cq in the camera frame known internally to OpenGL.
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Manually Generating an Isometric View – 1/3
Rotating about the y axis by 45o
Rotating about the x axis by 35.26o
Translating along the z direction by d
z
y
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2 z
y
22 2
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Manually Generating an Isometric View – 2/3
1000
3/33/33/3
06/63/66/6
02/202/2
1000
03/33/33/3
06/63/66/6
02/202/2
1000
100
0010
0001
1000
02/202/2
0010
02/202/2
1000
03/63/30
03/33/60
0001
1000
100
0010
0001
dd
dRTRM yx
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Manually Generating an Isometric View – 3/3
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
glTranslatef(0.0, 0.0, -d);
glRotatef(35.26, 1.0, 0.0, 0.0);
glRotate(45.0, 0.0, 1.0, 0.0);
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The LookAt Function – 1/2 The GLU library contains the function glLookAt to fr
om the required modelview matrix through a simple interface
Note the need for setting an up direction Still need to initialize
Can concatenate with modeling transformations Example: isometric view of cube aligned with axes
glMatrixMode(GL_MODELVIEW):glLoadIdentity();gluLookAt(-1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0);
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The LookAt Function – 2/2glLookAt(eyex, eyey, eyez, atx, aty, atz,
upx, upy, upz)
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Other Viewing APIs – 1/2 The LookAt function is only one possible
API for positioning the camera Others include
View reference point, view plane normal, view up (PHIGS, GKS-3D)
Yaw, pitch, roll Elevation, azimuth, twist Direction angles
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Other Viewing APIs – 2/2
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Simple Perspective Center of projection at the origin Projection plane z = d, d < 0
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Perspective EquationsConsider top and side views
xp =dz
x
/yp =
dz
y
/zp = d
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Homogeneous Coordinate Form
M =
0/100
0100
0010
0001
d
consider q = Mp where
1
z
y
x
dz
z
y
x
/
q = p =
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Perspective Division However w 1, so we must divide by w to
return from homogeneous coordinates This perspective division yields
the desired perspective equations We will consider the corresponding clipping
volume with the OpenGL functions
xp =dz
x
/yp =
dz
y
/zp = d
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Orthogonal Projections The default projection in the eye (camera) frame is
orthogonal For points within the default view volume
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
xp = xyp = yzp = 0
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Homogeneous Coordinate Representation
xp = xyp = yzp = 0wp = 1
pp = Mp
M =
1000
0000
0010
0001
In practice, we can let M = I and set the z term to zero later
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Projections in OpenGL
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OpenGL PerspectiveglFrustum(xmin,xmax,ymin,ymax,near,fa
r)
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Using Field of View With glFrustum it is often difficult to get the d
esired view gluPerpective(fovy, aspect, near, fa
r) often provides a better interface
aspect = w/h
front plane
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OpenGL Orthogonal ViewingglOrtho(xmin,xmax,ymin,ymax,near,far)
glOrtho(left,right,bottom,top,near,far)
near and far measured from camera
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Hidden-Surface Removal
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Z-Buffer Algorithm Keep track of the smallest depth or z value f
or each pixel Z value is initialized to the farthest distance Worst-case time complexity is O(n), where
n is the number of polygons glutInitDisplayMode(GLUT_DEPTH);
glEnable(GL_DEPTH_TEST);glClear(GL_DEPTH_BUFFER_BIT);
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Culling Removing all the faces pointing away from
the viewer. For example, rendering n cubes with culling can filter 3n polygons
glEnable(GL_CULL);
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Walking Through a Scene – 1/3void keys(unsigned char key, int x, int y)
{
if(key==‘x’) viewer[0]-=1.0;
if(key==‘X’) viewer[0]+=1.0;
if(key==‘y’) viewer[1]-=1.0;
if(key==‘Y’) viewer[1]+=1.0;
if(key==‘z’) viewer[2]-=1.0;
if(key==‘Z’) viewer[2]+=1.0;
glutPostRedisplay();
}
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Walking Through a Scene – 2/3void display(void)
{
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
glLoadIdentity();
gluLookAt(viewer[0],viewer[1],viewer[2],0.0,0.0,0.0,0.0,1.0,0.0);
glRotatef(theta[0],1.0,0.0,1.0);
glRotatef(theta[1],0.0,1.0,0.0);
glRotatef(theta[2],0.0,0.0,1.0);
colorcube();
glFlush();
glutSwapBuffers();
}
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Walking Through a Scene – 3/3void myReshape(int w, int h)
{
glViewport(0, 0, w, h);
glMatrixMode(GL_PROJECTION);
glLoadIdentity();
if(w<=h) glFrustum(-2.0, 2.0, -2.0*(GLfloat)h/(GLfloat)w,
2.0*(GLfloat)h/(GLfloat)w, 2.0, 20.0);
else glFrustum(-2.0, 2.0, -2.0*(GLfloat)w/(GLfloat)h,
2.0*(GLfloat)w/(GLfloat)h, 2.0, 20.0);
glMatrixMode(GL_MODELVIEW);
}
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Parallel-Projection Matrices
Perspective ViewOrthographic projection of distorted object
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Normalization Rather than derive a different projection
matrix for each type of projection, we can convert all projections to orthogonal projections with the default view volume
This strategy allows us to use standard transformations in the pipeline and makes for efficient clipping
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Pipeline View
modelviewtransformation
projectiontransformation
perspective division
clipping projection
nonsingular
4D 3D
against default cube 3D 2D
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Notes We stay in four-dimensional homogeneous coordinate
s through both the modelview and projection transformations Both these transformations are nonsingular Default to identity matrices (orthogonal view)
Normalization lets us clip against simple cube regardless of type of projection
Delay final projection until end Important for hidden-surface removal to retain depth infor
mation as long as possible
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Orthogonal NormalizationglOrtho(left,right,bottom,top,near,fa
r)normalization find transformation to convertspecified clipping volume to default
Canonical view volume
-1-1 -11 1 1
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Orthogonal Matrix Two steps
Move center to origin
T(-(left+right)/2, -(bottom+top)/2,(near+far)/2)) Scale to have sides of length 2
S(2/(left-right),2/(top-bottom),2/(near-far))
1000
200
02
0
002
nearfar
nearfar
farnear
bottomtop
bottomtop
bottomtop
leftright
leftright
leftright
P = ST =
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Final Projection Set z =0 Equivalent to the homogeneous coordinate
transformation
Hence, general orthogonal projection in 4D is
1000
0000
0010
0001
Morth =
P = MorthST
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Oblique Projections The OpenGL projection functions cannot produce
general parallel projections such as
However if we look at the example of the cube it appears that the cube has been sheared
Oblique Projection = Shear + Orthogonal Projection
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General Shear
top viewside view
cot
cot
tan
zyy
zxx
xx
z
p
p
p
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Shear Matrixxy shear (z values unchanged)
Projection matrix
General case:
1000
0100
0φcot10
0θcot01
H(,) =
P = Morth H(,)
P = Morth STH(,)
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Equivalency
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Effect on Clipping The projection matrix P = STH transforms
the original clipping volume to the default clipping volume
top view
DOP DOP
near plane
far plane
object
clippingvolume
z = -1
z = 1
x = -1x = 1
distorted object(projects correctly)
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Simple PerspectiveConsider a simple perspective with the COP at the
origin, the near clipping plane at z = -1, and a 90 degree field of view determined by the planes
x = z, y = z
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Perspective MatricesSimple projection matrix in homogeneous
coordinates (z=-1, and the center of projection is at the origin)
Note that this matrix is independent of the far clipping plane
0100
0100
0010
0001
M =
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Generalization
0100
βα00
0010
0001
Let N =
after persepective divison, the point (x, y, z, 1) goes to
x’’ = -x/zy’’ = -y/zz’’ = -(+/z)
If we apply an orthographic projection along the z-axis to N, we get:
0100
0000
0010
0001
MorthN=
And the projection of an arbitrary point pis, p’=MorthNp = , after perspective division:
z
y
x
0
xp = -x/zyp = -y/z
We get the same result regardless of and
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Picking and If we pick
minmax
minmax
zz
zz
minmax
minmax
zz
zz2
the zmin plane is mapped to z = -1the zmax plane is mapped to z =1and the sides are mapped to x = 1, y = 1
Hence the new clipping volume is the default clipping volume
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Normalization Transformation
original clipping
volumeoriginal object new clipping
volume
distorted objectprojects correctly
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Normalization and Hidden-Surface Removal
Although our selection of the form of the perspective matrices may appear somewhat arbitrary, it was chosen so that if z1 > z2 in the original clipping volume then the for the transformed points z1’ > z2’
Thus we hidden surface removal works if we first apply the normalization transformation
However, the formula z’’ = -(+/z) implies that the distances are distorted by the normalization which can cause numerical problems especially if the near distance is small
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OpenGL Perspective glFrustum allows for an unsymmetric vie
wing frustum (although gluPerspective does not)
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OpenGL Perspective Matrix The normalization in glFrustum requires
an initial shear to form a right viewing pyramid, followed by a scaling to get the normalized perspective volume. Finally, the perspective matrix results in needing only a final orthogonal transformation
P = NSH
our previously defined perspective matrix
shear and scale
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Normalization DetailsShear the point of ((xmin+xmax)/2,(ymin+ymax)/2, zmax) to (0, 0, zmax)
)
2(cot),
2(cot),(
max
maxmin1
max
maxmin1
z
yy
z
xxHH
So the resulting frustum is described by the planes:
max
min
max
minmax
max
minmax
2
2
zz
zz
zz
yyy
zz
xxx
Next, scale the volume so that x=z, y= z, so the scaling matrix is:S(2zmax/(xmax-xmin),2zmax/(ymax-ymin),1)Finally, we make use of the previous N.
x
z z=zmax
z=zmin
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Why do we do it this way? Normalization allows for a single pipeline
for both perspective and orthogonal viewing
We keep in four dimensional homogeneous coordinates as long as possible to retain three-dimensional information needed for hidden-surface removal and shading
We simplify clipping
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Projections and Shadows
Idea: put the projectionplane behind the object!For example, in this figure,assume (xl, yl, zl) is a lightsource, then we can set the projection plane on theplane of y=0
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Implementing Shadows – 1/2Move the light source to the originby a translation:T(-xl, -yl, -zl), then wecould use a simple perspective projectionthrough the origin, the projection matrix is
Finally we translate everything back.
001
0
0100
0010
0001
ly
M
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Implementing Shadows – 2/2GLfloat m[16];
/* shadow projection matrix */
For(i=0;i<15;i++) m[i]=0.0;
m[0]=m[5]=m[10]=1.0;
m[7]=-1.0/yl;
glColor3fv(polygon_color);
glBegin(GL_POLYGON);
. /* draw polygon normally */
.
glEnd();
glMatrixMode(GL_MODELVIEW);
glPushMatrix();
glTranslatef(xl, yl, zl);
glMultMatrixf(m);
glTranslate(-xl, -yl, -zl);
glColor3fv(shadow_color);
glBegin(GL_POLYGON);
. /* draw the polygon again */
.
glEnd();
glPopMatrix(); /* restore state */This works if objects cannot cast shadows on other objects!
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Summary and Notes Classical Viewing Parallel Projection versus Perspective
Projection OpenGL Projections Projection Normalization Shadows