Viewing

Lecture 5 Viewing and projections
Viewing
In OpenGL
The camera frame
Other view APIs
Projections
Perspective projections
Parallel projections
View volumes
Parallelpiped
Frustum

Viewing
Modeling coords
What we model object in
Independent of viewing
OpenGL uses the modeling part of the model-view matrix to convert from the modeling frame to the world frame
OpenGL camera
Placed at origin of world frame pointing in negative z direction

Viewing
If we model scene around origin the camera will not see all objects so we must
1) move camera back, or
2) move scene to in front of camera
Sequence of operations
1.Begin with model_view matrix = identity matrix
2.Define objects using glVertex
3.Apply model_view matrix to move world frame to camera frame (I.e. move objects to eye space)
*If we now position an object, is w.r.t. the repositioned world frame
These are equivalent operations

Camera positioning
OpenGL Example
This moves points Or we can think of it as world space positioning the camera
glMatrixMode (GL_MODEL_VIEW); glLoadIdentity (); glTranslate (0, 0, -d); glRotate (0, -90, 1, 0); -90deg z=-d 90deg z=+d M = TR = T(0,0,-d) R y (-90) M -1 = R -1 T -1 = R y (90) T(0,0,d)

Example: Find isometric view (p205) 45deg (-1,1,1)  (0,1,sqrt(2)) y y y z z z (1,1,1) d = sqrt(1*1+sqrt(2)*sqrt(2)) = sqrt(1+2) = sqrt(3)  cos  = sqrt(2)/sqrt(3) = sqrt(6)/3 sin  = sqrt(1)/sqrt(3) = sqrt(3)/3   = acos(sqrt(6)/3) = 35.26 1 0 0 0 0 sqrt(6)/3 sqrt(3)/3 0 0 -sqrt(3)/3 sqrt(6)/3 0 0 0 0 1 sqrt(2)/2 0 sqrt(2)/2 0 0 1 0 0 -sqrt(2)/2 0 sqrt(2)/2 0 0 0 0 1 R x R y = R =  y z (1,sqrt(2)) d 1 sqrt(2) 

Defining a camera frame
1. Position camera (VRP)
set_view_reference_point (x, y, z)
2. Specify view plane by its normal (VPN)
set_view_plane_normal (nx, ny, nz)
3. Specify camera up vector (VUP)
set_view_up (v upx, vupy, vupz)
(VUP must be orthogonal to VPN, so project to VP)
4. This gives us two orthogonal vectors, v and n
find the third, u = v x n
vup n VP u v vrp

Creating a camera frame
Given: p, n (unit), and up
Find: view matrix, V, which takes x,y,z to u,v,n
x,y,z are world coords, u,v,n are eye coords
V= RT =
u x u y u z 0 v x v y v z 0 n x n y n z 0 0 0 0 1 px py pz 1 pu pv pn 1 = V 1 0 0 -vrp x 0 1 0 -vrp y 0 0 1 -vrp z 0 0 0 1 Need to find u and v

Creating a camera frame
Find v:
Find u:
1. Project up onto n ([(up*n)/(n*n)]n) 2. Subtract the result (the comp. of up in the direction of n ) from up to get v , which is orthogonal to n and thus lies in the plane VP: v = up - [(up*n)/(n*n)]n v = (I - nn T /n T n) up 3. Normalize up n u v VP This is the projection onto VP. What does this remind you of? 1. Take the cross product: u = v x n 2. Normalize

OpenGL camera specification
Instead of arbitrarily specifying n, we can specify an eye position and a lookat position in world coords and compute n
n = eye - at (in OpenGL, vrp = eye)
In OpenGL, we’ll use this convenient function
gluLookAt (eye x , eye y , eye z , at x , at y , at z , up x , up y , up z )

Other view APIs
Flight simulation
Orientation of an airplane is called its attitude
Defined by 3 angles independent angles around x, y, and z axes (Euler angles)
pitch (x), yaw (y), and roll (z)

Other view APIs
Celestial navigation
Positions given in polar coordinates
elevation - angle between the optical axis and the horizontal plane of view
- sweeps optical axis up and down
azimuth - angle between the optical axis and some arbitrary reference direction in the horizontal plane
- sweeps optical axis left to right.
twist angle - rotates camera about optical axis
AZIMUTH ANGLE ELEVATION ANGLE

Projections
A projection is
A mapping from R n  R n
we often assume z=0, but we are still in 3D
Idempotent => P*P*…*P = P
subsequent projections have no effect
Two main types we will use in CG
Parallel
Perspective
Can be combined for a generalized projection

Projections
Taxonomy of projections

Projections COP (eye, origin of camera frame) As COP moves to infinity, rays become parallel & we say DOP (direction of proj.) Perspective Parallel Both perspective and parallel projections are planar geometric projections because the surface is a plane and the projectors are lines* view plane (VP) view volume, or frustum

Characteristics
Parallel lines of the object that are not parallel to view plane converge to a vanishing point
Closer objects look larger than farther objects
Natural view, used in rendering and animation
Does not preserve lengths or angles
Types
One, two, three point perspectives
defined by number of principal axes cut by proj.

One-point
One principle axis cut by projection plane
One axis vanishing point
Two-point
Two principle axes cut by projection plane
Two axis vanishing points
Three-point
Three principle axes cut by projection plane
Three axis vanishing points

Perspective projections Assume that the view plane (VP) is orthogonal to the z axis at z = d. Find the projection p of a point p: By similar triangles, x p /d = x/z => x p = (d*x)/z = x/(z/d) y p /d = y/z => y p = (d*y)/z = y/(z/d) p p =(x p ,y p ,z p ) p=(x,y,z) z z=d VP z=0 COP *d is a scale factor applied to x and y, *division by z causes distant objects to appear smaller than closer objects *perspective proj is irreversible (do you see why?)

The fact that many points map to one point is a problem
We need depth info for hidden surface removal
Homogeneous coordinates allow a fix
Use p = (x, y, z, w) instead of p = (x, y, z, 1)
1 0 0 0 0 1 0 0 0 0 1 0 0 0 1/d 0 x y z 1 x p y p z p 1 x y z z/d = = x/(z/d) y/(z/d) d 1 = Note: w = z/d cannot be zero (=> pts on plane z=0 do not project) (This division is not technically part of the projection.)

Characteristics:
Maps 3D points to view plane (VP) along lines parallel to VP
Preserves relative proportions of objects
used in drafting, esp. for plan views (top, side, etc.) because it gives accurate measurements of lengths and angles
does not give a realistic appearance
Defined by a projection vector
Two types, orthographic and oblique

Orthographic parallel projection
DOP is orthographic to projection plane
Elevations:
proj. plane is perpendicular to a principle axis.
front, top (plan), side
Axonometric:
proj. plane is not orthogonal to a principle axis
more than one face of an object will be in image
does not preserve distances or angles (in general, isometric is an exception)

Axonometric projections
Isometric
DOP makes equal angles with each principle axis
Most common axonometric projection
Maintains relative proportions
Dimetric
Proj. plane placed symmetrically to 2 faces
Trimetric
General case

Oblique parallel projection
DOP is not orthogonal to the projection plane; projection plane is normal to a principle axis.
less natural, eye and camera lens are  parallel to image plan, circles are projected to ellipses
Cavalier
DOP makes 45deg. angle with the proj. plane
Cabinet:
DOP makes a 63.4deg. angle with the proj. plane

Projections
Isometric projection - architectural
http://www.mda1.demon.co.uk/html/htmpd/smary400.htm
Projection examples - varied http:// cleo . murdoch . edu .au/ asu /pubs/ tlf /tlf99/ ns / ostrogonac .html
Projection tutorial http://mane. mech .Virginia.EDU/~engr160/Graphics/Projection.html

Assume the VP is at z = 0
Then the projection is easy,
p=(x,y,z) z z=0 VP DOP 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 x y z 1 x p y p z p 1 = p p =(x p ,y p ,z p ) x y 0 1 =

Generalized projections 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1/d 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 M perspective = M parallel = Assumes: COP at origin Assumes: DOP parallel to z axis A more robust formulation not only removes these restrictions but also integrates perspective and parallel projections into a single matrix

View volumes
Two types
View volume for orthographic projection is a right parallelpiped
View frustum for perspective is a clipped pyramid
Defined by six clip planes, left/right, top/ bottom, near/far (also hither/yon & front/back)

View volumes in OpenGL
Frustum
glFrustum(xmin, xmax, ymin, ymax, near, far)
near and far must be positive distances measured from COP, they describe planes parallel to z=0
frustum does not have to be a right frustum, i.e., xmin/xmax and ymin/ymax
Alters CTM
glMatrix(GL_PROJECTION);
glLoadIdentity();
glFrustum(xmin, xmax, ymin, ymax,near, far);

Alternatively, we can specify volume using the angle, or field, of view
gluPerspective (fovy, aspect, near, far)
fovy is angle between top and bottom clip planes
aspect is the aspect ratio (width/height) of VP
near and far are the same as before
Alters CTM

Perspective view frustum
Define frustum by field of view and near and far clip planes
s t near plane t=sin(  /2)   /2 1 Find x extents (t): fov =  left = -t, right = +t Find y extents (s): height/width = s/t (keep aspect ratio of window) => s = t * height/width t s

Volume
glOrtho (xmin, xmax, ymin, ymax, near, far)
near must be less than far , but the distances do not need to be positive w.r.t. COP
*we don’t have to worry about division by zero
arguments are otherwise the same as in glFrustum

Z-Buffering
We only want to render surfaces that are visible
Algorithms that do this are called hidden-surface removal algorithms
z-buffer algorithm: replace frame buffer color if current depth is less than stored depth
p g p b p r p p=? z VP 45 35 70 distance from VP p p=? 35 b frame buffer depth buffer

Z-Buffering
z-buffer algorithm
Part of projection process (image space)
(object space algorithms work on objects)
OpenGL
glutInitDisplayMode(GLU_RGB | GLUT_DEPTH) - initialize buffers
glEnable(GL_DEPTH_TEST) - enable vsp
glClear(GL_DEPTH_TEST) - clear buffer

Parallel projection matrices
Projection normalization
Convert all projections to orthogonal proj
1. Normalize view volume (will distort object)
2. Project orthogonally
The orthogonal projection of the distorted object will equal the original projection of the original object
orthogonal proj of distorted obj oblique proj of object (x,y,z) (x p ,y p ,z p ) x y z VP x y z VP

View volume normalization
1. Shear view volume to make ortho to VP
Find matrix that shears volume in x and y
Oblique projection characterized by angle projectors make with VP
cos  = (x-x p )/len sin  = z/len => tan  = z/(x-x p ) => x p = x-zcot  top view (from +y) co  = (y-y p )/len sin  = z/len => tan  = z/(y-y p ) => y p = y-zcot  side view (from +x) And, z p = 0 perspective view x y z VP x z (x,z) (x p ,0)  y z (z,y) (0,y p ) 

General parallel projection
1. Shear view volume to make ortho to VP
Find matrix that shears volume in x and y
Oblique projection characterized by angle projectors make with VP
Find x p
len = sqrt(x 2 +z 2 ) sub x p from x (moves x p to origin) cos  = (x-x p )/len sin  = z/len => tan  = z/(x-x p ) => cot  = (x-x p )/z => x-x p = zcot  => x p = x-zcot  top view (from +y) x z (x,z) (x p ,0)  x y z VP

General parallel projection
Find y p
Find z p
len = sqrt(y 2 +z 2 ) sub y p from y (moves y p to origin) cos  = (y-y p )/len sin  = z/len => tan  = z/(y-y p ) => cot  = (y-y p )/z => y-y p = zcot  => y p = y-zcot  side view (from +x) And, z p = 0 x y z VP y z (z,y) (0,y p ) 

OpenGL
Canonical view volume in OpenGL is x = y = z = +-1 (maps to 2D screen coords)
We specify a convenient clip volume with glOrtho(xmin,xmax,ymin,ymax,near,far)
Let GL map our volume to the canonical one
M (1,1,-1) (-1,-1,1) (xmax,ymax,zmin) (xmin,ymin,zmax)

2. Scale and translate view volume to make canonical: M = ST
Translate to origin
T ( - (xmax+xmin)/2, -(ymax+ymin)/2, -(zmax+zmin)/2 )
Scale to 2x2x2
S ( 2/(xmax-xmin), 2/(ymax-ymin), 2/(zmax-zmin) )
M (1,1,-1) (-1,-1,1) (xmax,ymax,zmin) (xmin,ymin,zmax)

General parallel projection H TS M ortho P=M ortho STH

General parallel projection 1. Align proj. vector with VP normal (shear in x,y along z) 1 0 k xz 0 0 1 k yz 0 0 0 1 0 0 0 0 1 H xy = where: so that: k xz = -px/pz x  = x - z(px/pz) k yz = -py/pz y  = y - z(py/pz) z  = z 1 0 -cot  0 0 1 -cot  0 0 0 1 0 0 0 0 1 = H n VP

General 3x3 shear matrix Read entr y k xy as ‘ a shear in x along y’ 1 k xy k xz k yx 1 k yz k zx k zy 1 K =

General parallel projection 2. Scale to normalized device coords (NDC) M = ST 1 0 0 T x 0 1 0 T y 0 0 1 T z 0 0 0 1 T = 1 0 0 - (xmax+xmin)/2 0 1 0 -(ymax+ymin)/2 0 0 1 -(zmax+zmin)/2 0 0 0 1 = ST VP n S x 0 0 0 0 S y 0 0 0 0 S z 0 0 0 0 1 S = 2/(xmax-xmin) 0 0 0 0 2/(ymax-ymin) 0 0 0 0 2/(zmax-zmin) 0 0 0 0 1 =

General parallel projection H TS M ortho P=M ortho STH

General perspective projection cop cop (1,1,1) (-1,-1,-1) P=M orth NPSH eye c c  eye H S xy P N z eye

General perspective proj. H cop cop 1. Shear view volume so centerline of proj is perp to VP (shear in x,y along z) where: so that: k xz = (l+r)/2n x  = x + z(l+r)/2n k yz = (t+b)/2n y  = y + z(t+b)/2n z  = z shear c = ((l+r)/2, (t+b)/2, n) to c  = (0,0,n) eye c c  eye 1 0 k xz 0 0 1 k yz 0 0 0 1 0 0 0 0 1 H xy =

General perspective proj. eye 2. Scale sides of frustum,i.e. normalize in x,y (want l = -1, r = 1, n = -1) 3. Persp proj creates rect. parallelpiped cop eye r =1 l=-1 n=-1 2n/(r-l) 0 0 0 0 2n/(t-b) 0 0 0 0 1 0 0 0 0 1 S xy = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1/d 0 P persp. = P persp eye S xy

General perspective proj. 4. Normalize in z (scale and translate far plane) *in text, N includes P persp P = H xy S xy P persp N z 1 0 0 0 0 1 0 0 0 0 (f+n)/(f-n) 2fn/(f-n) 0 0 1 1 N z = f z =1 Final projection matrix is: N z (1,1,1) (-1,-1,-1)

General perspective projection cop cop (1,1,1) (-1,-1,-1) P=M orth NPSTH eye c c  eye H S xy P N z eye

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