Cameras
Computer Graphics II
Autumn 2016-2017
CS4085
Cameras
Outline
1
Cameras
The Perspective Camera Model
CS4085
Cameras
The Perspective Camera Model
Outline
1
Cameras
The Perspective Camera Model
CS4085
Cameras
The Perspective Camera Model
View Volumes
Only a part of world is displayed at any one time: the view
volume
Culling is the process of determining what objects are not
visible
Objects that intersect the view volume boundaries are only
partially visible
Clipping is the process of intersecting an object with the
view volume
Visible data is displayed by projecting it on to a view plane
The viewport is the rectangular region of the view plane
that is drawn on the computer screen
The view frustum is defined by the infinite pyramid whose
apex is the eye-point, with four flat (non-parallel sides) and
truncated at the near plane and far plane
CS4085
Cameras
The Perspective Camera Model
View Volumes
Only a part of world is displayed at any one time: the view
volume
Culling is the process of determining what objects are not
visible
Objects that intersect the view volume boundaries are only
partially visible
Clipping is the process of intersecting an object with the
view volume
Visible data is displayed by projecting it on to a view plane
The viewport is the rectangular region of the view plane
that is drawn on the computer screen
The view frustum is defined by the infinite pyramid whose
apex is the eye-point, with four flat (non-parallel sides) and
truncated at the near plane and far plane
CS4085
Cameras
The Perspective Camera Model
View Volumes
Only a part of world is displayed at any one time: the view
volume
Culling is the process of determining what objects are not
visible
Objects that intersect the view volume boundaries are only
partially visible
Clipping is the process of intersecting an object with the
view volume
Visible data is displayed by projecting it on to a view plane
The viewport is the rectangular region of the view plane
that is drawn on the computer screen
The view frustum is defined by the infinite pyramid whose
apex is the eye-point, with four flat (non-parallel sides) and
truncated at the near plane and far plane
CS4085
Cameras
The Perspective Camera Model
View Volumes
Only a part of world is displayed at any one time: the view
volume
Culling is the process of determining what objects are not
visible
Objects that intersect the view volume boundaries are only
partially visible
Clipping is the process of intersecting an object with the
view volume
Visible data is displayed by projecting it on to a view plane
The viewport is the rectangular region of the view plane
that is drawn on the computer screen
The view frustum is defined by the infinite pyramid whose
apex is the eye-point, with four flat (non-parallel sides) and
truncated at the near plane and far plane
CS4085
Cameras
The Perspective Camera Model
View Volumes
Only a part of world is displayed at any one time: the view
volume
Culling is the process of determining what objects are not
visible
Objects that intersect the view volume boundaries are only
partially visible
Clipping is the process of intersecting an object with the
view volume
Visible data is displayed by projecting it on to a view plane;
this will be orthogonal to viewing direction
The viewport is the rectangular region of the view plane
that is drawn on the computer screen
The view frustum is defined by the infinite pyramid whose
apex is the eye-point, with four flat (non-parallel sides) and
truncated at the near plane and far plane
CS4085
Cameras
The Perspective Camera Model
View Volumes
Only a part of world is displayed at any one time: the view
volume
Culling is the process of determining what objects are not
visible
Objects that intersect the view volume boundaries are only
partially visible
Clipping is the process of intersecting an object with the
view volume
Visible data is displayed by projecting it on to a view plane
The viewport is the rectangular region of the view plane
that is drawn on the computer screen
The view frustum is defined by the infinite pyramid whose
apex is the eye-point, with four flat (non-parallel sides) and
truncated at the near plane and far plane
CS4085
Cameras
The Perspective Camera Model
View Volumes
Only a part of world is displayed at any one time: the view
volume
Culling is the process of determining what objects are not
visible
Objects that intersect the view volume boundaries are only
partially visible
Clipping is the process of intersecting an object with the
view volume
Visible data is displayed by projecting it on to a view plane
The viewport is the rectangular region of the view plane
that is drawn on the computer screen
The view frustum is defined by the infinite pyramid whose
apex is the eye-point, with four flat (non-parallel sides) and
truncated at the near plane and far plane
CS4085
Cameras
The Perspective Camera Model
Camera Model
Projection onto the (near) view plane is computed by
intersecting a ray with the view plane
The ray originates at e, the eye point, and passes through
world point x; the intersection point with the view plane is y
The combination of eye point, coordinate axes located at
eye point, view plane, view port and view frustum defines
the camera model
Camera coordinate system
#» #» #»
origin e = (0, 0, 0) in D − U − R ; 6= (0, 0, 0) in world
co-ords!
#»
unit-length direction vector D perp. to view plane
#»
closest point to observer is p = e + dmin D, dmin > 0
#»
U is unit-length camera up vector
#»
#» #» #»
R is unit-length right vector such that R = D × U (in RHCS)
CS4085
Cameras
The Perspective Camera Model
Camera Model
Projection onto the (near) view plane is computed by
intersecting a ray with the view plane
The ray originates at e, the eye point, and passes through
world point x; the intersection point with the view plane is y
The combination of eye point, coordinate axes located at
eye point, view plane, view port and view frustum defines
the camera model
Camera coordinate system
#» #» #»
origin e = (0, 0, 0) in D − U − R
#»
unit-length direction vector D perp. to view plane; points
away from observer so eye point is on negative side of
plane by convention
#»
closest point to observer is p = e + dmin D, dmin > 0
#»
U is unit-length camera up vector
#»
#» #» #»
R is unit-length right vector such that R = D × U (in RHCS)
CS4085
Cameras
The Perspective Camera Model
Camera Model
Projection onto the (near) view plane is computed by
intersecting a ray with the view plane
The ray originates at e, the eye point, and passes through
world point x; the intersection point with the view plane is y
The combination of eye point, coordinate axes located at
eye point, view plane, view port and view frustum defines
the camera model
Camera coordinate system
#» #» #»
origin e = (0, 0, 0) in D − U − R
#»
unit-length direction vector D perp. to view plane
#»
closest point to observer is p = e + dmin D, dmin > 0
#»
U is unit-length camera up vector
#»
#» #» #»
R is unit-length right vector such that R = D × U (in RHCS)
CS4085
Cameras
The Perspective Camera Model
Camera Model
Projection onto the (near) view plane is computed by
intersecting a ray with the view plane
The ray originates at e, the eye point, and passes through
world point x; the intersection point with the view plane is y
The combination of eye point, coordinate axes located at
eye point, view plane, view port and view frustum defines
the camera model
Camera coordinate system
#» #» #»
origin e = (0, 0, 0) in D − U − R
#»
unit-length direction vector D perp. to view plane
#»
closest point to observer is p = e + dmin D, dmin > 0
#»
U is unit-length camera up vector chosen to be parallel to
opposing edges of viewport
#»
#» #» #»
R is unit-length right vector such that R = D × U (in RHCS)
CS4085
Cameras
The Perspective Camera Model
Camera Model
Projection onto the (near) view plane is computed by
intersecting a ray with the view plane
The ray originates at e, the eye point, and passes through
world point x; the intersection point with the view plane is y
The combination of eye point, coordinate axes located at
eye point, view plane, view port and view frustum defines
the camera model
Camera coordinate system
#» #» #»
origin e = (0, 0, 0) in D − U − R
#»
unit-length direction vector D perp. to view plane
#»
closest point to observer is p = e + dmin D, dmin > 0
#»
U is unit-length camera up vector
#»
#» #» #»
R is unit-length right vector such that R = D × U (in RHCS)
CS4085
Cameras
The Perspective Camera Model
Camera Model
wtl
vtl
vbl
wtr
wbl
vtr
p
wbr
vbr
e
View plane vertices, v.. and far plane, w..
#»
Both normals point into frustum; near plane D, far plane
#»
−D
CS4085
Cameras
The Perspective Camera Model
Frustum Vertices
#»
#»
#»
View plane vertices are vbl = e + dmin D + umin U + rmin R,
and (in coordinates form)
vtl = e + (dmin , umax , rmin ),
vbr = e + (dmin , umin , rmax ),
vtr = e + (dmin , umax , rmax )
Far plane vertices rely on “similar triangle” scaling factor
dmax
dmin
Far plane vertices are
wbl = e + ddmax
(dmin , umin , rmin ),
min
dmax
dmin (dmin , umax , rmin ),
wbr = e + ddmax
(dmin , umin , rmax ),
min
dmax
wtr = e + dmin (dmin , umax , rmax )
wtl = e +
CS4085
Cameras
The Perspective Camera Model
Frustum planes
#»
Near plane has a point p = e + dmin D; the vector between this
#»
and any point x on this plane, x − p, is orthogonal to normal, D.
So
#»
#»
#»
#»
D t x = D t (e + dmin D) = D t e + dmin
#»
Similarly for point x on far plane and its normal − D
#»
#»
#»
#»
− D t x = − D t (e + dmax D) = −( D t e + dmax )
On left plane, three points are e, vtl and vbl . The normal
#»
pointing into frustum is given by (no U component)
#»
#»
#»
(vbl − e) × (vtl − e) =(dmin D + umin U + rmin R)×
#»
#»
#»
(dmin D + umax U + rmin R)
..
=
.
#»
#»
=(umax − umin )(dmin R − rmin D)
CS4085
Cameras
The Perspective Camera Model
Frustum planes (contd.)
#»
When made unit-length, the left plane normal, N l is
#»
#»
dmin R − rmin D
q
2 + r2
dmin
min
and the equation of points on this plane is
#»
N l · (x − e) = 0
We can repeat this for right plane using (vtr − e) × (vbr − e) and
get
#»
#»
#»
#»
−dmin R + rmax D
Nr = q
, N r · (x − e) = 0
2 + r2
dmin
max
and likewise for top and bottom faces
CS4085
Cameras
The Perspective Camera Model
Camera Model (concl.)
We usually choose a symmetric view frustum so that
umax = −umin and rmax = −rmin
This implies four independent parameters
dmin , dmax , umax and rmax
Alternatively
#»
we can specify the field of view in the U direction and the
aspect ratio of view port window
Field of view is 2θu then and aspect ratio, ρ, is
ρ = rmax /umax
We get umax = dmin tan θu and then rmax = ρumax
Then dmin , dmax , θu and ρ completely specify the frustum
again
CS4085
Cameras
The Perspective Camera Model
Camera Model (concl.)
We usually choose a symmetric view frustum so that
umax = −umin and rmax = −rmin
This implies four independent parameters
dmin , dmax , umax and rmax
Alternatively
#»
we can specify the field of view in the U direction and the
aspect ratio of view port window
Field of view is 2θu then and aspect ratio, ρ, is
ρ = rmax /umax
We get umax = dmin tan θu and then rmax = ρumax
Then dmin , dmax , θu and ρ completely specify the frustum
again
CS4085
Cameras
The Perspective Camera Model
Camera Model (concl.)
We usually choose a symmetric view frustum so that
umax = −umin and rmax = −rmin
This implies four independent parameters
dmin , dmax , umax and rmax
Alternatively
#»
we can specify the field of view in the U direction and the
aspect ratio of view port window
Field of view is 2θu then and aspect ratio, ρ, is
ρ = rmax /umax
We get umax = dmin tan θu and then rmax = ρumax
Then dmin , dmax , θu and ρ completely specify the frustum
again
CS4085
Cameras
The Perspective Camera Model
Camera Model (concl.)
We usually choose a symmetric view frustum so that
umax = −umin and rmax = −rmin
This implies four independent parameters
dmin , dmax , umax and rmax
Alternatively
#»
we can specify the field of view in the U direction and the
aspect ratio of view port window
#»
U
umax
e
θu
dmin
dmax
#»
D
umin
Field of view is 2θu then and aspect ratio, ρ, is
ρ = rmax /umax
CS4085
We get umax = dmin tan θu and
then rmax = ρumax
Cameras
The Perspective Camera Model
Camera Model (concl.)
We usually choose a symmetric view frustum so that
umax = −umin and rmax = −rmin
This implies four independent parameters
dmin , dmax , umax and rmax
Alternatively
#»
we can specify the field of view in the U direction and the
aspect ratio of view port window
Field of view is 2θu then and aspect ratio, ρ, is
ρ = rmax /umax
We get umax = dmin tan θu and then rmax = ρumax
Then dmin , dmax , θu and ρ completely specify the frustum
again
CS4085
Cameras
The Perspective Camera Model
Camera Model (concl.)
We usually choose a symmetric view frustum so that
umax = −umin and rmax = −rmin
This implies four independent parameters
dmin , dmax , umax and rmax
Alternatively
#»
we can specify the field of view in the U direction and the
aspect ratio of view port window
Field of view is 2θu then and aspect ratio, ρ, is
ρ = rmax /umax
We get umax = dmin tan θu and then rmax = ρumax
Then dmin , dmax , θu and ρ completely specify the frustum
again
CS4085
Cameras
The Perspective Camera Model
Camera Model (concl.)
We usually choose a symmetric view frustum so that
umax = −umin and rmax = −rmin
This implies four independent parameters
dmin , dmax , umax and rmax
Alternatively
#»
we can specify the field of view in the U direction and the
aspect ratio of view port window
Field of view is 2θu then and aspect ratio, ρ, is
ρ = rmax /umax
We get umax = dmin tan θu and then rmax = ρumax
Then dmin , dmax , θu and ρ completely specify the frustum
again
CS4085