Viewing and Projections
Dr. Amy Zhang
Reading
Hill, Chapter 5 and 7
Red Book, Chapter 3, “Viewing”
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2
3D Graphics Pipeline
The big picture…
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Outline
Camera Models
Viewing Transformation
Projection Matrix
OpenGL Transformation Pipeline
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Cameras
Cameras have an optical system:
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Filters
Lenses
Aperture
The projection surface may be flat or curved, oriented at
various angles with respect to the incoming light.
Examples: A camera or the eye.
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Camera Obscura
The first cameras – a dark box with a small hole in it
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The Pinhole Camera
An abstract camera model
Models the geometry of perspective projection
Used in most of computer graphics
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Pinhole Optics
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Perspective
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Perspective Derivation
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Consider the projection of a point onto the projection
plane:
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By similar triangles we can compute how much the x and
y-coordinates are scaled
Looking down y axis:
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We get:
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This is clearly a non-linear transformation
BUT: We can split it into a linear part followed by a
nonlinear part
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Homogeneous Coordinates
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Remember homogeneous coordinates:
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To get a homogeneous point we divide all
the coordinates by w:
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This is called the perspective divide
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Perspective Projections
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We can now rewrite the perspective projection as a
linear transformation:
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After division by the 4th component we get:
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The Reason for Lenses *
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The Lens Model
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Lens, aperture, and image plane
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Focal length f: the distance from lens to image plane
A point in focus: the image of a point is on the image
plane
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An out-of-focus point
The circle of confusion r
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The Gaussian / thin lens formula:
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A near point in out-of-focus
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The depth of focus dfocus and the depth of field (DOF)
dfield
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Decreasing the aperture size reduces the size of the
blur for points not in the focused plane, so that the
blurring is imperceptible, and all points are within the
dfield.
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Viewing and Projection
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In OpenGL we distinguish between:
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Viewing: placing the camera
Projection: describing the viewing frustum of the camera (and
thereby the projection transformation)
Perspective divide: computing homogeneous points
Outline
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Camera Models
Viewing Transformation
Projection Matrix
OpenGL Transformation Pipeline
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OpenGL Transformations
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The viewing transformation V transforms a point from
world space to eye space:
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Placing the Camera
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It is most natural to position the camera in world
space as if it were a real camera
Identify the eye point where the camera is located
Identify the look-at point that we wish to appear in the
center of our view
Identify an up-vector vector that we wish to be
oriented upwards in our final image
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Look-At Positioning
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We specify the view frame using the look-at vector a and
the camera up vector up
The vector a points in the negative viewing direction
In 3D, we need a third vector that is perpendicular to
both up and a to specify the view frame
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Where does it point to?
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The result of the cross product is a vector, not a scalar, as
for the dot product
Depending on the basis vectors i, j, and k, the new vector
follows the right or left handed rule
In OpenGL, the cross product a x b yields a right hand
side (RHS) vector perpendicular to a and b
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Computing Cross Products
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We can compute the cross product using yet another
matrix-vector multiplication:
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The matrix is sometimes called the skew-symmetric matrix
of the vector (in this case a)
Cross products produce vectors for both vector and
point inputs
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Constructing a Frame
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The cross product between the up and the look-at vector
will get a vector that points to the right.
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Finally, using the vector a and the vector r we can
synthesize a new vector u in the up direction:
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World and Camera Frames
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The relation between the world and the camera is
expressed as:
We move the eye (camera) by updating E
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Rotation
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Rotation first:
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Translation
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Translation to the eye point:
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Composing the Result
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The final camera transformation is:
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Why?
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The Viewing Transformation
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Expressing P in eye coordinates:
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The Viewing Transformation
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As a single 4x4 matrix:
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Where these are normalized vectors:
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gluLookAt()
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OpenGL provides a very helpful utility function that
implements the look‐at viewing specification:
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These parameters are expressed in world coordinates
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Outline
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Camera Models
Viewing Transformation
Projection Matrix
OpenGL Transformation Pipeline
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OpenGL Transformations
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The projection transformation P transforms a point from
eye space to clip space:
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Projection Transformations
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Projections fall into two categories:
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Parallel projections:The camera is placed at an infinite distance
from the viewplane; lines of projection are parallel to each
other
Perspective projections: Lines of projection converge at a point
Parallel Projections
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The simplest form of parallel projection is simply along
lines parallel to the z-axis onto the xy-plane
This form of projection is called orthographic
For other parallel projections see, e.g.:
http://www.mtsu.edu/~csjudy/planeview3D/tutorialparallel
.html
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Orthographic Frustum
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The user specifies the orthographic viewing frustum by
specifying minimum and maximum x/y coordinates
It is necessary to indicate a range of distances along the zaxis by specifying near and far planes
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Orthographic Projections to NDC
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Normalized Device Coordinate (NDC) makes up a coordinate
system that describes positions on a virtual plotting device
Here is the orthographic world-to-clip transformation:
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Move center to origin
T(-(left+right)/2, -(bottom+top)/2,(near+far)/2))
Scale to have sides of length 2
S(2/(right-left),2/(top-bottom),2/(far-near))
P = ST =
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2
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 right  left
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0
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0
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0
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0
2
top  bottom
0
0
right  left 
right  left 
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top  bottom
0
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top  bottom
2
far  near 
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far  near
far  near 
0
1
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0
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Orthographic Projection in OpenGL
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This matrix is constructed with the following OpenGL
call:
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And the 2D version (another GL utility function):
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Just a call to glOrtho() with near = -1 and far = +1
Properties of Parallel Projections
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Not realistic looking
Good for exact measurements
A kind of affine transformation
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Parallel lines remain parallel
Ratios are preserved
Angles (in general) not preserved
Most often used in CAD, architectural drawings, etc.,
where taking exact measurement is important
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Isometric Games
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A special kind of parallel projection called isometric projection
is often used in games
It’s essentially a shear and an orthographic projection
Easier to compute than a full perspective transformation
Diablo
SimCity
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The Sims
Perspective Projections
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Artists (Donatello, Brunelleschi, and Da Vinci) during the
renaissance discovered the importance of perspective for
making images appear realistic
Parallel lines intersect at a point
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Perspective Viewing Frustum
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Just as in the orthographic case, we specify a perspective
viewing frustum
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Values for left, right, top, and bottom are specified at the
near depth.
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Perspective Viewing Frustum
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OpenGL provides a function to set up this perspective
transformation:
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There is also a simpler OpenGL utility function:
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fov = vertical field of view in degrees
aspect = image width / height at near depth
Can only specify symmetric viewing frustums where the
viewing window is centered around the –z axis.
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OpenGL Clip Space
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In OpenGL clip space, the viewing frustum is mapped to a
cube that extends from -1 to 1 in x, y, and z.
OpenGL also flips the z axis to create a left handed
coordinate system during projection
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OpenGL Perspective Matrix
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Mapping the perspective viewing frustum in OpenGL to
clip space involves some affine transformations
OpenGL uses a clever composition of these
transformations with the perspective projection matrix:
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What happens to the z value?
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We get the following expression for z from our
perspective matrix:
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This mapping is non‐linear:
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Uniform differences in 3D (z) depth values do not correspond
to uniform differences in z’ values:
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The number of discernable depths is greater near the near
plane than the far plane
 Tip: Choose the near plane as far away as possible
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Properties of Perspective Projections
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The perspective projection is an example of a projective
transformation
Here are some properties of projective transformations:
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Lines map to lines
Parallel lines do not necessarily remain parallel
Ratios are not preserved
One of the advantages of perspective projection is that
size varies inversely with distance – looks realistic
A disadvantage is that we can't judge distances as exactly
as we can with parallel projections
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Outline
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Camera Models
Viewing Transformation
Projection Matrix
OpenGL Transformation Pipeline
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OpenGL Transformations
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The rest of the OpenGL transformation pipeline:
3D:
2D:
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Clipping & Perspective Division
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The scene's objects are clipped against
the clip space bounding box
This step eliminates any objects (and
pieces of objects) that are not visible in
the image
Hill describes an efficient clipping
algorithm for homogeneous clip space
Perspective division divides all
homogeneous coordinates through w
Clip space becomes Normalized Device
Coordinate (NDC) space after the
perspective division
Viewport Transformation
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One last transformation from NDC
space to the viewport on the screen
The viewport transformation also maps
NDC depth values from the range [-1,1]
to [0, 1]
The resulting screen space is still 3D
with depth values between 0 and 1
Viewport Transformation
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OpenGL provides a function to set up the viewport
transformation:
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Screen Coordinate Systems
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3D Graphics Pipeline
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The rasterization step scan converts the object into
pixels
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3D Graphics Pipeline
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A z-buffer depth test resolves visibility of the objects on a
per-pixel basis and writes the pixels to the frame buffer
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The end
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Questions and answers
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