Computer Graphics CS630

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Computer GraphicsCS630

• Lecture 3 Transformation And Coordinate Systems

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What is a Transformation?

• Maps points (x, y) in one coordinate system to
  points (x', y') in another coordinate system

x' ax by c y' dx ey f
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Transformations

• Simple transformation
• Translation
• Rotation
• Scaling

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Transformations

• Deformable transformations
• Shearing
• Tapering
• Twisting
• Etc..
• Issues
• Can be combined
• Are these operations invertible?

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Transformations

• Why use transformations?
• Position objects in a scene (modeling)
• Change the shape of objects
• Create multiple copies of objects
• Projection for virtual cameras
• Animations

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Classes of Transformation

• Rigid-Body/ Euclidean transformation
• Similarity Transforms
• Linear Transforms
• Affine Transforms
• Projective Transforms

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Rigid-Body / Euclidean Transforms

• Preserves distances
• Preserves angles

Rigid / Euclidean
Identity
Translation
Rotation
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How are Transforms Represented?
x' ax by c y' dx ey f
c f
x y
x' y'
a b d e


p' M p t
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Translation




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Properties of Translation




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Scaling
Uniform scaling iff
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Rotations (2D)
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Rotations 2D

• So in matrix notation

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Rotations (3D)
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Properties of Rotations
order matters!
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Combining Translation Rotation
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Combining Translation Rotation
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Homogeneous Coordinates

• Add an extra dimension
• in 2D, we use 3 x 3 matrices
• In 3D, we use 4 x 4 matrices
• Each point has an extra value, w

x y z w
a e i m
b f j n
c g k o
d h l p
x' y' z' w'

p' M p
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Homogeneous Coordinates

• Most of the time w 1, and we can ignore it

x' y' z' 1
x y z 1
a e i 0
b f j 0
c g k 0
d h l 1

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Homogeneous Coordinates
can be represented as
where
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Translation Revisited
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Rotation Scaling Revisited
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Combining Transformations
where
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Transforming Tangents
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Transforming Normals
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Surface Normal

• Surface Normal unit vector that is locally
  perpendicular to the surface

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Why is the Normal important?

• It's used for shading makes things look 3D!

object color only
Diffuse Shading
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Visualization of Surface Normal

• x Red y Green z Blue

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How do we transform normals?
nWS
nOS
World Space
Object Space
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Transform the Normal like the Ray?

• translation?
• rotation?
• isotropic scale?
• scale?
• reflection?
• shear?
• perspective?

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More Normal Visualizations
Incorrect Normal Transformation
Correct Normal Transformation
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Transforming Normals
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Rotations about an arbitrary axis
Rotate by around a unit axis
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An Alternative View

• We can view the rotation around an arbitrary axis
  as a set of simpler steps
• We know how to rotate and translate around the
  world coordinate system
• Can we use this knowledge to perform the rotation?

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Rotation about an arbitrary axis

• Translate the space so that the origin of the
  unit vector is on the world origin
• Rotate such that the extremity of the vector now
  lies in the xz plane (x-axis rotation)
• Rotate such that the point lies in the z-axis
  (y-axis rotation)
• Perform the rotation around the z-axis
• Undo the previous transformations

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Rotation about an arbitrary axis

• Step 1
• Rotate x-axis

y
(a,b,c)
x
x
(a,b,c)
z
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Closer Look at Y-Z Plane

• Need to rotate ? degrees around the x-axis

y
?
z
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Equations for ?
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Rotation about the Y-axis

• Using the same analysis as before, we need to
  rotate ? degrees around the Y-axis

y
x
(a,b,c)Rx (?) (a,b,c)T
z
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Equations for ?
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Rotation about the Z-axis

• Now, it is aligned with the Z-axis, thus we can
  simply rotate ? degrees around the Z-axis.
• Then undo all the transformations we just did

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Equation summary
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Deformations

• Transformations that do not preserve shape
• Non-uniform scaling
• Shearing
• Tapering
• Twisting
• Bending

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Shearing
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Tapering
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Twisting
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Bending
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Quick Recap

• Computer Graphics is using a computer to generate
  an image from a representation.

computer
Model
Image
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Modeling

• What we have been studying so far is the
  mathematics behind the creation and manipulation
  of the 3D representation of the object.

computer
Model
Image
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What have we seen so far?

• Basic representations (point, vector)
• Basic operations on points and vectors (dot
  product, cross products, etc.)
• Transformation manipulative operators on the
  basic representation (translate, rotate,
  deformations) 4x4 matrices to encode all
  these.

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Why do we need this?

• In order to generate a picture from a model, we
  need to be able to not only specify a model
  (representation) but also manipulate the model in
  order to create more interesting images.

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Overview

• The next set of slides will deal with the other
  half of the process.
• From a model, how do we generate an image

computer
Model
Image
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Scene Description
Scene
Materials
Lights
Camera
Objects
Background
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Graphics Pipeline
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Graphics Pipeline

• Modeling transforms orient the models within a
  common coordinate frame (world space)

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Graphics Pipeline
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Graphics Pipeline

• Maps world space to eye space
• Viewing position is transformed to origin
  direction is oriented along some axis (usually z)

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Graphics Pipeline

• Transform to Normalized Device Coordinates (NDC)
• Portions of the object outside the view volume
  (view frustum) are removed

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Graphics Pipeline

• The objects are projected to the 2D image place
  (screen space)

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Graphics Pipeline
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Graphics Pipeline

• Z-buffer - Each pixel remembers the closest
  object (depth buffer)

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Graphics Pipeline

• Almost every step in the graphics pipeline
  involves a change of coordinate system.
  Transformations are central to understanding 3D
  computer graphics.

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Intuitively
World Space
Object Space
Camera Space Projection NDC
Rasterization
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Coordinate Systems

• Object coordinates
• World coordinates
• Camera coordinates
• Normalized device coordinates
• Window coordinates

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Object Coordinates

• Convenient place to model the object

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World Coordinates

• Common coordinates for the scene

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Positioning Synthetic Camera
What are our degrees of freedom in camera
positioning? To achieve effective visual
simulation, we want 1) the eye point to be in
proximity of modeled scene 2) the view to be
directed toward region of interest, and 3) the
image plane to have a reasonable twist
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Eye Coordinates
Eyepoint at origin u axis toward right of image
plane v axis toward top of image plane view
direction along negative n axis
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Transformation to Eye Coordinates
Our task construct the transformation M that
re-expresses world coordinates in the viewer frame
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Machinery Changing Orthobases
Suppose you are given an orthobasis u, v, n What
is the action of the matrix M with rows u, v, and
n as below?
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Applying M to u, v, n
Two equally valid interpretations, depending on
reference frame 1 Think of uvn basis as a rigid
object in a fixed world space Then M rotates
uvn basis into xyz basis 2 Think of a fixed axis
triad, with labels from xyz space Then M
reexpresses an xyz point p in uvn coords! It is
this second interpretation that we use today to
relabel world-space geometry with eye space
coordinates
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Positioning Synthetic Camera
Given eyepoint e, basis u, v, n Deduce M that
expresses world in eye coordinates Overlay
origins, then change bases
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Positioning Synthetic Camera
Check does M re-express world geometry in eye
coordinates?
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Positioning Synthetic Camera
Camera specification must include World-space
eye position e World-space lookat direction -n
Are e and -n enough to determine the camera DOFs
(degrees of freedom)?
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Positioning Synthetic Camera
Are e and -n enough to determine the camera
DOFs? No. Note that we were not given u and
v! (Why not simply require the user to specify
them?)
We must also determine u and v, i.e., camera
twist about n. Typically done by specification
of a world-space up vector provided by user
interface, e.g., using gluLookat(e, c,
up) Twist constraint Align v with world up
vector (How?)
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Positioning Synthetic Camera
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Where are we?
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What is Projection?
Any operation that reduces dimension (e.g., 3D to
2D)
Orthographic Projection Perspective Projection
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Orthographic Projection

• focal point at infinity
• rays are parallel and orthogonal to the image
  plane

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Comparison
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Simple Perspective Camera

• camera looks along z-axis
• focal point is the origin
• image plane is parallel to xy-plane at distance d
• d is call focal length

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Similar Triangles

• Similar situation with x-coordinate
• Similar Triangles point x,y,z projects to
  (d/z)x, (d/z)y, d

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Projection Matrix

• Projection using homogeneous coordinates
• transform x, y, z to (d/z)x, (d/z)y, d

• 2-D image point
• discard third coordinate
• apply viewport transformation to obtain physical
  pixel coordinates

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Perspective Projection
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Perspective Projection
z 0 not allowed (what happens to points on
plane z 0?) Operation well-defined for all
other points
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Perspective Projection
Matrix formulation using homogeneous 4-vectors
Finally, recover projected point using homogenous
convention Divide by 4th element to convert
4-vector to 3-vector
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Camera Coordinates

• Coordinate system with the camera in a convenient
  pose

y
x
z
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View Volume and Normalized Device Coordinates
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Normalized Device Coordinates

• Device independent coordinates
• Visible coordinate usually range from

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Perspective Projection

• Taking the camera coordinates to NDC

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Perspective Projection
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Perspective Projection NDC
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Window Coordinates

• Adjusting the NDC to fit the window

is the lower left of the window
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Window Coordinates

• Adjusting the NDC to fit the window

is the lower left of the window
height
width
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Window Coordinates
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Summary Object Coordinate to Device Coordinate

• Take your representation (points) and transform
  it from Object Space to World Space (Mwo)
• Take your World Space point and transform it to
  Camera Space (Mcw)
• Perform the remapping and projection onto the
  image plane in Normalized Device Coordinates
  (Mw_p Mpc)
• Perform this set of transformations on each point
  of the polygonal object (M Mw_pMpcMcwMwo)