Objectives

1
CSC461 Lecture 19 Computer Viewing

• Objectives
• Introduce computer views and positioning camera
• Look at alternate viewing APIs
• Introduce the mathematics of projection

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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 Projection

• Default projection is orthogonal

clipped out
2
z0
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Moving the Camera Frame

• If we want to visualize object with both positive
  and negative 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
  determined by the model-view matrix
• A translation glTranslatef(0.0,0.0,-d)
• d gt 0

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Moving Camera back from Origin
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Moving the Camera

• We can move the camera to any desired position by
  a sequence of rotations and translations
• Example side view
• Rotate the camera
• Move it away from origin
• Model-view matrix
• Concatenation of rotation and translation C TR

• OpenGL code last transformation specified is
  first applied
• glMatrixMode(GL_MODELVIEW)
• glLoadIdentity()
• glTranslatef(0.0, 0.0, -d)
• glRotatef(90.0, 0.0, 1.0, 0.0)

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Camera Positioning

• Where the camera is
• View-reference point (VRP) in the world system
• How the camera is oriented
• View-plane normal (VPN) specifying the
  orientation of the projection plan
• View-up vector (VUP) specifying the up direction
• With VRP, VPN and VUP, can build a frame
• The origin VRP
• Three independent vectors
• VPN, VUP, and
• VRNVUP

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The LookAt Function

• The function glLookAt to form the required
  model-view matrix through a simple interface
• Syntax glLookAt(
• eyex,eyey,eyez,
• atx,aty,atz,
• upx,upy,upz)

• VRP eye, VPN at-eye, VUPup
• 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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Constructing LookAt View

• Create an isometric view of the cube
• Start with default setting
• Transformations
• Rotate about the y-axis -45 degrees
• Rotate about the x-axis 35.26 degrees
• Translate along the z-axis by the distance d
• Result
• C T(0,0,-d)Rx(35.26)Ry(-45)
• Equivalent to
• gluLookAt(1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0,
  0.0)

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Other Viewing APIs

• The LookAt function is only one possible API for
  positioning the camera in OpenGL
• Others include not provided by OpenGL
• View reference point, view plane normal, view up
  (PHIGS, GKS-3D)
• Yaw, pitch, roll flight simulation
• Elevation, azimuth, twist rotation about other
  objects instead of points
• Direction angles
• All these viewing can be constructed by
  specifying concatenation of translation and
  rotations

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Orthogonal Projections

• The default projection in the eye (camera) frame
  is orthogonal
• A special case of parallel projections in which
  the projectors are perpendicular to the view
  plane
• For points within the default view volume
• xp x, yp y, zp 0

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Homogeneous Coordinate Representation
pp Mp

• xp x
• yp y
• zp 0
• wp 1

M
In practice, we can let M I and set the z term
to zero later
Normalization

• 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

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Simple Perspective Projection

• Center of projection at the origin
• All projectors pass through the origin
• Simple projection Projection plane z d, d lt 0
• General projection Projection plane can have any
  orientation w.r.t. the front

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Simple Perspective Equations

• Consider top and side views

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Characteristics

• Nonlinear equations
• Non-uniform foreshortening the images of objects
  from the COP are reduced in size compared to the
  images of objects closer to the COP
• Can be considered as a transformation of from (x,
  y, z) to (xp, yp, zp)
• Preserves lines but not affine
• Irreversible because all points along a
  projector project into the same point

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Homogeneous Coordinate Form
Consider q Mp where M transforms p to q
M

• Represent points (x, y, z) as (wx, wy, wz, w),
  w?0
• The difference is the coefficient w
• When w1, homogeneous coordinates
• Can represent a broader class of transformations

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Perspective Division

• However w ? 1, so we must divide by w to return
  to homogeneous coordinates
• This perspective division yields
• the desired perspective equations
• d/z is the foreshortening factor
• We will consider the corresponding clipping
  volume with the OpenGL functions

xp
yp
zp d
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Projection Pipeline

• Set the model-view matrix
• Apply the projection matrix
• Perform a perspective division