Chapter 7: Three-Dimensional Viewing

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Chapter 7 Three-Dimensional Viewing

• Chapter 5 Camera with parallel projection
• Now Camera with perspective projection
• 7.2 The Camera Revisited
• Eye, view volume, view angle, near plane,far
  plane, aspect ratio,viewplane.
• Perspective view P determined by finding where
  line from eye to P intersects viewplane.

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Chapter 7 Three-Dimensional Viewing

• 7.2.1 Setting the View Volume
• Projection matrix
• gluPerspective(viewAngle, aspectRatio, N, F)
• 7.2.2 Positioning and Pointing the Camera
• Move camera away from default position and point
  in given direction Rotation and translation in
  modelview matrix.

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Chapter 7 Three-Dimensional Viewing

• General camera with Arbitrary Orientation and
  Position
• Transformation? Modelview matrix?
• Attach explicit coordinate system to camera
• Pitch, heading, yaw, roll
• Selfstudy pp.361-366

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Chapter 7 Three-Dimensional Viewing

• 7.3 Building a Camera in a Program
• Selfstudy.
• 7.4 Perspective Projections of 3D Objects
• Vertex v ? modeling transformations ? camera
  position and orientation Now in eye coordinates
• Vertex P in eye coordinates must be projected
  onto point (x,y) on near plane.

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Chapter 7 Three-Dimensional Viewing

• 7.4.1 Perspective Projection of a Point
• Fundamental operation Projecting 3D point into
  2D coordinates on a plane.
• Construct local coordinate system on near plane
• (x,y) (N(Px/(-Pz)), N(Py,/(-Pz))

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Chapter 7 Three-Dimensional Viewing

• Note
• -Pz achieves perspective foreshortening
• Pz0 if P lies on plane z0 Clip before
  projecting
• If Pz lies behind eye, also clipped before
  projecting.
• N scales picture (size only)
• Straight lines project to straight lines (proof
  given)
• Selfstudy Example 7.4.2 IMPORTANT!

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Chapter 7 Three-Dimensional Viewing

• 7.4.2 Perspective Projection of a Line
• Lines parallel in 3D project to lines, but not
  necessarily parallel.
• Lines that passed behind the eye cause passage
  through infinity should be clipped.
• Perspective projections produce geometrically
  realistic pictures, except for very long lines
  parallel to viewplane.

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Chapter 7 Three-Dimensional Viewing

• Projecting Parallel Lines
• 3D line P(t) A c(t)
• Yields (parametric form) for projection
• If Act parallel to viewplane, then cz0,
  andp(t) N(Axcxt,Aycyt)/(-Az)Slope is cy/cx
• If two lines in 3D are parallel to each other and
  to the viewplane, they project two parallel
  lines.

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Chapter 7 Three-Dimensional Viewing

• Suppose c not parallel to viewplane (cz!0)
• For large t, p(?)(Ncx/(-cz), Ncy/(-cz))
  vanishing point
• Depends only on direction c ? all parallel lines
  share same vanishing point, i.e. project to lines
  that are not parallel.

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Chapter 7 Three-Dimensional Viewing

• Lines that pass behind the eye
• B projects to Bwrong side of viewplane
• Let C move from A to B asC moves, its
  projection slidesfurther to right until it
  spurtsoff to infinity
• When C moves behind eye, projection appears to
  left on viewplane.
• Selfstudy Example 7.4.3,
• Anomaly of Viewing Long Parallel Lines

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Chapter 7 Three-Dimensional Viewing

• Perspective and the Graphics Pipeline
• Adding pseudodepth if two points project to the
  same point, we only need to know which is nearer
• Efficiency (x,y,z) (N(Px/(-Pz)),
  N(Py,/(-Pz), (aPzb)/(-Pz))
• Choose 1lta,blt1
• Let pseudodepth be 1 when Pz -N, 1 when Pz
  -F.
• Then a -((FN)/(F-N)) and b
  (-2FN)/(F-N).
• Due to precision problems, pseudodepth values may
  be equal for two different pints as Pz
  approaches F.

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Chapter 7 Three-Dimensional Viewing

• Using Homogeneous Coordinates
• Point (Px,Py,Pz,1) vector (vx,vy,vz,0)
• Extend Point has family of homogeneous
  coordinates (wPx,wPy,wPz,w) for any w except w0.
• To convert point from ordinary to homogeneous
  coordinates, append 1.
• To convert point from homogenous to ordinary
  coordinates, divide all components by last, and
  discard last component.

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Chapter 7 Three-Dimensional Viewing

• Transforming points in homogeneous coordinates
• If matrix M has last row (0 0 0 1), affine
  transformation MPQ, last component of Q is w.
• If M doesnt have last row (0 0 0 1), MPQ gives
  point can divide by last component to find
  coordinates perspective division.
• If M doesnt have last row (0 0 0 1), not affine
  but perspective transformation.
• Note perspective projection perspective
  transformation orthographic projection

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Chapter 7 Three-Dimensional Viewing

• Geometric Nature of Perspective Transformation
• Lines through eye map intolines parallel to
  z-axis.
• Lines perpendicular toz-axis map into lines
  perpen-dicular to z-axis.
• Transformation warps objects into new shapes.
• Perspective transformation warps objects so that,
  when viewed with an orthographic projection, they
  appear the same as the original objects do when
  viewed with a perspective projection.

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Chapter 7 Three-Dimensional Viewing

• Transformed View Volume Canonical View Volume
• top to ytop
• bottom to ybott
• left to xleft
• right to xright
• parallelepiped with dimensions related to
  cameras properties
• Scale and shift into canonical view volume (cube
  from 1 to 1 in each dimension)
• Transformation matrix known as projection matrix.
• OpenGL glFrustrum()

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Chapter 7 Three-Dimensional Viewing

• Projection matrix

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Chapter 7 Three-Dimensional Viewing

• 7.4.4 Clipping Faces against View Volume
• As Cyrus-Beck, but in 4D.
• Clip AC against six infiniteplanes.
• For each wall, test whetherA and C same side.
  If not, clip.
• Calculate six boundary coordinates (BC) for A and
  C. All 6 positive point inside CVV, else
  outside.
• Both same side trivial accept, reject. Else
  clip.

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Chapter 7 Three-Dimensional Viewing

• Selfstudy Rest of clipping algorithm,
  pp.387-389.
• Selfstudy Why clip against CVV?
• Selfstudy Why clip in Homogeneous Coordinates?
• Selfstudy The Viewport Transformation.

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Chapter 7 Three-Dimensional Viewing

• 7.5 Producing Stereo Views
• Not for exam purposes.
• 7.6 Taxonomy of Projections

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Chapter 7 Three-Dimensional Viewing

• 7.6.1 One-, Two-, Three Point Perspective
• Suppose n-axis of camera is perpendicular to one
  principal axis or another therefore vanishing
  point at infinity.
• Count number of finite vanishing points, i.e.
  number of principal exis not perp. to n.
• One point Perspective
• n perp. to two principal axes two of (nx,ny,nz)
  must be 0.

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Chapter 7 Three-Dimensional Viewing

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Chapter 7 Three-Dimensional Viewing

• 7.6.2 Parallel Projections
• Perspective projection Points projected along
  projectors that converge on eye
• Parallel projection All projectors have same
  direction d.
• Two types obliqueorthographic

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Chapter 7 Three-Dimensional Viewing

• Orthographic Projections
• dxdy0
• Orthographic projection in OpenGL READ.
• Types of Orthographic Projections
• Multiview
• Top, front, side views
• n made parallel to each of k, i, j in turn.

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Chapter 7 Three-Dimensional Viewing

• Axonometric
• n not parallel to any principal axis
• Foreshortening factor sin(?)
• Isometric all 3 principal axes foreshortened
  equally
• Dimetric 2 foreshortened equally
• Trimetric all 3 foreshortened unequally

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Chapter 7 Three-Dimensional Viewing

• Oblique projections Selfstudy.

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Chapter 7 Three-Dimensional Viewing

• Programming Task 5 Implement Case Study 7.1
  (Flying a camera through a scence), p. 405, in
  Hill.