Computer Graphics

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Computer Graphics

• 3D Graphics

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3-D coordinates

• Normal
• (x,y,z)
• Homogeneous
• (x,y,z,h)

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3-D World Coordinate Systems

• Same as for 2-D, with the addition of a z-axis
• Two orientations are generally used
• Right-handed
• Left-handed
• For each, the thumb points in the x direction,
  the pointer finger is y, and the middle finger
  is z

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3-D World Coordinate Systems
Right-handed
Left-handed
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Projections

• Perspective
• One-point
• Two-point
• Three-point

• Parallel
• Orthographic
• Top (plan)
• Front elevation
• Side elevation
• Axonometric
• Isometric
• Oblique
• Cabinet
• Two-point

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Parallel projection

• Orthographic
• Normals and projections in same direction
• Side, front and top views
• Oblique
• Not necessarily the same direction

• Side
• Front
• Top

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Orthographic Parallel projection

• Axonometric
• Projection planes are not normal to principal
  axis
• Show several faces at once
• Isometric
• Projection-plane normal makes equal angles with
  each principal axis

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Perspective projection
y
z-axis vanishing point
x
z
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Viewing Volume
view reference point

• Starting with a projection reference point (PRP)
  we trace four lines, one through each corner of
  the window
• The window in located on the view plane
• The resulting pyramid is called a view volume

projection reference point
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Projecting an image

• The first step in rendering 3-D images, is to
  project the object onto the 2-D view plane

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Projecting an image

• The distance from the PRP to the view plane is
  called the viewing distance

z
yp
y
viewingdistance (d)
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Projecting an image

• The position of a point, as projected on the view
  plane, is computed by ...

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Projecting an image

• Using matrices
• We create an intermediate vector
• Dividing this vector by z /d yields our projected
  point

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Frustum View Volume

• The viewing volume is turned into a frustum by
  adding front and back planes
• Only objects within the frustum are visible

back
view plane
front
prp
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Parallelpiped View Volume

• Adjacent sides are at right angles to each other
• A frustum to parallelpiped transformation is
  accomplished via scaling or shearing and scaling

prp
front
vp
back
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Parallelpiped View Volume

• Simple case only scaling
• Caution!Division by zero up ahead!
• Go review earlier discussion

yp
y
z
d
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Generalized Transformation

• Q What happens when the viewing vector is not
  perpendicular to the viewplane?
• A First Shear and then Scale!

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First, Shear
vp
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First, Shear
vp
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Mshear
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Then, Scale
vp
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Then, Scale
vp
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Mscale
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Normalized View Volume
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World to Eye

• We have so far assumed that the world and eye
  coordinate systems are the same
• If not, we need to add one last transformation as
  a pre-processing step to what we have done so far

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Viewing Transformation Pipeline
Question?What operations move us from one system
to the next?