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CS 455 – Computer Graphics . Viewing Transformations I. Motivation. Want to see our “virtual” 3-D world on a 2-D screen. Graphics Pipeline. Object Space. Model Transformations. World Space. Viewing Transformation. Eye/Camera Space. Projection & Window Transformation. Screen Space.

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cs 455 computer graphics

CS 455 – Computer Graphics

Viewing Transformations I

motivation
Motivation

Want to see our “virtual” 3-D world on a 2-D screen

graphics pipeline
Graphics Pipeline

Object Space

Model Transformations

World Space

Viewing Transformation

Eye/Camera

Space

Projection & Window

Transformation

Screen Space

viewing transformations
Viewing Transformations
  • Projection: take a point from m dimensions to n dimensions where n < m
  • There are essentially two types of viewing transforms:
    • Orthographic: parallel projection
      • Points project directly onto the view plane
      • In eye/camera space (after viewing

tranformation): drop z

    • Perspective: convergent projection
      • Points project through the origin

onto the view plane

      • In eye/camera space (after viewing

tranformation): divide by z

parallel projections
Parallel Projections
  • We will first deal with orthographic projection
  • Get the concept and models down
    • Projection direction is parallel to projection plane normal
    • Center of projection (COP) is at infinity
    • Parallel lines remain parallel
    • All angles are preserved for faces parallel to the projection plane

p1

p1’

Center of

projection

at infinity

p2

p2’

Projectors

orthographic projection
Orthographic Projection
  • Points project orthogonally onto (i.e., normal to) the view plane:
    • Projection lines are parallel

y

z

x

projection environment

y

d

x

z

Projection Environment
  • We will use a right-handed view system
  • The eyepoint or camera position is on the +z axis, a distance d from the origin
  • The view direction is parallel to the z axis
  • The view plane is in the xy plane and passes through the origin
parallel projection

y

(x, y, z)

(x’, y’, z’)

x

z

Parallel Projection
  • A point in 3-space projects onto the viewplane via a projector which is parallel to the z axis
  • What is (x’, y’, z’)?
parallel projection1

(x’, y’, z’)

Parallel Projection
  • Looking down the y axis:

(x, y, z)

x

z

  • So z’ = 0, x’ = x
parallel projection2

(x’, y’, z’)

Parallel Projection
  • Looking down the x axis:

y

(x, y, z)

z

So y’ = y

parallel projection3
Parallel Projection
  • Thus, for parallel, orthographic projections,
  • x’ = x, y’ = y, z’ = 0
  • So, to perform a parallel projection on an object, we need to multiply it by some matrix that has this effect

What is M?

i.e., we simply

drop the z

coordinate

perspective projection
Perspective Projection
  • In the real-world, we see things in perspective:
    • Parallel lines do not look parallel
    • They converge at some point
perspective projection1

y

x

z

Perspective Projection
  • Points project through the focal point (e.g., eyepoint) onto the view plane:
    • Projection lines are convergent
perspective projection2
Perspective Projection
  • Center of projection (COP) is no longer at infinity
  • Projectors form a view frustum that is a pyramid with the tip at the COP

eye

view plane

perspective projection3
Perspective Projection
  • We will start with the projection plane parallel to the XY plane and perpendicular to the Z axis
  • Lines parallel to the X or Y axis remain parallel
  • X and Y distances become shorter as Z becomes more negative, e.g. a cube viewed in perspective:

y

x

perspective projection computation
Perspective Projection Computation
  • Assume the projection plane is normal to the Z axis, located at Z = 0.
  • Assume the center of projection (COP, eyepoint) is located at Z = d
  • What is P’(x’, y’, z’)?

P(x, y, z)

projection plane

y

P’(x’, y’, z’)

x

Center of Projection

z

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