Cs 455 computer graphics
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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