3D Transformations, Schemes and Mind Maps of Computer science

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CS 430

Computer Graphics

3D Transformations

World Window to Viewport Transformation

Week 2, Lecture 3 David Breen, William Regli and Maxim Peysakhov Department of Computer Science Drexel University

Outline

• World window to viewport transformation
• 3D transformations
• Coordinate system transformation

4

Window-to-Viewport

Transformation

• Given a window and a viewport, what is the transformation from WCS to VPCS? Three steps: - Translate - Scale - Translate 1994 Foley/VanDam/Finer/Huges/Phillips ICG

5

Transforming World

Coordinates to Viewports

• 3 steps
  1. Translate
  2. Scale
  3. Translate Overall Transformation: 1994 Foley/VanDam/Finer/Huges/Phillips ICG

P '= M

wv

P

3D Transformations

10

Representation of 3D

Transformations

• Z axis represents depth
• Right-Handed System
  • When looking “down” at the origin, positive rotation is CCW
• Left-Handed System
  • When looking “down”, positive rotation is in CW
  • More natural interpretation for displays, big z means “far” (into screen) 1994 Foley/VanDam/Finer/Huges/Phillips ICG

3D Transformations:

Scale & Translate

• Scale
  • Parameters for each axis direction
• Translation

3D Transformations:

Rotation

• One rotation for each world coordinate axis

• Also can be expressed as the

Rodrigues Formula

P

rot = P cos( ϑ ) + ( n × P )sin( ϑ ) + n ( n ⋅ P )( 1 − cos( ϑ ))

Rotation Around an

Arbitrary Axis

Improved Rotations

• Euler Angles have problems
  • How to interpolate keyframes?
  • Angles aren’t independent
  • Interpolation can create Gimble Lock, i.e. loss of a degree of freedom when axes align
• Solution: Quaternions!

€ p = ( 0 ,  x )

slerp – Spherical linear interpolation Need to take equals steps on the sphere A & B are quaternions