• 3D vectors are expressed as coordinate triples that define a relative location.
  
  

• Defining a coordinate system requires four 3D vectors:
  
  

  1 to locate the origin
  1 X unit vector
  1 Y unit vector
  1 Z unit vector
  
  

• A new coordinate system can therfore be expressed via a 4 x 3 transforamtion matrix
  
  
• Homogeneous transforms or 4 x 4 matrices are sometimes used
  
  

• Multiple transforms can be expressed by multiplying matrices for the component tranforms together, so that any number of nested transforms can ultimately be collapsed into a single transformation matrix
  
  
• Hardware can do matrix multiplications very quickly
  
  

• Since matrix multiplication is not commutative, the order in which transformations are applied can be significant
  

Basic transformation set

• Translate
  
  

• Scale
  
  

• Rotate
  
  

• Rotation axis can be arbitrarily defined in space (i.e. not one of the three coordinate axes)
  
  

• Transformations often are used within modeling operations
  
  
  
  • Rotation - rotational sweep
    
  • Translation - extrusion
    
  • Translation in object space - extrude along a path
    
  • Scaling - reflection
    
    
• Modeling transformations can affect normals
  
  

Non-linear transformations - deformations

• Coordinate locations are transformed according to math functions
  
  

• Both geometry and cameras are positioned with local coordinate systems within the global coordinate system of the scene
  
  
• Transforming either will affect the location of objects as they appear on the image plane.
  
  

Scene Graphs

• Directed Acyclic Graphs (DAG)
  
  
• Heirarchical structure used to describe entire scenes within several common "scene graph" APIs
  (e.g. JAVA3D, VRML, Performer, Inventor)
  
  

Back to course outline

This file was last modified on Sept. 5, 2000.