3-D Transformations

3-D Transformations

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(x, y, z)

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

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• 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

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• 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

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

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Basic transformation set

• Translate

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• Scale

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• Rotate

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

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• 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

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Non-linear transformations - deformations

• Coordinate locations are transformed according to math functions

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• 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.

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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)