CS488 - Introduction to Computer Graphics - Lecture 5

Comments and Questions

Euclidean Space

An affine space plus a metric, d(P1, P2), that has translational invariance

The familiar dot product (defined on vectors, Why?) is your baby.

The dot product also defines ahgles between vectors.

Cartesian Space

A Euclidean space plus a (usually orthonormal) frame.

Using the frame (often called `standard' or `distinguished' ) we can write uniquely

Properties of Affine Spaces

Affine Combinations

Affine Transformations

The general idea is that we apply a transformation to a point to position it.

There are two ways to see this process

One Point, Two Frames

  1. P = x i + y j + z k + O = x' i' + y' j' + z' k' + O'
  2. Express the second frame in terms of the first.

    i' = aii i + aij j + aik k, etc.

    O' = O + bi i + bj j + bk k.

  3. Then collect terms.

Two Points, One Frame

Matrix Representations of Affine Transformations

View Transformations

Return to the rendering pipeline.

  1. Several objects positioned in the world
  2. Viewer's eye looks at the world. Possibilities

What happened?

  1. Point Po in object coordinates.
  2. Transform Po by T1 to get it into world coordinates: Pw = T1( Po )
  3. Transform Pw by V to get it into view coordinates: Pv = V( Pw ) = V( T1( Po ) ) = V T1 Po.

To move the point transform it again.

  1. Pw = T2 T1 Po
  2. Pv = V T2 T1 Po

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