Lecture 004

Spatial Transformations

Rotation

Orthogonal Transformation: the transformation that preserve distance and origin

• represented by matrices $Q^T Q = I$

• Rotation: additionally preserve orientation ($\det Q > 0$)

• Reflection: reverse orientation ($\det Q < 0$)

Scaling

Scaling: preserve direction while change magnitude

Negative Scaling: a sequence of reflections (how many reflections depends on the dimension of the space)

• in 2D: Negative Scaling is Rotation

• in 3D: Negative Scaling is Reflection

Symmetric Matrix: a matrix $A$ such that $A = A^T$.

Nonuniform Scaling: a symmetric matrix $A := R^TDR$

• rotate to new axes ($R$)

• apply a diagonal scaling ($D$)

• rotate back to original axes ($R^T$)

Spectral Theeorem: all symmetric matrix represent nonuniform scaling