Lecture 004

Spatial Transformations

Linear Transformations

Linear Transformations

Linear Transformation Inariant

Linear Transformation Inariant

Rotation

Rotation Invariant

Rotation Invariant

Rotation in 3D

Rotation in 3D

The transpose of a rotation is its inverse: observe klzzwxh:0004, which means klzzwxh:0005

The transpose of a rotation is its inverse: observe R^TR = I, which means R^T = R^{-1}

Orthogonal Transformation: the transformation that preserve distance and origin

Rotation vs. Reflection

Rotation vs. Reflection

Scaling

Scaling: preserve direction while change magnitude

Negative Scaling in 2D and 3D

Negative Scaling in 2D and 3D

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

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

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

Spectral Theeorem: all symmetric matrix represent nonuniform scaling

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