2D Rotation matrices (5): exponential map

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 27:15

Summary:

This video will end the study of 2D (planar) rotation matrices, introducing the exponential map which formalizes the idea that “products” of rotation matrices are equivalent to “addition” of angles, so angles are a sort of “logarithms of rotations”.

Basically, 2D rotation matrices are exponentials of skew-symmetric matrices $R(𝜃)\; =\; exp(𝜃J)$, being $J$ the matrix $J\; =\; [0,-1;\; 10]$.

Next, as $J^2=\; -I$, i.e., skew symmetric matrices square to a multiple of identity, it can operate “as if $J$ were the imaginary unit”. In fact, it is, in a sense, indistinguishable from the imaginary unit. For instance, the Euler formula is recovered: $exp(J𝜃)\; =\cos <!--FUNCTION\; APPLICATION-->𝜃\; +\; J\; \cdot \sin <!--FUNCTION\; APPLICATION-->𝜃$. This idea is further explored in video [imisMEN]. Basically, the imaginay unit is the “rotor” generator in 2D.

The Euler formula can be changed to $I\; +\; J\; \cdot \sin <!--FUNCTION\; APPLICATION-->𝜃\; +J^2\cdot \; (1\; -\cos <!--FUNCTION\; APPLICATION-->𝜃)$, which will be the root of the Euler-Rodrigues formula in 3D rotations, to be discussed in sequel videos.

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