special_orthogonal_group

introduction

%%visits: 2 The structure of the special_orthogonal_group is called a sphere with diametrically opposed points being equal. This is basically saying that the poles are equal and you can move the poles around anywhere you want it too.

This is not in Euclidean space. (! :todo: check if true) ## intuition If N is odd then O(N) ≅ ℤ₂ × SO(N)

This is geometrically a [[rotation_matrix]], but the “diametrically opposed” points mean that they are mirrored.

SO(3) is structurally a funny shape. It is not a shape in Euclidean space. And it can be shown as matrices in the form $\begin{pmatrix} 1 &0&0\\0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta \end{pmatrix}$ or something [[similar_matrices|similar]] ## rigour special_orthogonal_grou := ${A O(n)|(A) = 1} $

Theorem: If A ∈ SO(3) ⟹ ∃n̂ ∈ ℝ³ : An̂ = n̂ ## exam clinic {{file:../figures/screenshot_20220115_173952.png}} ## examples and non-examples ## resources tags :math:

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