Finite Groups in Two and Three Dimensions
If T ∈ O(ℛ2), then T is completely determined by its action on the basis vectors e l = (1, 0) and e 2 = (0, 1). If Te 1 = (µ, v), then µ 2 + v 2 = 1 and Te 2 = ±(−v, µ), since T preserves length and orthogonality. Choose θ, 0 ≤ θ < 2π, such that cos θ = µ and sin θ = v.
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