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Unformatted text preview: ert from a, b, c, and d to r, s, θ, and ϕ, note that reﬂection
across the x-axis equals reﬂection across the y -axis, followed by a rotation of π .
(See Exercise 1.) That is, we need reﬂection only across the y -axis.
From the form of the matrix in eq (2.1), we know
a = r cos(θ) b = −s sin(ϕ) c = r sin(θ) d = s cos(ϕ) (2.15) Then
r=± a2 + c2 and s = b2 + d2 . (2.16) where sign of r is negative if and only if the transformation involves a reﬂection.
This happens if and only if S × R and S ′ × R′ point in the opposite direction,
where S ′ = Q2 − Q1 and R′ = Q3 − Q1 .
Also from eq (2.15) we have
tan(θ) = c/a and tan(ϕ) = −b/d. (2.17) We are not quite ﬁnished with this analysis, because arctan is single-valued only
in the range [−π/2, π/2], but θ and ϕ can lie anywhere in the range [−π, π ]. To
deal with this, we consider the signs of a, b, c, and d. Attending to appropriate
details, here are the cases.
a = 0 and c >...
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