Also we can determine this by taking the appropriate

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. If If If If a>0 a<0 a = 0 and c >...
View Full Document

This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

Ask a homework question - tutors are online