18.1 Concepts 501
EXAMPLE 18.1Rotation as a coordinate transformation
Consider a point P in the xy-plane, with coordinates(x, y)with respect to the coordinate
system Oxy. As discussed in Section 8.5, an anticlockwise rotation through angle θ
about the Ozaxis moves the point to position P′with coordinates(x′, y′), as in Figure
18.1, such that (equations (8.41))
x′ 1 = 1 x 1 cos 1 θ 1 − 1 y 1 sin 1 θ
y′ 1 = 1 x 1 sin 1 θ 1 + 1 y 1 cos 1 θ
(18.8)
An alternative way of interpreting these equations (the ‘passive’ instead of ‘active’
interpretation) is as the change of coordinates of the fixedpoint P when the
coordinate system itself undergoes a clockwiserotation through angle θ, from Oxy
to Ox′y′as in Figure 18.2.
The corresponding matrix equation in both cases is
(18.9)
and the transformation matrix
(18.10)
completely characterizes the coordinate transformation.
0 Exercises 1, 2
cos sin
sin cos
θθ
θθ
−
x
y
x
y
′
′
=
−
cos sin
sin cos
θθ
θθ
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θ •P(x,y)
P
′
(x
′
,y
′
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o
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y
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Figure 18.1 Figure 18.2