1.1 Operations and Points
3
Fig. 1.2 Counterclockwise
rotation of the point P 1 by an
angle α in the xy plane
in the opposite sense to that of the rotation of the hands on his watch. A synonym
for counterclockwise here is right-handed. If the reader orients his/her thumb in
the direction of the rotational pole, the palm of his/her right hand will indicate the
counterclockwise direction. The transformation can be obtained as follows. Let r be
the length of the radius-vector, r, from the origin to the point P 1 , and let φ 1 be the
angular coordinate of the point measured in the horizontal plane starting from the
x-direction, as shown in Fig. 1.2. The coordinates of P 1 are then given by
x 1 = r cos φ 1
y 1 = rsinφ 1
z 1 = 0
(1.3)
Rotating the point will not change its distance from the origin, but the angular coordinate will increase by α. The angular coordinate of P 2 will thus be given by
φ 2 = φ 1 + α. The coordinates of the image point in terms of the coordinates of the
original point are thus given by
x 2 = r cos φ 2 = r cos(φ 1 + α)
= r cos φ 1 cos α − rsinφ 1 sin α
= x 1 cos α − y 1 sin α
y 2 = rsinφ 2 = rsin(φ 1 + α)
= r cos φ 1 sin α + rsinφ 1 cos α
= x 1 sin α + y 1 cos α
z 2 = 0
(1.4)
In this way the coordinates of P 2 are obtained as functions of the coordinates of P 1
and the rotation angle. This derivation depends simply on the trigonometric relationships for sums and differences of angles. We may also express this result in the
form of a matrix transformation. For this, we put the coordinates in a column vector