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Figure 2.2 shows two coordinate frames, with frame o₁x₁y₁ obtained by rotating frame o₀x₀y₀ by an angle θ. Perhaps the most obvious way to represent the relative orientation of these two frames is merely to specify the angle of rotation θ. There are two immediate disadvantages to such a representation. First, there is a discontinuity in the mapping from relative orientation to the value of θ in a neighborhood of θ = 0. In particular, for θ = 2π − ε, small changes in orientation can produce large changes in the value of θ, for example, a rotation by ε causes θ to “wrap around” to zero. Second, this choice of representation does not scale well to the three-dimensional case.

Figure 2.2 Coordinate frame o₁x₁y₁ is oriented at an angle θ with respect to o₀x₀y₀.

A slightly less obvious way to specify the orientation is to specify the coordinate vectors for the axes of frame o₁x₁y₁ with respect to coordinate frame o₀x₀y₀:

in which and are the coordinates in frame o₀x₀y₀ of unit vectors x₁ and y₁, respectively.¹ A matrix in this form is called a rotation matrix. Rotation matrices have a number of special properties that we will discuss below.

In the two-dimensional case, it is straightforward to compute the entries of this matrix. As illustrated in Figure 2.2,

which gives

(2.1)

Note that we have continued to use the notational convention of allowing the superscript to denote the reference frame. Thus, is a matrix whose column vectors are the coordinates of the unit vectors along the axes of frame o₁x₁y₁ expressed relative to frame o₀x₀y₀.

Although we have derived the entries for in terms of the angle θ, it is not necessary that we do so. An alternative approach, and one that scales nicely to the three-dimensional case, is to build the rotation matrix by projecting the axes of frame o₁x₁y₁ onto the coordinate axes of frame o₀x₀y₀. Recalling that the dot product of two unit vectors gives the projection of one onto the other, we obtain

which can be combined to obtain the rotation matrix

Thus, the columns of specify the direction cosines of the coordinate axes of o₁x₁y₁ relative to the coordinate axes of o₀x₀y₀. For example, the first column (x₁ · x₀, x₁ · y₀) of specifies the direction of x₁ relative to the frame o₀x₀y₀. Note that the right-hand sides of these equations are defined in terms of geometric entities, and not in terms of their coordinates. Examining Figure 2.2 it can be seen that this method of defining the rotation matrix by projection gives the same result as we obtained in Equation (2.1).

If we desired instead to describe the orientation of frame o₀x₀y₀ with respect to the frame o₁x₁y₁ (that is, if we desired to use the frame o₁x₁y₁ as the reference frame), we would construct a rotation matrix of the form

Since the dot product is commutative, (that is, xi · yj = yj · xi), we see that

In a geometric sense, the orientation of o₀x₀y₀ with respect to the frame o₁x₁y₁ is the inverse of the orientation of o₁x₁y₁ with respect to the frame o₀x₀y₀. Algebraically, using the fact that coordinate axes are mutually orthogonal, it can readily be seen that

The above relationship implies that and it is easily shown that the column vectors of are of unit length and mutually orthogonal (Problem 2–4). Thus is an orthogonal matrix. It also follows from the above that (Problem 2–5) . If we restrict ourselves to right-handed coordinate frames, as defined in Appendix B, then (Problem 2–5).

More generally, these properties extend to higher dimensions, which can be formalized as the so-called special orthogonal group of order n.

Definition 2.1.

The special orthogonal group of order n, denoted SO(n), is the set of n × n real-valued matrices

(2.2)

Thus, for any the following properties hold

 

 The columns (and therefore the rows) of are mutually orthogonal

 Each column (and therefore each row) of is a unit vector

 

The special case, SO(2), respectively, SO(3), is called the rotation group of order 2, respectively 3.

To provide further geometric intuition for the notion of the inverse of a rotation matrix, note that in the two-dimensional case, the inverse of the rotation matrix corresponding to a rotation by angle θ can also be easily computed simply by constructing the rotation matrix for a rotation by the angle − θ: