2.1. We could specify the coordinates of the point p with respect to either frame o0x0y0 or frame o1x1y1. In the former case we might assign to p the coordinate vector (5, 6) and in the latter case ( − 3, 3). So that the reference frame will always be clear, we will adopt a notation in which a superscript is used to denote the reference frame. Thus, we write
Geometrically, a point corresponds to a specific location in space. We stress here that p is a geometric entity, a point in space, while both
Since the origin of a coordinate frame is also a point in space, we can assign coordinates that represent the position of the origin of one coordinate frame with respect to another. In Figure 2.1, for example, we may have
Thus, o01 specifies the coordinates of the point o1 relative to frame 0 and o10 specifies the coordinates of the point 00 relative to frame 1. In cases where there is only a single coordinate frame, or in which the reference frame is obvious, we will often omit the superscript. This is a slight abuse of notation, and the reader is advised to bear in mind the difference between the geometric entity called p and any particular coordinate vector that is assigned to represent p. The former is independent of the choice of coordinate frames, while the latter obviously depends on the choice of coordinate frames.
While a point corresponds to a specific location in space, a vector specifies a direction and a magnitude. Vectors can be used, for example, to represent displacements or forces. Therefore, while the point p is not equivalent to the vector v1, the displacement from the origin o0 to the point p is given by the vector v1. In this text, we will use the term vector to refer to what are sometimes called free vectors, that is, vectors that are not constrained to be located at a particular point in space. Under this convention, it is clear that points and vectors are not equivalent, since points refer to specific locations in space, but a free vector can be moved to any location in space. Thus, two vectors are equal if they have the same direction and the same magnitude.
When assigning coordinates to vectors, we use the same notational convention that we used when assigning coordinates to points. Thus, v1 and v2 are geometric entities that are invariant with respect to the choice of coordinate frames, but the representation by coordinates of these vectors depends directly on the choice of reference coordinate frame. In the example of Figure 2.1, we would obtain
In order to perform algebraic manipulations using coordinates, it is essential that all coordinate vectors be defined with respect to the same coordinate frame. In the case of free vectors, it is enough that they be defined with respect to “parallel” coordinate frames, that is, frames whose respective coordinate axes are parallel, since only their magnitude and direction are specified and not their absolute locations in space.
Using this convention, an expression of the form
2.2 Representing Rotations
In order to represent the relative position and orientation of one rigid body with respect to another, we attach coordinate frames to each body, and then specify the geometric relationship between these coordinate frames. In Section 2.1 we saw how one can represent the position of the origin of one frame with respect to another frame. In this section, we address the problem of describing the orientation of one coordinate frame relative to another frame. We begin with the case of rotations in the plane, and then generalize our results to the case of rotations in a three-dimensional space.
2.2.1 Rotation in the Plane
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
