This chapter begins the proper study of the closely related subjects of how to transform vector coordinates across bases, and how to quantify vehicle attitude. The direction-cosine matrix appears, and I discuss its properties. I then cover several in-depth examples of using it to describe aircraft attitude. Transforming coordinates leads to a discussion of the meaning of position, which serves to introduce homogeneous coordinates. I end with an example of calculating the motion of a ship.

Results of previous chapters come together here in the equations that model a vehicle’s position and attitude given a knowledge of, for example, its angular turn rates. These equations can seem perplexing at first glance, and so I derive them in careful steps, again making strong use of vectors and the frame dependence of the time derivative. I end with a detailed example of applying these equations to a spinning top.

An important set of coordinates to understand is that of our oblate Earth. I derive the equations transforming latitude/longitude/height to and from the ECEF cartesian axes. I use the model aircraft of a previous chapter as an aid to visualise the rotation sequences that are useful for calculating NED or ENU coordinates at a given point on or near Earth’s surface. I use these in a detailed example of sighting a distant aircraft. This leads to a description of the ‘DIS standard’ designed for such scenarios. I also use these ideas in a detailed example of estimating Earth’s gravity at a given point, which is necessary for implementing inertial navigation systems.