Next, we want to explore properties of rotation matrices. To remind you, rotation matrices are three by three matrices that have the properties of orthogonality and the fact that the determinant is equal to plus one. Matrices of these kinds constitute the special orthogonal group. We use SO3 to refer to this group. It's the special orthogonal group in three dimensions. Rotation matrices are a great way of describing rotations, but they're not intuitively very obvious. So, sometimes we prefer to use coordinates to describe SO3. So in addition to rotation matrices, we'll discuss a set of angles called Euler angles that are often used to describe rotations. We will also look at a parametrization that explicitly describes the axis of rotation and the angle of rotation. It's also possible to use coordinates called exponential coordinates, although we will not discuss this in this course. Finally, it's also possible to use quaternions, again we won't discuss this in this class. To motivate the discussion on coordinates, let's first start looking at coordinates of an object we're familiar with, the sphere. In fact, let's think about the coordinates that we might use to describe the location on the Earth's surface. Well, we know that coordinates in the Earth's surface are described using latitudes and longitudes, but are these descriptors unique? In other words, given any point on the Earth's surface, is there a unique combination of latitudes and longitudes that describe that point? You might be tempted to say yes, but the answer is no, if you consider the two poles. The poles have a unique latitude but the longitudes are not well-defined. In fact, any longitude works to describe the North Pole or the South Pole. It's only the latitude that's well-defined. We're interested in a coordinate chart enrich every point on the Earth's surface maps to a pair of coordinates and these coordinates are unique. Since this is hard to do, what we generally do is we use a collection of coordinates or a collection of charts. So, for instance, we can agree that as we get close to the North Pole or the South Pole, we use a different nomenclature to describe the area around the North Pole and the area around the South Pole. So, if you have a set of coordinates, there should be a one-to-one map between the coordinates and the Earth's surface. So, one question you might want to ask is, what's the minimum number of charts you need to cover the earth's surface? Remember, the chart has to lend itself to a coordinate system that is one to one. The answer to this question is two. Similarly, we want to ask ourselves the question, how many coordinates do we need to describe a general rotation? We saw before that if you consider rotation about a single axis, you only need one parameter. In this example, looking at rotations about the x axis, the angle theta parameterizes the rotation, and of course, from that we get the rotation matrix. How would we do that for a general rotation? So, in this example, it's not clear how to describe the orientation of a rigid body in the final configuration with respect to the initial configuration. So, here's position one and here is position two. In this example, the rigid body has moved from position one to position two through a rotation. How do we describe position two with respect to position one? How many coordinates do we need to describe this rotation?