![]() ![]() Represent these two successive rotations? This turns out to be very difficult How do we calculate an equivalent set of 3 Euler angles (c1,c2,c3) which will ![]() We apply an additional rotation represented by another 3 Euler angles (b1,b2,b3), If we have a 3D rotation represented by 3 Euler angles (a1,a2,a3), and then ![]() We tend to give the different angles different names depending on the situation where we are using them. For example, if we are using a right hand coordinate system, then if the thumb of our right hand indicates the direction of the positive rotation axis then the fingers show the direction of rotation. I don't mean the left or right handedness of the coordinate system I mean the direction along the rotation axis that we consider to be positive. This has the additional advantage that if we need to calculate the physics (dynamics) of the object then the inertia tensor is in the frame of the object. However, as we can see from the examples above, it usually makes sense to measure the angles in the coordinates of the object itself. Our first intuition would probably measure all angles relative to the ground especially if we were writing a computer simulation or game where everything is represented in absolute coordinates. So the number of permutations of options goes up from 12 to 24. We could measure each of the 3 angles relative to the ground (or some absolute coordinate system) or relative to the object being rotated. Relative to rotating object or absolute coordinates. So altogether we have 12 possible angle sequences. This means that we can also use any of the following sequences: xyx, xzx, yxy, yzy, There are more because the angles are not independent for instance rotating 90° around x followed by 90° around y then back -90° around x is the same as a single rotation of 90° around z, so we can form any 3D rotation by combining rotations in just 2 planes, but we may have to apply 2 separate rotations in one of the planes. ![]() so we could represent the order of 3D rotations as: xyz, yzx, zxy or reversing the order zyx, xzy, yxz which gives 6 permutations. Order of combining angles.Īs a quick shorthand lets use x to represent a pure rotation about the x axis and y to represent a pure rotation about the y axis and so on. Unfortunately there is not a widely accepted set of standards so I have tried to combine what standards there are so that we can have consistency across the site. Relative to rotating object or absolute coordinates.It is hard to find an example where all 3 angles are measured relative to the ground.īecause, when we are combining rotations, order is important so there are many types of Euler angles. However the third angle 'tilt' is around the axis of the telescope. In this example we might think that it would make sense to measure the angles with reference to the ground. In this example we would naturally measure the angles in the frame of reference of the aircraft. See this page for an explanation of the conventions and standards used on this site. The standards we choose depends on the properties we want, I have discussed the Other websites and books that you come across will use different standards, I cant be compatible with them all, so when I setup the site I chose the conventions which seemed the most popular, that's the best I can do. Right hand coordinate system (for both coordinates and rotations) and angles measured in the frame of reference of the rotating object. Standards Used on this pageĪs we shall see, there is no single set of conventions and standards in this area, therefore I have chosen to adopt the following to be consistent across this site. On this page we will also consider theoretical issues such as the number of degrees of freedom in rotations. If you are reading this page in order to write a 3D computer program I suggest you read enough of this page to convince yourself of the problems with Euler angles and to get an intuitive understanding of 3D rotations and then move on to quaternion or matrix algebra representations. When we first start to think about 3D rotations this seems the natural way to proceed but our intuition can be deceptive and there are a lot of problems that arise when we use Euler angles to do calculations. We use the term "Euler Angle" for any representation of 3 dimensional rotations where we decompose the rotation into 3 separate angles. ![]()
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