Euclidean Symmetries in Mathematics by Gidget Haddad

By Gidget Haddad

This can be a good written thorough learn with an useful method of Euclidean Symmetries in Mathematics
Table of Contents
Chapter 1 - Euclidean Group
Chapter 2 - Dihedral Group
Chapter three - Euclidean aircraft Isometry
Chapter four - Euler Angles
Chapter five - Euler's Rotation Theorem
Chapter 6 - Rotation illustration (Mathematics)
Chapter 7 - Orthogonal Group
Chapter eight - aspect teams in 3 Dimensions

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Others Euler angles, normally in the Tait-Bryan convention, are also used in robotics for speaking about the degrees of freedom of a wrist. They are also used in Electronic stability control in a similar way. Gun fire control systems require corrections to gun-order angles (bearing and elevation) to compensate for deck tilt (pitch and roll). In traditional systems, a stabilizing gyroscope with a vertical spin axis corrects for deck tilt, and stabilizes the optical sights and radar antenna. However, gun barrels point in a direction different from the line of sight to the target, to anticipate target movement and fall of the projectile due to gravity, among other factors.

These two circles intersect in two (opposite) points of which one, say A, is chosen. This point lies on the initial circle and thus is transported to a point a on the second circle. On the other hand, A lies also on the translated circle, and thus corresponds to a point α on the initial circle. Notice that the arc aA must be equal to the arc Aα. Now Euler needs to construct point O in the surface of the sphere that is in the same position in reference to the arcs aA and αA. If such a point exists then:   it is necessary, that the distances OA and Oa are equal to each other; the arcs Oa and OA must be equal, it is necessary that the angles OaA and OAα are equal.

That is, the elements of a rotation matrix are not all completely independent. For example, the rotation matrix in dimension 2 has only one degree of freedom, since all four of its elements depend on a single angle of rotation. A rotation matrix in dimension 3 (which has nine elements) has three degrees of freedom, corresponding to each independent rotation, for example by its three Euler angles or a magnitude one (unit) quaternion. In SO(4) the rotation matrix is defined by two quaternions, and is therefore 6-parametric (three degrees of freedom for every quaternion).

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