## How do you calculate Euler angle of a rotation matrix?

Given a rotation matrix R, we can compute the Euler angles, ψ, θ, and φ by equating each element in R with the corresponding element in the matrix product Rz(φ)Ry(θ)Rx(ψ). This results in nine equations that can be used to find the Euler angles. Starting with R31, we find R31 = − sin θ. are valid solutions.

**Do quaternions have units?**

Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis.

### Why are quaternions 4D?

Why is quaternion algebra 4D and not 3D? Complex algebra is 2D and what is known as quaternion algebra jumps to 4D. Using 1,i,j, and k as the base (where complex uses 1 and i (or j if you are an EE)) which results in a 4-axis space.

**Why do quaternions represent rotations?**

The representation of a rotation as a quaternion (4 numbers) is more compact than the representation as an orthogonal matrix (9 numbers). Furthermore, for a given axis and angle, one can easily construct the corresponding quaternion, and conversely, for a given quaternion one can easily read off the axis and the angle.

#### How do you calculate LERP?

lerp(a, b, x) = a + (b -a ) * x; From this formula you can get that the third parameter is actually just the percentage of the B. You then multiply it with B to calculate the percent value, and sum it with the percentage of A (which is equal to 100% minus the percent of B).

**How do you combine quaternion rotations?**

To rotate a vector v=ix+jy+kz by a quaternion q you compute vq=qvq−1. So if q and q′ are two rotation quaternions, to rotate by q then q′ you calculate (vq)q′=q′qvq−1q′−1=q′qv(q′q)−1=vq′q.

## What is the formula for rotations?

Rotation Formula

Type of Rotation | A point on the Image | A point on the Image after Rotation |
---|---|---|

Rotation of 90° (Clockwise) | (x, y) | (y, -x) |

Rotation of 90° (Counter Clockwise) | (x, y) | (-y, x) |

Rotation of 180° (Both Clockwise and Counterclockwise) | (x, y) | (-x, -y) |

Rotation of 270° (Clockwise) | (x, y) | (-y, x) |

**What is derivative of a rotation matrix?**

Derivative of a rotation matrix. A rotation of theta about the vector L is equal to a skew-symmetric matrix computed on the vector Omega multiplied by the original rotational matrix. Omega in this case is the angular velocity vector. It is the rate of change of angle multiplied by the vector direction about which the rotation is occurring.

### What is the formula for interpolated quaternion?

– qm = interpolated quaternion – qa = quaternion a (first quaternion to be interpolated between) – qb = quaternion b (second quaternion to be interpolated between) – t = a scalar between 0.0 (at qa) and 1.0 (at qb) – θ is half the angle between qa and qb From: http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/slerp/

**What is a rotation of theta about the vector L?**

A rotation of theta about the vector L is equal to a skew-symmetric matrix computed on the vector Omega multiplied by the original rotational matrix. Omega in this case is the angular velocity vector.

#### Is the inverse of an orthogonal matrix equal to its transpose?

One of the properties of an orthogonal matrix is that it’s inverse is equal to its transpose so we can write this simple relationship R times it’s transpose must be equal to the identity matrix. First off I’m going to consider the simple case of a rotation by the angle theta about the X-axis.