D39_Eulerian Angles and Quaternions

Table of Contents

1. Euler Angles

2. Quaternion

3. Negative Quaternion

4. Unit Quaternion

5. Cross product of quaternions

6. Difference of Quaternion

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1. Euler Angles

Consists of three angles (x, y, z), which are used to describe the rotation of an object in a specific coordinate system

The rotation adjusted by the Inspector panel is the Euler angle transform.eulerAngles

Eulerian angles are intuitive and easy to understand, but one disadvantage is the universal lock (please encyclopedia)

Unity uses y→x→z

2. Quaternion

Quaternion

A quaternion consists of a scalar and a 3D element [w,(x,y,z)]

Meaning: Represents a rotation in 3D space

Given the rotation, assuming that the rotation is about the N axis, the rotation is θ degrees, and the N axis is (x, y, z), then it constitutes a quaternion

Q=[cos(θ/2),sin(θ/2)*x,sin(θ/2)*y,sin(θ/2)*z]

Constructing a quaternion from axis-angle pairs Quaternion.AngleAxis

Euler angle conversion to quaternion Quaternion.Euler

Quaternion to Euler Angles .eulerAngles

3. Negative Quaternion

q[w,(x,y,z)] -q[-w,(-x,-y,-z)]

Geometric meaning: q and -q represent the same amount of rotation

4.Unit Quaternion

q[1,0,0,0] -q[-1,0,0,0]

Quaternion.identity

Geometric meaning: represents a quaternion without a rotation angle

5. QuaternionThe cross product of

The result obtained by multiplying a quaternion by a quaternion is still a quaternion

Geometric meaning: The new quaternion obtained by the cross product of two quaternions is the superposition of the two quaternion rotations

transform.rotation*=Quaternion.AngleAxis();

The cross product of a quaternion and a vector can get a new vector that rotates according to the rotation direction of the quaternion

Note: Only the quaternion * vector operator is overloaded in the quaternion, and there is no overloaded vector * quaternion operator, so only

Vector3 dir=Quaternion.AngleAxis()*Vector3.forward;

Quaternion.LookRotation(); Passing a vector as a parameter can get the angle required to turn the positive direction of the object to the vector

6. Quaterniondifference

Quaternion.Slerp(a,b,t);