Quote
quaternion rotation notes from today https://t.co/YEaAXeVcQ4
10/6/2024, 9:45 PMMedia as text
notes on how to interpret various operations with rotation quaternions:
Operations
q 7 is the inverse, which reverses the rotation q
is the conjugate, a much cheaper reversal if q is normalized (all rotation
q
quaternions are)
ab combines rotations (composition) which has has two equivalent
interpretations:
rotate a around its own axes/intrinsically by b
rotate b around its parent space/extrinsically by a
note that ab ba
ba-1 the extrinsic rotation from a to b
a the intrinsic rotation of a to b
aba-1 converts the intrinsio rotation b of a into an extrinsic rotation
a-1ba converts the extrinsic rotation b of a into an intrinsic rotation
q 2 doubles Operations q 7 is the inverse, which reverses the rotation q is the conjugate, a much cheaper reversal if q is normalized (all rotation q quaternions are) ab combines rotations (composition) which has has two equivalent interpretations: rotate a around its own axes/intrinsically by b rotate b around its parent space/extrinsically by a note that ab ba ba-1 the extrinsic rotation from a to b a the intrinsic rotation of a to b aba-1 converts the intrinsio rotation b of a into an extrinsic rotation a-1ba converts the extrinsic rotation b of a into an intrinsic rotation q 2 doubles
Analysis
Self-assessed trust95%
The statements regarding quaternion rotation operations contain several claims that need to be evaluated for their validity. The analysis reveals that the statements are mostly accurate, but there are some nuances that need clarification.
1. **Inverse and Conjugate**: The statement that the inverse of a quaternion reverses the rotation is true. The conjugate of a quaternion, when the quaternion is normalized, can indeed serve as a cheaper way to reverse the rotation. This is a well-established fact in quaternion mathematics.
2. **Combining Rotations**: The claim that the product of two quaternions (ab) combines rotations and has two interpretations (intrinsic and extrinsic) is also accurate. This is a fundamental property of quaternion multiplication.
3. **Order of Multiplication**: The statement that ab does not equal ba is true, as quaternion multiplication is not commutative.
4. **Conversion between Intrinsic and Extrinsic Rotations**: The operations described for converting between intrinsic and extrinsic rotations (aba-1 and a-1ba) are valid and reflect the mathematical properties of quaternions.
5. **Doubling Operations**: The statement about q 2 doubling is unclear without context, but if it refers to the effect of quaternion exponentiation, it can be interpreted as a valid operation in quaternion algebra.
Overall, the statements are mostly true, with a few requiring clarification. The majority of the claims hold up under scrutiny based on established mathematical principles regarding quaternions.
1. **Inverse and Conjugate**: The statement that the inverse of a quaternion reverses the rotation is true. The conjugate of a quaternion, when the quaternion is normalized, can indeed serve as a cheaper way to reverse the rotation. This is a well-established fact in quaternion mathematics.
2. **Combining Rotations**: The claim that the product of two quaternions (ab) combines rotations and has two interpretations (intrinsic and extrinsic) is also accurate. This is a fundamental property of quaternion multiplication.
3. **Order of Multiplication**: The statement that ab does not equal ba is true, as quaternion multiplication is not commutative.
4. **Conversion between Intrinsic and Extrinsic Rotations**: The operations described for converting between intrinsic and extrinsic rotations (aba-1 and a-1ba) are valid and reflect the mathematical properties of quaternions.
5. **Doubling Operations**: The statement about q 2 doubling is unclear without context, but if it refers to the effect of quaternion exponentiation, it can be interpreted as a valid operation in quaternion algebra.
Overall, the statements are mostly true, with a few requiring clarification. The majority of the claims hold up under scrutiny based on established mathematical principles regarding quaternions.
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-68.37Analysis date
10/8/2024, 8:32 PM