Quaternions — definition q = (q₀, q₁, q₂, q₃), unit quaternion constraint

3.5.6 · Physics › Guidance, Navigation & Control (GNC)

Quaternion KIYA hai?

Hum ise aksar compactly $q = (q_{0}, q_{v})$ likhte hain.

Hamilton ka multiplication rule scratch se derive karna

Hum sirf yeh assume karte hain ki units $ij k = - 1$ satisfy karti hain aur har ek $- 1$ pe square hoti hai. Baaki sab kuch follow hota hai.

Yeh step kyun? $ij k = - 1$ se, right-multiply karo $k$ se: $ij k \cdot k = - k \Rightarrow ij (k^{2}) = - k \Rightarrow ij (- 1) = - k \Rightarrow ij = k .$

Yeh step kyun? $ij = k$ ko $i$ se left-multiply karo: $i (ij) = ik \Rightarrow (i^{2}) j = ik \Rightarrow - j = ik \Rightarrow k i = j (after cycling) .$

Do quaternions ka full multiplication phir term-by-term expand karke milta hai:

$p \times q$ term hi wajah hai ki quaternion multiplication (jaise rotation composition) non-commutative hai.

KYUN half-angle $θ /2$ ? Kyunki kisi vector ko rotate karne ke liye sandwich $v^{'} = q \otimes v \otimes q^{- 1}$ use hota hai, jo $q$ effectively do baar apply karta hai, isliye har factor ko sirf aadha angle carry karna chahiye. (Iska matlab yeh bhi hai ki $q$ aur $- q$ same rotation dete hain — "double cover".)

Conjugate, inverse, aur yeh kyun matter karta hai

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Recall Cover me: ek unit quaternion mein kitne free parameters hote hain?

4 numbers − 1 constraint = 3, jo 3 rotational DOF se match karta hai.

Recall Cover me: unit constraint aur identity quaternion likho.

$q_{0}^{2} + q_{1}^{2} + q_{2}^{2} + q_{3}^{2} = 1$ ; identity $= (1, 0, 0, 0)$ .

Recall Feynman: quaternions ko ek 12-saal ke bachche ko explain karo.

Socho tum ek khilona ghuma rahe ho. Ghuman describe karne ke liye tumhe kehna hoga: spinning stick kis taraf point kar raha hai (yeh ek arrow hai, 3 numbers) aur kitna twist kiya tumne (1 number). Ek quaternion 4 numbers ka ek neat chhota dibba hai jo dono ko hold karta hai. Ek rule hai: agar tum chaaron numbers ko square karke add karo, tumhe exactly 1 milna chahiye — jaise kehna "yeh arrow exactly ek meter lamba hai." Agar dibba add-up mistakes se thoda galat ho jaaye, tum sirf usse shrink ya stretch karo taaki total 1 ho jaaye. Phir yeh hamesha ek real spin describe karta hai, aur yeh kabhi "jam" nahi karta jaise purane methods karte hain.

Flashcards

What are the four components of a quaternion?

Scalar part

q_{0}

aur vector part

(q_{1}, q_{2}, q_{3})

q = q_{0} + q_{1} i + q_{2} j + q_{3} k

State the unit-quaternion constraint.

q_{0}^{2} + q_{1}^{2} + q_{2}^{2} + q_{3}^{2} = 1

Why 4 numbers for a 3-DOF rotation?

4 params − 1 unit constraint = 3 DOF; extra number gimbal-lock singularity remove karta hai.

How is a rotation of angle θ about unit axis n̂ encoded?

q = (cos \frac{θ}{2}, \overset{n}{^} sin \frac{θ}{2})

Why the half-angle θ/2?

Vector ko rotate karne mein double product

q v q^{- 1}

use hota hai, jo

q

do baar apply karta hai, isliye har factor aadha angle carry karta hai.

Do q and −q represent the same rotation?

Haan —

q v q^{- 1}

mein signs cancel ho jaate hain; yeh SO(3) ka double cover hai.

Give the quaternion conjugate.

q^{*} = (q_{0}, - q_{1}, - q_{2}, - q_{3})

Inverse of a unit quaternion?

q^{- 1} = q^{*}

(kyunki

∥ q ∥ = 1

Hamilton's fundamental relation?

i^{2} = j^{2} = k^{2} = ij k = - 1

Scalar part of

p \otimes q

p_{0} q_{0} - p \cdot q

Vector part of

p \otimes q

p_{0} q + q_{0} p + p \times q

Is quaternion multiplication commutative?

Nahi —

p \times q

term ise non-commutative banata hai.

Geometrically, where do unit quaternions live?

4D space mein unit 3-sphere

S^{3}

pe.

How do you fix drift after integration?

Renormalize karo:

q \leftarrow q / ∥ q ∥

Identity quaternion?

(1, 0, 0, 0)

— "kuch mat karo" wala rotation.

Connections

Euler Angles and Gimbal Lock — woh problem jo quaternions solve karte hain.
Rotation Matrices SO(3) — 9 numbers, 6 constraints wala alternative.
Quaternion Kinematics — dq/dt — $q$ kaise evolve karta hai angular velocity $ω$ ke saath.
Axis-Angle Representation — half-angle form ka direct source.
Spacecraft Attitude Determination — jahan GNC yeh sab use karta hai.
Complex Numbers as 2D Rotations — quaternions $e^{i θ}$ ke 3D analogue hain.