3.5.6 · D1Quaternions — definition q = (q₀, q₁, q₂, q₃), unit quaterni
Ek 3D rotation poori tarah do cheezein se describe hoti hai: ek direction jiske around spin karein (ek arrow) aur kitna twist karein (ek angle). Quaternion bas 4 numbers ka ek seedha-saada daabba hai jo dono ek saath store karta hai, aur ek rule se honest rehta hai — charon numbers ko square karke add karo to exactly 1 aana chahiye.
Neeche sab kuch woh vocabulary hai jo tumhe chahiye is sentence ka poora matlab samajhne se pehle: axis kya hai, angle kya hai, "squared and added to 1" ka geometrically kya matlab hai, aur 4 numbers 3-way freedom ko kyun describe karte hain.
Is page par kuch bhi assume nahi kiya gaya . Hum har woh symbol banate hain jo parent note (parent topic) use karta hai, ek-ek karke, har ek ko ek picture se anchor karke.
Sabse simple object se shuru karo: ek akela number, jaise 3 − 2 line par ek point ki tarah picture karo. Yeh ek scalar hai — ek akela number bina kisi direction ke, bas ek size (aur ek sign).
Ab teen numbers ek saath stack karo: ( x , y , z ) arrow picture karo jo origin se shuru hokar x y z vector hai. Hum yaad dilane ke liye uske upar ek chhota hat ya arrow banate hain ki woh kuch point kar raha hai: v
Definition Scalar vs vector
Scalar s magnitude (size) hota hai.Vector v = ( x , y , z ) magnitude aur direction dono hote hain.
Topic ko dono kyun chahiye: ek rotation ek scalar carry karta hai (kitna twist karte ho) AUR ek vector (spin-axis kis taraf point karta hai). Quaternion q = ( q 0 , q v )
Arrow ko phir se dekho. Yeh kitna lamba hai? Agar yeh x y x y
Intuition Kyun square karte hain, add karte hain, aur square-root lete hainEk right triangle ka rule hai ( leg ) 2 + ( leg ) 2 = ( hypotenuse ) 2 x 2 + y 2 z Squaring negative signs hata deta hai (length negative nahi ho sakti aur left-vs-right matter nahi karna chahiye); square root squaring ko undo karke ek honest length deta hai.
Ek unit vector ek arrow hai jis ki length exactly 1 hoti hai, jise hat se likha jaata hai: n ^ direction carry karna hai — yeh koi size add kiye bina ek taraf point karta hai.
Kisi bhi arrow ko unit vector banana ho toh use apni length se divide karo ("normalize " karo): v ^ = v / ∥ v ∥
Topic ko iska kyun zaroori hai: ek spin-axis ek pure direction hai — "is taraf spin karo." Isliye axis n ^ = ( n x , n y , n z ) n x 2 + n y 2 + n z 2 = 1
Ek angle θ kitna turn kiya . Ek clock ki suii ko ek position se doosri position par sweep karte hua picture karo. Ek poora chakkar 36 0 ∘ 2 π
Definition Sine aur cosine, circle parEk unit circle (radius 1) banao. Ek point ko angle θ
cos θ horizontal coordinate.sin θ vertical coordinate.
Kyunki point radius-1 circle par hai, Pythagoras woh identity deta hai jo tum baar baar use karoge:
cos 2 θ + sin 2 θ = 1.
Quaternion product formula do vector products use karta hai. Dono se milo.
Definition Dot product (scalar-valued)p ⋅ q = p x q x + p y q y + p z q z number hota hai (ek scalar).
Picture: do arrows kitna same taraf point karte hain. Same direction → bada positive; perpendicular → zero; opposite → negative.
Definition Cross product (vector-valued)p × q = ek naya arrow jo p aur q dono ke perpendicular ho. vector hota hai, jo dono inputs ke plane se sideways nikal raha hota hai.
Picture: apne right-hand ki ungliyaan p q p × q
Tumne pehle ek imaginary unit dekha hoga: i i 2 = − 1 Complex Numbers as 2D Rotations ). Quaternions mein teen hote hain: i , j , k
Definition Imaginary unitsHar ek − 1 i 2 = j 2 = k 2 = − 1. ij = k , j k = i , k i = j Picture: i , j , k
Poora quaternion phir likha jaata hai q = q 0 + q 1 i + q 2 j + q 3 k q = ( q 0 , q v ) q v = ( q 1 , q 2 , q 3 ) q 0 i , j , k
Common mistake "Vector part( q 1 , q 2 , q 3 ) Kyun sahi lagta hai: dono 3-number bundles hain.
Fix: axis n ^ unit vector hai; quaternion ka vector part n ^ sin 2 θ scaled sin 2 θ
Iisse pehle ki hum "rotation apply karo" bhi bol sakein, hume woh operations chahiye jo quaternions par act karte hain. Inhe abhi milte hain — neeche kuch bhi inhen use nahi karta jab tak yahan define na ho jaayein.
Definition Hamilton product⊗ ⊗ do quaternions multiply karte ho. Yeh ordinary number multiplication NAHI hai — yeh Section 4 ke dot aur cross products use karke scalar aur vector parts ko mix karta hai:
p ⊗ q = ( p 0 q 0 − p ⋅ q , p 0 q + q 0 p + p × q ) . Picture: 4-ka-daabba andar do, 4-ka-daabba bahar ek milta hai. p ⋅ q p × q sign flip karta hai (p × q = − q × p non-commutative hai: generally p ⊗ q = q ⊗ p
q ∗ Conjugate sirf vector part ka sign flip karta hai:
q ∗ = ( q 0 , − q v ) = ( q 0 , − q 1 , − q 2 , − q 3 ) . Picture: twist-amount rakhna, lekin spin-axis ko opposite taraf point karana — yeh rotation ka undo button hai.
q − 1 Inverse woh quaternion hai jo, multiply karne par, "kuch na karo" wala identity ( 1 , 0 ) q − 1 = ∥ q ∥ 2 q ∗ . Unit quaternion ke liye (∥ q ∥ = 1 1 q − 1 = q ∗ (bas 3 vector signs negate karo). Kyun matter karta hai: rotation reverse karna sasta hona chahiye. Unit quaternions ke saath yeh bas teen sign flips hain — koi division nahi, koi matrix kaam nahi. Isliye spacecraft flight computers (Spacecraft Attitude Determination ) inhe pasand karte hain.
Ab jab ⊗ q − 1 θ /2
Intuition Rotation sandwichArrow v sandwich karna padta hai:
v ′ = q ⊗ v ⊗ q − 1 . v dono taraf se touch karta hai, isliye iska rotation effect do baar apply hota hai. Agar ek q θ 2 θ q aadha angle, θ /2 θ
Ab headline formula mein har symbol define ho chuka hai. Ise dheere se padho:
Intuition Kyun 4 numbers, 1 rule = 3 freedoms — 3-sphere pictureChaar numbers ( q 0 , q 1 , q 2 , q 3 ) ∑ q i 2 = 1 4 − 1 = 3 point hai jo ek sphere ki surface par pinned hai , hamesha centre se distance 1 par. Ek surface ka dimension us space se ek kam hota hai jisme woh baith ti hai — ek 2D sphere-surface 3D mein rehti hai, aur hamara "3-sphere" surface 4D mein rehta hai. Tum 4D draw nahi kar sakte, isliye neeche wali figure honest 3D stand-in dikhata hai: ek ordinary ball jiska surface woh jagah hai jahan har valid rotation rehta hai.
Definition Do signs ek hi rotation hain (double cover)Sandwich v ′ = q ⊗ v ⊗ q − 1 q − q dono sides par aate hain aur cancel ho jaate hain:
( − q ) ⊗ v ⊗ ( − q ) − 1 = q ⊗ v ⊗ q − 1 . q − q identical 3D rotation produce karte hain. Har real rotation isliye do quaternion points se name hota hai (3-sphere par antipodal). "Do-naam-ek-rotation" wala yeh fact S O ( 3 ) double cover kehlata hai — sphere rotations ka set do baar cover karta hai.
q − q Kyun sahi lagta hai: yeh alag 4-tuples hain, toh surely alag outputs honge.
Fix: sandwich sign ko square kar deta hai. − q doosra naam hai.
θ = 0 Zero rotation par, q = ( cos 0 , n ^ sin 0 ) = ( 1 , 0 ) n ^ ka koi asar nahi aur woh undefined hai — jo sahi bhi hai: "kuch spin mat karo" ko parwah nahi ki tum kis axis ke around nahi ghume. Yeh ( 1 , 0 ) identity quaternion hai, kuch-na-karo wala element.
θ = 18 0 ∘ π cos 2 π = 0 q = ( 0 , n ^ ) pure vector quaternion, scalar part zero. Yahan q − q n ^ − n ^ n ^ 18 0 ∘ − n ^ 18 0 ∘
Quaternion q = q0 plus vector part
cos and sin on unit circle
Cross product - a new arrow
Hamilton product times operator
Double cover q and minus q
Unit constraint sum of squares = 1
Represents a real 3D rotation
Yeh map seedha Axis-Angle Representation , Rotation Matrices SO(3) , Euler Angles and Gimbal Lock , aur baad mein Quaternion Kinematics — dq/dt aur Spacecraft Attitude Determination mein feed hota hai.
Right side cover karo aur khud se test karo — tum parent note ke liye ready ho jab sab cleanly reveal ho jaayein.
Scalar kya hai... ek akela number; picture = ek line par ek point, sirf magnitude.
Vector ( x , y , z ) origin se ek arrow; magnitude aur direction dono.
v = ( x , y , z ) ∥ v ∥ = x 2 + y 2 + z 2 (3D Pythagoras).
Unit vector kya hai... ek arrow jis ki length exactly 1 ho, n ^
Vector normalize karne ke liye... use apni length se divide karo,
v ^ = v / ∥ v ∥ .
cos θ sin θ unit circle ke around angle θ
Inhe jodne wali identity kya hai... cos 2 θ + sin 2 θ = 1
Dot product p ⋅ q ek number : p x q x + p y q y + p z q z
Cross product p × q ek vector dono ke perpendicular; order swap karne par sign flip hota hai.
Units kya obey karte hain... i 2 = j 2 = k 2 = − 1 ij = k , j k = i , k i = j
⊗ Hamilton (quaternion) product; non-commutative; scalar part
p 0 q 0 − p ⋅ q , vector part
p 0 q + q 0 p + p × q .
Conjugate q ∗ ( q 0 , − q v ) — vector part flip karo; undo direction.
Unit quaternion ka inverse kya hai...q − 1 = q ∗ ∥ q ∥ = 1
Rotation sandwich kya hai... v ′ = q ⊗ v ⊗ q − 1 q
Half-angle θ /2 do-sided sandwich q
q − q same rotation (double cover); signs sandwich mein cancel ho jaate hain.
θ = 0 ( 1 , 0 ) , identity; axis undefined hai (kuch spin nahi kiya).
θ = 18 0 ∘ q 0 = cos 9 0 ∘ = 0
Vector part vs axis... vector part = n ^ sin 2 θ scaled ), bare axis nahi.
Kyun 4 numbers, 1 rule... 4 − 1 = 3
