#L10 · Linear Algebra & Manifolds · Lv.5View in 3D Mind-Map

Quaternions & 3D Spherical Interpolation (SLERP)

William Rowan Hamilton · 1843 CE

q = \cos\frac{\theta}{2} + (u_x i + u_y j + u_z k)\sin\frac{\theta}{2},\quad i^2=j^2=k^2=ijk=-1

\mathbf{p}' = q\,\mathbf{p}\,q^{-1},\quad \operatorname{Slerp}(q_1, q_2, t) = \frac{\sin((1-t)\Omega)}{\sin\Omega}q_1 + \frac{\sin(t\Omega)}{\sin\Omega}q_2

Extends complex numbers to 4D hypercomplex numbers (w, x, y, z) to represent 3D rotations without Gimbal Lock and smoothly interpolate orientations via SLERP.

🧮 Live Formula Calculator & Plotter

Rotation θ°

Axis uz

\mathbf{q} = 0.707 + 0.000\mathbf{i} + 0.707\mathbf{k} \quad (\|\mathbf{q}\|=1)

스칼라부 w = cos(θ/2)

0.7071

벡터부 (qx, qz)

(0.000, 0.707)

단위 노름 ||q||

1.0000

Prerequisite Path (26 STEP)

Full Path Analyzer

Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 2#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 3#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 4#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 5#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

STEP 6#L01Vector Operations, L1/L2 Norms & Normalization

\|\mathbf{v}\|_2 = \sqrt{\sum_{i=1}^{n} v_i^2}

STEP 7#A06Logarithm Laws & Change of Base

\log_a(xy) = \log_a x + \log_a y

STEP 8#A08Binomial Theorem & Pascal's Triangle

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

STEP 9#A07Arithmetic & Geometric Series Sum

S_n = \frac{a(1 - r^n)}{1 - r}

STEP 10#C02Derivative Definition & Instantaneous Rate

f'(x) = \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

STEP 11#G022D/3D Euclidean Distance Formula

d(P

STEP 12#A04Quadratic Formula & Discriminant

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

STEP 13#G04Unit Circle Trigonometry & Identities

\cos^2\theta + \sin^2\theta = 1

STEP 14#C01Limits, Continuity & Euler's Number e

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828

STEP 15#G05Law of Sines & Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos C

STEP 16#L04Matrix Multiplication & Linear Transformation

(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}

STEP 17#L02Dot Product & Cosine Similarity

\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta

STEP 18#L033D Cross Product & Surface Normal Vector

\mathbf{a} \times \mathbf{b} = (a_y b_z - a_z b_y

STEP 19#C04Fundamental Theorem of Calculus (FTC)

\int_a^b f(x)\

STEP 20#G06Trigonometric Addition & Double-Angle Formulas

\cos(\alpha\pm\beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta

STEP 21#L05Determinant & Matrix Inverse

\det(A) = ad - bc

STEP 22#A09Complex Numbers & Polar Plane

z = a + bi = r(\cos\theta + i\sin\theta)

STEP 23#L072D/3D Rotation Matrix & Orthogonal Group SO(3)

R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

STEP 24#C06Taylor & Maclaurin Series Expansion

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

STEP 25#C07Euler's Formula & Euler's Identity

e^{i\theta} = \cos\theta + i\sin\theta

STEP 26#L10Quaternions & 3D Spherical Interpolation (SLERP)

q = \cos\frac{\theta}{2} + (u_x i + u_y j + u_z k)\sin\frac{\theta}{2}

Geometric & Intuitive Insight

2D 평면 회전을 복소수 e^(iθ) = cos θ + i sin θ 로 표현했듯, 3D 공간에서 단위 회전축 u = (u_x, u_y, u_z) 둘레로 θ만큼 회전하는 것은 q · p · q⁻¹ 양쪽 곱셈 때문에 반각(θ/2)을 쓰는 4차원 단위 사원수가 됩니다!

Step-by-Step Derivation

Step 1.1단계: 해밀턴이 더블린 브로엄 다리에 새긴 기본 관계식 i² = j² = k² = ijk = -1 로부터 ij = k, ji = -k (외적 성질)가 도출됩니다.
Step 2.2단계: 회전축 단위벡터 u 에 대해 단위 사원수 q = cos(θ/2) + u sin(θ/2) 를 구성합니다.
Step 3.3단계: 3D 점 p를 순허수 사원수로 보고 q p q⁻¹ 을 계산하면, 축 u 에 수직인 성분이 정확히 θ/2 + θ/2 = θ 만큼 회전합니다.

AI & Coding Real-World Use

Three.js/Unity/Unreal skeletal animation (Quaternion SLERP) and spacecraft/drone attitude control

#Three.js Quaternion#Gimbal Lock Free#Aerospace Attitude

const quatFromAxisZ = (rad: number) => ({ w: Math.cos(rad/2), x: 0, y: 0, z: Math.sin(rad/2) });

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