#L07 · Linear Algebra & Manifolds · Lv.3View in 3D Mind-Map

2D/3D Rotation Matrix & Orthogonal Group SO(3)

Leonhard Euler & Olinde Rodrigues · 1775 CE

R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix},\quad R^{-1} = R^T

\det(R) = \cos^2\theta + \sin^2\theta = 1

Rotates vectors by angle θ without distortion (det R = 1), where the inverse is simply the transpose R⁻¹ = Rᵀ—the workhorse of 3D game engines and robotics.

🧮 Live Formula Calculator & Plotter

Input x

Input y

Rotation θ°

(x', y') = (1.41,\; 4.24), \quad \det R(45^\circ) = 1

회전 후 좌표 (x', y')

(1.414, 4.243)

보존된 벡터 길이 ||v||

4.472

회전각 θ

45°

Prerequisite Path (15 STEP)

Full Path Analyzer

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

STEP 1#A02Polynomial Expansion & Factoring

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

STEP 2#A03Laws of Exponents & Radicals

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

STEP 3#L01Vector Operations, L1/L2 Norms & Normalization

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

STEP 4#G01Pythagorean Theorem

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

STEP 5#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 6#A01Linear Equation & Balance Axiom

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

STEP 7#G022D/3D Euclidean Distance Formula

d(P

STEP 8#G05Law of Sines & Law of Cosines

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

STEP 9#L02Dot Product & Cosine Similarity

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

STEP 10#L033D Cross Product & Surface Normal Vector

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

STEP 11#G04Unit Circle Trigonometry & Identities

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

STEP 12#L04Matrix Multiplication & Linear Transformation

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

STEP 13#G06Trigonometric Addition & Double-Angle Formulas

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

STEP 14#L05Determinant & Matrix Inverse

\det(A) = ad - bc

STEP 15#L072D/3D Rotation Matrix & Orthogonal Group SO(3)

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

🚀 Next Unlocks (Derived Formulas) (2)

#L09Singular Value Decomposition (SVD) & PCA #L10Quaternions & 3D Spherical Interpolation (SLERP)

Geometric & Intuitive Insight

x축 기저벡터 (1, 0)을 θ만큼 회전시키면 (cos θ, sin θ)가 되어 1열에 들어가고, y축 기저벡터 (0, 1)을 θ만큼 회전시키면 (-sin θ, cos θ)가 되어 2열에 들어갑니다!

Step-by-Step Derivation

Step 1.1단계: 기저벡터 e₁=(1,0) 과 e₂=(0,1) 이 각도 θ 회전 후 이동한 좌표를 행렬의 열벡터로 배치합니다.
Step 2.2단계: 임의의 점 (x, y) = x·e₁ + y·e₂ 에 선형성을 적용하면 x' = x cos θ - y sin θ, y' = x sin θ + y cos θ
Step 3.3단계: 행렬식 det(R) = cos²θ - (-sin²θ) = 1 이므로 넓이가 보존되며, R(-θ) = R(θ)^T = R(θ)⁻¹ 입니다.

AI & Coding Real-World Use

Three.js/Unreal/Unity 3D object & camera rotation and LiDAR point-cloud alignment (ICP)

#3D Engine Rotation#LiDAR Point Cloud#Orthogonal Matrix SO(3)

const rot2D = (x:number, y:number, r:number) => ({ x: x*Math.cos(r) - y*Math.sin(r), y: x*Math.sin(r) + y*Math.cos(r) });

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