#G02 · Geometry & Trigonometry · Lv.2View in 3D Mind-Map

2D/3D Euclidean Distance Formula

René Descartes · 1637 CE

d(P, Q) = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}

\|P - Q\|_2 = \sqrt{\sum_{i=1}^{n} (p_i - q_i)^2}

Applies the Pythagorean theorem across Cartesian coordinates (x, y, z) to measure exact straight-line distance in 2D, 3D, and n-dimensional spaces.

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Delta x

Delta y

Delta z

d = \sqrt{(3)^2 + (4)^2 + (12)^2} = 13.000

3D 거리 d (L₂)

13.0000

2D 바닥 투영 거리

5.0000

맨해튼 거리 L₁

19.0

Prerequisite Path (5 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#G01Pythagorean Theorem

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

STEP 5#G022D/3D Euclidean Distance Formula

d(P

🚀 Next Unlocks (Derived Formulas) (6)

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Geometric & Intuitive Insight

직육면체 상자의 바닥 대각선 길이 제곱은 Δx² + Δy² 이고, 여기에 높이 Δz² 을 한 번 더 피타고라스 정리로 더하면 3차원 공간 대각선 길이 √(Δx² + Δy² + Δz²) 이 됩니다.

Step-by-Step Derivation

Step 1.1단계: xy평면 투영 거리 d_xy² = (x₂ - x₁)² + (y₂ - y₁)² 을 피타고라스 정리로 구합니다.
Step 2.2단계: 바닥 대각선 d_xy 와 수직 높이 (z₂ - z₁) 이 이루는 직각삼각형에 다시 피타고라스 정리를 적용합니다.
Step 3.3단계: d² = d_xy² + (z₂ - z₁)² = Δx² + Δy² + Δz² 이므로 제곱근을 취해 거리를 얻습니다.

AI & Coding Real-World Use

KNN / K-Means clustering distance metric in ML and A* pathfinding heuristic in 3D games

#K-Means & KNN ML#A* Pathfinding#3D Spatial Query

const dist3D = (dx: number, dy: number, dz: number) => Math.hypot(dx, dy, dz);

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