#G10 · Geometry & Trigonometry · Lv.3View in 3D Mind-Map

Parametric Bézier Curve Formula

Pierre Bézier & Paul de Casteljau · 1962 CE

\mathbf{B}(t) = \sum_{k=0}^{n} \binom{n}{k}(1-t)^{n-k}t^k \mathbf{P}_k \quad (0 \le t \le 1)

\mathbf{B}_2(t) = (1-t)^2\mathbf{P}_0 + 2(1-t)t\mathbf{P}_1 + t^2\mathbf{P}_2

Blends control points using Bernstein binomial polynomials as t varies from 0 to 1, generating smooth curves across SVG, TrueType fonts, CAD, and 3D animation paths.

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Parameter t

Control P1y

B_2(0.50) = 0.25P_0 + 0.50P_1(8) + 0.25P_2 = 4.500

베지에 곡선 높이 B_y(0.5)

4.500

번스타인 가중치 합 ∑B_k

1.000

제어점 P₁ 가중치 2(1-t)t

50.0%

Prerequisite Path (9 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#G01Pythagorean Theorem

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

STEP 4#A01Linear Equation & Balance Axiom

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

STEP 5#A04Quadratic Formula & Discriminant

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

STEP 6#A08Binomial Theorem & Pascal's Triangle

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

STEP 7#G022D/3D Euclidean Distance Formula

d(P

STEP 8#G07Conic Sections (Parabola, Ellipse, Hyperbola)

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

STEP 9#G10Parametric Bézier Curve Formula

\mathbf{B}(t) = \sum_{k=0}^{n} \binom{n}{k}(1-t)^{n-k}t^k \mathbf{P}_k \quad (0 \le t \le 1)

Geometric & Intuitive Insight

선분 P₀P₁ 위의 내분점과 선분 P₁P₂ 위의 내분점을 구한 뒤, 그 두 점을 다시 비율 t:(1-t)로 선형보간(Lerp)하여 이으면 마법처럼 부드러운 포물선 곡선이 그려집니다(드 카스텔조 알고리즘).

Step-by-Step Derivation

Step 1.1단계: 1차 선형보간 L₀(t) = (1-t)P₀ + tP₁ 과 L₁(t) = (1-t)P₁ + tP₂ 를 구합니다.
Step 2.2단계: 두 점을 다시 보간하면 B₂(t) = (1-t)L₀(t) + tL₁(t) = (1-t)²P₀ + 2(1-t)tP₁ + t²P₂
Step 3.3단계: 이를 n차로 반복하면 이항정리 계수 nCk (1-t)^(n-k) t^k 가 가중치로 나타납니다.

AI & Coding Real-World Use

SVG paths, CSS cubic-bezier easing, TrueType fonts, and autonomous vehicle / 3D camera path planning

#SVG & Font Rendering#CSS cubic-bezier#3D Camera Splines

const quadBezier = (p0:number, p1:number, p2:number, t:number) => (1-t)*(1-t)*p0 + 2*(1-t)*t*p1 + t*t*p2;

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