#G07 · Geometry & Trigonometry · Lv.2View in 3D Mind-Map

Conic Sections (Parabola, Ellipse, Hyperbola)

Apollonius & Johannes Kepler · 200 BCE

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\quad e = \frac{\sqrt{a^2 - b^2}}{a}

y^2 = 4px \;(\text{Parabola}),\quad \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \;(\text{Hyperbola})

Algebraic equations of curves formed by slicing a cone—governing Keplerian planetary/satellite orbits, parabolic dish antennas, and quadratic optimization contours.

🧮 Live Formula Calculator & Plotter

Semi-major a

Semi-minor b

c = \sqrt{5^2 - 3^2} = 4.00, \quad e = \frac{c}{a} = 0.800

Eccentricity e

0.8000

Focal Distance ±c

±4.000

Ellipse Area πab

47.12

Prerequisite Path (7 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#A04Quadratic Formula & Discriminant

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

STEP 6#G022D/3D Euclidean Distance Formula

d(P

STEP 7#G07Conic Sections (Parabola, Ellipse, Hyperbola)

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

🚀 Next Unlocks (Derived Formulas) (2)

#G08Spherical Coordinates & Sphere Volume/Area #G10Parametric Bézier Curve Formula

Geometric & Intuitive Insight

두 초점 F, F' 으로부터의 거리 합이 2a로 일정한 점들의 자취가 타원입니다. 두 초점이 하나로 합쳐지면(이심률 e=0) 완벽한 원이 됩니다.

Step-by-Step Derivation

Step 1.1단계: 두 초점 F(c, 0), F'(-c, 0) 으로부터 거리의 합이 2a 인 점 P(x, y) 의 방정식을 세웁니다.
Step 2.2단계: √((x-c)² + y²) + √((x+c)² + y²) = 2a 의 양변을 제곱하여 근호를 정리합니다.
Step 3.3단계: b² = a² - c² 으로 치환하면 x²/a² + y²/b² = 1 표준형 방정식이 도출됩니다.

AI & Coding Real-World Use

Kepler satellite orbit simulation and Multivariate Gaussian covariance ellipses in ML

#Kepler Orbit Sim#Mahalanobis Ellipse#Antenna Optics

const eccentricity = (a: number, b: number) => Math.sqrt(1 - (b*b)/(a*a));

Geometry & Trigonometry — Related Formulas in This Cluster

Pythagorean Theorem

Lv.1 · Game Physics Distance

#G01

2D/3D Euclidean Distance Formula

Lv.2 · K-Means & KNN ML

#G02

Pi, Radian Measure & Circular Sector

Lv.1 · Three.js Radians

#G03

Unit Circle Trigonometry & Identities

Lv.2 · LLM RoPE Encoding

#G04

Law of Sines & Law of Cosines

Lv.2 · Inverse Kinematics IK

#G05

Trigonometric Addition & Double-Angle Formulas

Lv.3 · Rotation Matrix Proof

#G06