Explore the geometry of ellipses, hyperbolas, and parabolas. Compute foci, vertices, eccentricity, asymptotes, and directrix. Interactive graph with real-time updates — a complete toolkit for analytic geometry and beyond.
From Kepler's laws of planetary motion to satellite dish design, conic sections describe natural and engineered trajectories. This interactive analyzer empowers you to explore ellipses (orbits, whispering galleries), hyperbolas (navigation, cooling towers), and parabolas (reflectors, projectile motion).
General quadratic equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0 — but our tool focuses on translated, axis-aligned canonical forms ensuring clarity and precision in learning.
Each conic is defined by eccentricity e: ellipse (0 < e < 1), parabola (e = 1), hyperbola (e > 1). The distance from center to focus is c, where c² = |a² − b²| for ellipse/hyperbola. For parabolas, p determines focus and directrix.
Accuracy assurance: All calculations are performed using double-precision (IEEE 754) arithmetic. The underlying formulas have been cross-checked against standard calculus textbooks (Larson, Stewart). The graphing engine dynamically samples each curve with >300 points to ensure smooth representation. If any parameter is invalid (e.g., non-positive a or b, or p = 0), an error message will guide you to correct it.
Ellipses have two foci; the sum of distances from any point on ellipse to the foci is constant (2a). Eccentricity measures elongation. Our calculator accurately computes both foci, major/minor vertices, and directrices (x = ±a/e for horizontal ellipse).
Hyperbolas consist of two disconnected branches. The difference of distances to the foci is constant. Asymptotes provide direction guidance. The tool calculates asymptote equations and foci for both horizontal/vertical orientations.
A parabola reflects parallel rays to the focus, making it ideal for antennas and headlights. Using vertex (h,k) and direction, we compute focus, directrix line, and axis of symmetry.
Earth orbits the Sun in an ellipse with eccentricity ≈ 0.0167. Our analyzer can simulate such near‑circular ellipses. Hyperbolic trajectories are used for interplanetary space probes (gravity assists). Parabolic paths approximate escape velocity.
Horizontal ellipse centered at (0,0): x²/a² + y²/b² = 1. Foci at (±c,0) where c = √(a²−b²). Eccentricity e = c/a. Directrices x = ±a/e. Our calculator solves these instantly and validates the geometric definition.
| Conic | Canonical Form | Eccentricity e | Focal Parameter |
|---|---|---|---|
| Ellipse | (x−h)²/a² + (y−k)²/b² = 1 (a ≥ b for horiz) | c/a (c² = |a²−b²|) | 0 < e < 1 |
| Hyperbola | (x−h)²/a² − (y−k)²/b² = 1 | c/a (c² = a²+b²) | e > 1 |
| Parabola | (x−h)² = 4p(y−k) (vertical) | 1 | p = focal distance |