Interactively adjust parameters for circles, ellipses, parabolas, and hyperbolas in real time. Visualize foci, directrices, eccentricity, polar form, and the correspondence with conic cross-sections on a single screen.
Curve Selection
x²/16 + y²/6.25 = 1
Presets
Results
Eccentricity e
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Focal distance c
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Semi-latus rectum l = b²/a
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Curve type
Ellipse
CurveFoci F₁, F₂DirectrixAsymptotes (hyperbola)
Cvcartesian
Polar
Polar form r = l/(1 + e·cosθ). The blue point is the moving point for θ.
Cvcone
Relationship between cone section angle and eccentricity. The curve type is set by the cutting-plane inclination α compared with the cone half-angle φ.
🙋 Are circles, ellipses, parabolas, and hyperbolas all 'the same family'?
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I learned about circles and ellipses in class, but then the term 'conic sections' came up... I can't believe circles and parabolas are related?
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They are actually legitimate siblings. When you cut a right circular cone with a plane, the shape of the cross-section changes depending on how you cut. Cut perpendicular to the cone's axis → a circle. Tilt it a bit → an ellipse. Cut at an angle parallel to the slant edge (generatrix) of the cone → a parabola. Cut steeper than that → a hyperbola. Check out the 'Cone Cross-Section' tab to get a visual feel for the cutting angles.
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What is eccentricity e? When I move the slider, the shape changes, but what does that number actually mean?
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Eccentricity is the ratio e = r/d of the distance to a focus and the distance to the directrix. It classifies all conic sections in a unified way: e=0 is a circle, 01 is a hyperbola. As the eccentricity of an ellipse approaches 1, it becomes more and more elongated and eventually becomes a parabola—try experiencing that continuous change with the slider.
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When I select the 'Halley's Comet Preset' (e=0.967), it becomes an extremely elongated ellipse. Are planetary orbits also ellipses?
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That's exactly Kepler's First Law: 'Planets move in elliptical orbits with the Sun at one focus.' Earth's eccentricity is e≈0.017, nearly circular. Halley's Comet has e≈0.967, an extremely elongated ellipse; at perihelion (closest point) it's right next to the Sun, and at aphelion it goes beyond the orbit of Pluto. That's why its orbital period is about 75 years.
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Please explain why parabolas are used in satellite dishes and flashlight reflectors.
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Parabolas have the optical property that 'light emitted from the focus becomes parallel rays after reflection.' The proof can be shown using the law of reflection and tangent properties, but intuitively: light from focus F reflects off each point on the parabola, and no matter where it reflects, it travels parallel to the axis. Conversely, 'collecting all parallel radio waves (satellite broadcasts) into a single point—the focus' is the principle of receiving antennas.
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What does 'asymptote' mean for a hyperbola?
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It's a straight line that the curve approaches as it goes to infinity but never intersects. For the hyperbola x²/a² - y²/b² = 1, the asymptotes are y = ±(b/a)x. The two branches of the hyperbola spread out sandwiched between these asymptotes. In particular, when a=b (rectangular hyperbola), the asymptotes are y=±x (45 degrees). This type can also be written as xy = constant, and the curve representing 'isothermal change of a gas PV=constant' is exactly this.
Frequently Asked Questions
Because the cross-sectional shape when cutting a right circular cone with a plane yields the four types of conic sections. Depending on the angle α between the cutting plane and the cone axis, and the half-apex angle φ of the cone: α=90° (perpendicular to axis) → circle, φ<α<90° → ellipse, α=φ (parallel to generatrix) → parabola, α<φ → hyperbola.
It is defined as the ratio e = r/d of the distance to a focus and the distance to the directrix for any point on the curve. Intuitively, it's an indicator of 'how much the curve deviates from a circle.' e=0 is a perfect circle (no deviation), e→1 makes the ellipse infinitely elongated (degenerating into a parabola), and e>1 gives a hyperbola (splitting into two branches).
Kepler's First Law (1609) showed that 'planets move in elliptical orbits with the Sun at one focus.' Earth e≈0.017 (nearly circular), Mars e≈0.093, Halley's Comet e≈0.967 (elongated ellipse). If the velocity equals escape velocity, e=1 (parabolic orbit); if greater, e>1 (hyperbolic orbit—approaches the Sun only once). The Voyager probes' Jupiter flyby utilized hyperbolic orbits.
A parabolic reflector has the property of 'converting all light (electromagnetic waves) emitted from the focus into a parallel beam along the axis.' This property can be mathematically proven from the tangent and the law of reflection. Conversely, it also functions as a receiving antenna that concentrates all parallel radio waves (satellite broadcasts) into a single focal point. Satellite dish antennas, radio telescopes, flashlight reflectors, and car headlights operate on this principle.
The two branches of the hyperbola x²/a² - y²/b² = 1 exist in the region bounded by the asymptotes y = ±(b/a)x, approaching them at infinity but never intersecting. A rectangular hyperbola (a=b) can also be written as xy = a²/2, with asymptotes y=±x. Isothermal expansion (PV=constant) and inverse proportion graphs take this form. While the foci of an ellipse lie inside the major axis, the foci of a hyperbola lie 'outside' the two branches (c = √(a²+b²)).
What is Conic Sections Explorer?
Conic Sections Explorer is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.
By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.
Physical Model & Key Equations
The simulator is based on the governing equations behind Conic Sections Explorer. Understanding these equations is key to interpreting the results correctly.
Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.
Real-World Applications
Engineering Design: The concepts behind Conic Sections Explorer are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.
Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.
CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.
Common Misconceptions and Points of Caution
Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.
Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.
Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.