Latus Rectum Calculator — Parabola, Ellipse & Hyperbola

About the Latus Rectum Calculator — Parabola, Ellipse & Hyperbola

The latus rectum is one of the most important yet often overlooked measurements of a conic section. It is the chord through a focus that is perpendicular to the principal axis of the curve. For a parabola y² = 4ax, the latus rectum has length 4a. For an ellipse or hyperbola, it equals 2b²/a. This single measurement encodes the "width" of the conic at its focus and connects eccentricity, semi-axes, and focal distance in one elegant quantity.

Understanding the latus rectum is essential in orbital mechanics (where it determines the shape of planetary orbits), optics (parabolic reflector design), and pure conic geometry. The semi-latus rectum — half the latus rectum — appears directly in the polar equation of any conic: r = ℓ/(1 + e cos θ), where ℓ is the semi-latus rectum and e is eccentricity.

This calculator supports all three main conics: parabola, ellipse, and hyperbola. Enter the defining parameters and get the latus rectum, semi-latus rectum, eccentricity, foci, directrix, and a visual eccentricity gauge from the same setup. Presets for real-world conic sections — satellite dishes, planetary orbits, navigation systems — let you explore applied mathematics.

When This Page Helps

The latus rectum connects multiple conic properties — foci, directrix, eccentricity, and axis lengths — in a single measurement. Calculating it by hand requires careful formula selection depending on the conic type, and mistakes are common (especially confusing a and b for ellipses vs. hyperbolas).

This calculator handles all three conic types with clear visual feedback, saving time for students working through analytic geometry, engineers designing reflective surfaces, and anyone studying orbital mechanics or optics.

How to Use the Inputs

1. Select the conic section type: parabola, ellipse, or hyperbola.
2. For a parabola, enter the parameter a (focus distance from vertex).
3. For an ellipse or hyperbola, enter the semi-major (transverse) and semi-minor (conjugate) axes.
4. Or click a preset for a real-world conic section.
5. View the latus rectum, semi-latus rectum, eccentricity, foci, and directrix.
6. Check the eccentricity gauge for a visual classification of the conic.
7. Consult the formula table for all conic section relationships.

Example Calculation

Tips & Best Practices

• For a circle (e = 0), the latus rectum equals the diameter.
• The latus rectum of Earth's orbit around the Sun is approximately 149.45 million km.
• In a parabolic reflector, the latus rectum determines the "opening width" at the focus — critical for antenna and satellite dish design.
• Remember: for ellipses, a > b (semi-major > semi-minor). If your b > a, swap them.
• The semi-latus rectum appears directly in the angular momentum equation of orbits: ℓ = h²/(GM).
• A smaller latus rectum means a "narrower" conic at the focus; a larger one means "wider."

The Latus Rectum in Orbital Mechanics

In celestial mechanics, every orbit is a conic section with the central body at one focus. The semi-latus rectum ℓ is one of the six classical orbital elements and determines the size of the orbit independently of its shape. It relates to the specific angular momentum h and gravitational parameter μ via ℓ = h²/μ. For Earth orbiting the Sun, ℓ ≈ 149.45 million km — slightly less than 1 AU because Earth's orbit is slightly elliptical (e ≈ 0.0167).

Parabolic Reflectors and the Latus Rectum

Parabolic mirrors and antennas work because all rays parallel to the axis reflect through the focus. The latus rectum determines the diameter of the reflector at the focal plane, which directly affects the concentration of energy. A satellite dish with a shorter focal length (smaller a) has a shorter latus rectum, meaning a "deeper" dish shape, while a larger focal length produces a shallower profile.

Comparing Conics Through the Latus Rectum

By keeping the semi-latus rectum constant and varying eccentricity, you can morph smoothly between conics: a circle (e = 0), ellipse (0 < e < 1), parabola (e = 1), and hyperbola (e > 1). This is the geometric insight behind the polar equation r = ℓ/(1 + e cos θ), which unifies all conic sections into a single formula parameterized by ℓ and e. The latus rectum is thus the natural "scale parameter" of any conic section.

Frequently Asked Questions

• What is the latus rectum of a conic section?

  The latus rectum is the chord through a focus drawn perpendicular to the principal axis. Its length characterizes the "width" of the conic at the focus.

• What is the semi-latus rectum?

  The semi-latus rectum (ℓ) is half the latus rectum. It appears in the polar equation of conics: r = ℓ/(1 + e cos θ).

• How is the latus rectum related to eccentricity?

  For any conic, the semi-latus rectum ℓ = a(1 − e²) for an ellipse and ℓ = a(e² − 1) for a hyperbola. For a parabola (e = 1), ℓ = 2a.

• Why is the latus rectum important in orbital mechanics?

  In Kepler orbits, the semi-latus rectum determines the shape and size of the orbit independently of the eccentricity. It appears in the vis-viva equation and orbital energy formulas.

• What happens to the latus rectum as eccentricity approaches 0?

  As e → 0, the ellipse becomes a circle and the latus rectum approaches the diameter (2a = 2b). The latus rectum equals the diameter when the conic is a circle.

• Can the latus rectum be longer than the major axis?

  For an ellipse, no — the latus rectum LR = 2b²/a is always ≤ 2b ≤ 2a. For a hyperbola, the latus rectum can exceed 2a when b > a.