How this calculator works
Every conic section can be described by how far it bends away from a circle. The eccentricity is that measure: a circle has 0, an ellipse is between 0 and 1, a parabola is exactly 1, and a hyperbola is greater than 1. The linear eccentricity is the distance from the centre to a focus. The semi-latus rectum is half the chord through a focus parallel to the directrix, and the focal parameter is the distance from a focus to the corresponding directrix. A horizontal or vertical choice only rotates the figure; the four scalar properties are the same for either orientation.
Formula
Ellipse: c = √(a² - b²), e = c/a, l = b²/a, p = b²/c. Hyperbola: c = √(a² + b²), e = c/a, l = b²/a, p = b²/c. Parabola with focal length f: l = 2f, e = 1, p = 2f. Circle of radius r: l = r, e = 0, c = 0.
Worked example
An ellipse with semi-major axis 5 m and semi-minor axis 3 m has c = √(25 - 9) = 4 m, e = 0.8, l = 9/5 = 1.8 m and p = 9/4 = 2.25 m. A hyperbola with a = 5 m and b = 3 m has c = √34 ≈ 5.831 m and e ≈ 1.1662.
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Frequently asked questions
Why is the semi-latus rectum useful?
It fixes the shape near a focus and appears directly in the polar equation of a conic, r = l / (1 + e cos θ). That equation describes circles, ellipses, parabolas and hyperbolas with one formula.
Does horizontal or vertical change the answers?
No. The orientation decides which axis is the major or transverse axis in the drawing. Eccentricity, linear eccentricity, semi-latus rectum and focal parameter do not depend on that rotation.
Why is there no focal parameter for a circle?
A circle has no directrix, so the distance from focus to directrix is infinite. The calculator reports the semi-latus rectum and the two zero eccentricities instead.
Why does the ellipse reject a smaller semi-major axis?
By definition the semi-major axis is the longer one. If your values are the other way around, swap them before calculating.