Free · Analytic geometry
Conic Axis, Focus and Directrix Parameters
Find eccentricity, focus distance, semi-latus rectum and focal parameter for all six captured conic orientations.
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
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Formula and method
An ellipse with semi-axes5 and3 has c=4,e=0.8,ℓ=1.8 and focal parameter2.25.
Ellipse c²=a²−b²; hyperbola c²=a²+b²; e=c/a; ℓ=b²/a; p=b²/c. For y²=4ax, e=1 and ℓ=p=2a.
Worked example
Enter these known values and leave the other values blank.
- Conic section
- Ellipse (horizontal)
- Semi-major axis (a)
- 5
- Semi-minor axis (b)
- 3
- Eccentricity (e)
- 0.8
- Linear eccentricity (c)
- 4
- Semi-latus rectum (l)
- 1.8
- Focal parameter (p)
- 2.25
Assumptions and limitations
- Results use finite 64-digit arithmetic. With extremely disparate axes, an ellipse or hyperbola eccentricity may be numerically indistinguishable from 1; this is a numerical resolution limit and does not turn the shape into a parabola. An arithmetic expression such as 1 + a very small offset may round to 1 before calculation; exact decimal literals can retain that offset. Near-equal literal axes use a factored focal-distance formula to avoid subtraction of rounded squares. Inputs already rounded to equal axes are withheld with the ordering error. Tiny nonzero derived lengths use scientific notation when ordinary rounding cannot represent them.
- All length inputs use the same unit. a and b are semi-axis lengths, not full diameters. Hyperbola b is a conjugate semi-axis, despite the legacy field key.
- An ellipse requires a>b>0. For a circle use the separate circle mode; the ellipse focal parameter has no finite limit when the foci merge. Hyperbola semi-axes and parabola focal distance must be positive.
- Focal parameter p here is the perpendicular distance from a focus to its corresponding directrix. It differs from semi-latus rectum ℓ. A circle has no finite directrix or focal parameter in this convention.
- The six modes are standard axis-aligned conics centered at the origin, except the parabola has its vertex at the origin and opens right. Rotations, translations and arbitrary quadratic fitting are not included.
- Derived conic quantities retain their captured output-only controls. Clear the input axes or switch shape to start a different calculation; output fields are not independent inverse-solving inputs.
- Every picture is a fixed original reference schematic, not a plot of your numbers.
Common questions
What is the difference between focal parameter and semi-latus rectum?
For an ellipse or hyperbola, p=b²/c measures focus-to-directrix distance, while ℓ=b²/a is half the chord through a focus perpendicular to the main axis.
Why is a circle a separate mode?
Its two foci coincide at the center, e=0 and ℓ equals its radius. The finite-directrix ellipse formula p=b²/c divides by zero at c=0.
References
Correct horizontal ellipse focus-to-directrix distance to b²/c, matching the vertical geometry. For both ellipse orientations use the algebraic identity a²-b²=(a-b)(a+b), e=c/a and p=b²/c to avoid cancellation for near-equal exact decimal axes at the unchanged 64-digit working precision. All source equation identities, controls, six modes and forward-only directions are preserved; no physical magnitude cap is added. Evaluate the captured positive semi-axis and strict ellipse-axis ordering requirements by sign instead of a fuzzy comparison with zero. This retains the mathematical domain while handling extremely small signed values correctly; it adds no physical magnitude cap.