Understanding the Ellipse
An ellipse is the set of all points whose total distance from two fixed foci stays constant. This calculator starts with those two foci. It then uses one extra size clue to build the equation. The clue can be the semi major axis, full major axis, semi minor axis, eccentricity, or one point on the curve.
Why Foci Matter
The foci define the direction and center of the ellipse. The center is the midpoint between them. Half the distance between the foci is called c. The semi major axis is a. The semi minor axis is b. A valid ellipse always needs a to be greater than c. When that rule fails, the shape cannot close.
Advanced Result Details
The tool supports tilted ellipses, not only horizontal or vertical ones. It builds a local coordinate system along the line joining the foci. This makes the standard equation easy to read. It also creates an expanded centered equation for tilted cases. The result includes center, distance between foci, axes, eccentricity, vertices, covertices, directrices, area, and perimeter estimate.
Use Cases
Students can verify analytic geometry homework. Teachers can prepare examples. Designers can examine oval layouts from two control points. Survey and drafting users can check whether a proposed ellipse is possible before drawing it. The export options help save the result for later comparison.
Accuracy Notes
The calculator uses the common Ramanujan approximation for perimeter. It is very accurate for most practical ellipses. The point mode estimates a from the sum of distances between the point and both foci. This matches the geometric definition directly. Inputs should use the same unit system throughout.
Helpful Interpretation
A smaller eccentricity means the ellipse is rounder. A value near one means it is longer and narrower. Directrices are included for users who compare focus based and conic section forms. The equation can be copied into notes, reports, or teaching material. Use the example table to test several clue types before entering your own values.