Understanding an Ellipse from Vertices and Eccentricity
An ellipse is a stretched circle with two fixed focus points. Its longest width is the major axis. The two endpoints of that axis are the vertices. When those vertices and eccentricity are known, the whole ellipse can be described.
Core Geometry
This calculator treats the given vertices as opposite ends of the major axis. It first finds the midpoint. That point becomes the center. It then measures the distance between the vertices. Half of that distance is the semi-major axis, called a.
Eccentricity describes how stretched the ellipse is. A value near zero gives a rounder shape. A value near one gives a thinner shape. The focus distance c equals eccentricity times a. The semi-minor axis b comes from the relation b squared equals a squared minus c squared.
Rotated Ellipse Support
The tool also finds rotation. The major axis may point in any direction on the coordinate plane. The calculator uses a unit vector along the vertices. A second perpendicular vector gives the minor-axis direction. These vectors allow foci, co-vertices, and equations to be reported for tilted ellipses.
Area is found by multiplying pi, a, and b. Circumference has no simple exact elementary formula. This page uses a trusted Ramanujan approximation. It is accurate for normal design, classroom, and checking work.
Point Testing and Exports
Use the optional point test to check a coordinate. The calculator substitutes that point into the rotated ellipse equation. A value below one means the point is inside. A value near one means it is on the curve. A value above one means it is outside.
This calculator is useful for geometry homework, drafting checks, orbital sketches, and analytic geometry notes. It also helps when an ellipse is not aligned with the x-axis. The CSV export stores the numeric summary. The PDF export gives a neat report for printing or sharing.
Always enter eccentricity between zero and one. The two vertices must not be identical. Use consistent units for all coordinates. If coordinates are in meters, then axes and focus distances are also in meters. Area will use square meters. Review the equation after calculating, especially for rotated shapes. Small rounding differences can appear. Increase decimals when comparing exact textbook answers or CAD based coordinate checks during final verification steps and reviews.