MATHEMATICS LAB · GEOMETRY

Study of the Ellipse

Change the center and semiaxes to explore the curve, foci, and directrices. Discover intersections and tangents with explained calculations.

The semiaxes must satisfy a ≥ b > 0. Axis-aligned ellipses are supported (not rotated). When a = b, the ellipse is a circle: the foci coincide and the directrices are undefined.

1. Define the ellipse

Center, semiaxes, or coefficients

Equation formMajor axis

Examples:

Display options

Controls and information

Grid Axes and foci Directrices

Drag O to move the ellipse; drag the handles a and b to change the semiaxes. When the tangent is enabled, drag T along the curve. Use +, − and “Fit” to zoom.

Ellipse and center O Foci F₁, F₂ Directrices Line ℓ and intersections Tangent and T

3. Additional options

Line and tangent are added to the graph

Intersection with a line ℓLine equation

Tangent at a point T on the ellipseParameter angle of T (degrees) Also show the normal at T

Disabled elements are not drawn. The parameter angle locates T using x = h + rₓ cos θ and y = k + rᵧ sin θ; it is not the polar angle measured from the center.

2. Interactive ellipse graph

Axes at a 1:1 scale

Ellipse, semiaxes and foci

4. Results

Updated values

5. Geometric construction

Meaning of foci and semiaxes

6. Explained calculations

Formulas and numeric substitutions

What you can explore in this version

Ellipse and eccentricity

Center, major and minor axes, vertices, foci, directrices and eccentricity, including translated ellipses.

Intersections

Two, one or no intersection points with a line; axis intersections and tangency cases.

Tangent and normal

Tangent and normal at point T, with an updated graph and explained calculations.

Nondegenerate real ellipses with axis-aligned principal axes are supported, including the general equation without an xy term. The perimeter is approximated using Ramanujan's formula. Rotated ellipses are not yet supported.