Ellipse Equation Calculator

Find the standard form equation of an ellipse and visualize it interactively.

Ellipse Parameters

Center X (h):

Center Y (k):

Semi-major Axis (a):

Semi-minor Axis (b):

Orientation:

Equation:

Interactive Ellipse Visualization

Understanding Ellipse Equations

An ellipse is a closed curve where the sum of the distances from any point on the curve to two fixed points (foci) is constant. The standard equation of an ellipse centered at (h, k) is given by:

$${\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1}$$ (for horizontal major axis)
$${\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1}$$ (for vertical major axis)

Here, (h, k) is the center of the ellipse, 'a' is the semi-major axis (half of the longest diameter), and 'b' is the semi-minor axis (half of the shortest diameter). The orientation (horizontal or vertical) depends on whether the major axis is along the x or y direction. This calculator helps you find this equation by inputting the center and the lengths of the semi-major and semi-minor axes. Ellipses are fundamental in various fields like astronomy (planetary orbits) and optics (reflective surfaces).