Ellipse Foci Calculator

Advanced Ellipse Foci Calculator

Formula Used

For a standard ellipse, the focal distance is found with c² = a² - b². Here, a is the semi-major axis. The value b is the semi-minor axis. The distance from the center to each focus is c.

For a horizontal ellipse, the foci are (h - c, k) and (h + c, k). For a vertical ellipse, the foci are (h, k - c) and (h, k + c). Eccentricity is calculated with e = c / a.

How to Use This Calculator

1. Select the input method that matches your known values.
2. Enter the center and semi-axis values, vertices, or equation coefficients.
3. Choose decimal precision for rounded output.
4. Press the calculate button to view foci and related ellipse values.
5. Use the CSV or PDF button to save the final result.

Example Data Table

Understanding Ellipse Foci

What the Foci Mean

The foci of an ellipse are two fixed points inside the curve. They define the shape. For every point on the ellipse, the sum of distances to both foci stays constant. This constant is equal to the full major axis length.

Why the Major Axis Matters

The major axis is the longest axis of the ellipse. Its half length is called the semi-major axis. The foci always lie on this axis. A horizontal major axis gives left and right foci. A vertical major axis gives upper and lower foci.

Finding the Center

Most ellipse problems start with the center. The center is written as h, k. It marks the middle of the ellipse. Once the center is known, the calculator offsets that point by the focal distance. That gives both focus coordinates.

Using Standard Form

In standard form, the larger denominator belongs to the semi-major axis. The smaller denominator belongs to the semi-minor axis. Their square roots give a and b. The calculator then applies c squared equals a squared minus b squared.

Using Vertex Data

Sometimes vertices are easier to use. The midpoint of the vertices gives the center. Half the distance between them gives the semi-major axis. The entered minor value finishes the ellipse model. This is useful for geometry, drawings, and layout checks.

Equation Conversion

A general equation can also be used when it has no xy term. The calculator completes the square in both variables. It then converts the equation into standard form. After that, the same focus formula is used.

Extra Results

This tool also reports vertices, co-vertices, eccentricity, directrices, and the distance between foci. These values help compare ellipse shapes. A small eccentricity means the ellipse is close to a circle. A larger eccentricity means the ellipse is more stretched.

Practical Uses

Ellipse foci are used in math, drafting, optics, architecture, and orbital models. Students can check homework steps. Designers can verify curve dimensions. The downloadable files make it easy to save work for reports, worksheets, or client notes.

FAQs

1. What are ellipse foci?

Ellipse foci are two fixed points inside an ellipse. The sum of distances from any ellipse point to both foci remains constant.

2. What formula finds the foci?

Use c² = a² - b². Then place c units from the center along the major axis.

3. What is the semi-major axis?

The semi-major axis is half of the longest width of the ellipse. The foci always lie on that axis.

4. Can a circle have two different foci?

No. A circle is a special ellipse where a equals b. Both foci meet at the center.

5. How do I know the ellipse direction?

Compare the semi-axis values. The larger value shows the major axis direction, unless you manually select the direction.

6. Does this handle rotated ellipses?

This version handles horizontal and vertical ellipses. It does not solve rotated ellipses with an xy term.

7. What is eccentricity?

Eccentricity measures how stretched the ellipse is. It equals c divided by a and stays between zero and one.

8. Why download CSV or PDF?

CSV is helpful for spreadsheets. PDF is useful for reports, printing, assignments, and sharing final results.