Enter Ellipse Values
Ellipse Graph
Example Data Table
| Center | Semi-major | Semi-minor | Direction | Vertices | Foci |
|---|---|---|---|---|---|
| (0, 0) | 6 | 4 | Horizontal | (6, 0), (-6, 0) | (4.472, 0), (-4.472, 0) |
| (2, 3) | 5 | 3 | Vertical | (2, 8), (2, -2) | (2, 7), (2, -1) |
| (-1, 2) | 8 | 5 | Horizontal | (7, 2), (-9, 2) | (5.245, 2), (-7.245, 2) |
Formula Used
The standard ellipse formula depends on the major axis direction.
Horizontal ellipse
((x - h)² / a²) + ((y - k)² / b²) = 1
Vertices are (h ± a, k). Foci are (h ± c, k).
Vertical ellipse
((x - h)² / b²) + ((y - k)² / a²) = 1
Vertices are (h, k ± a). Foci are (h, k ± c).
Focal distance
c = √(a² - b²)
Eccentricity and area
e = c / a
Area = πab
How To Use This Calculator
Enter the center point first. The center is written as (h, k). Use zero when the ellipse is centered at the origin. Then enter the two radius values. The larger value becomes the semi-major axis. The smaller value becomes the semi-minor axis.
Select the major axis direction. You may choose horizontal, vertical, or auto detect. Auto detect uses the larger radius as the major axis. Press the calculate button. The result appears above the form. You will see vertices, co-vertices, foci, focal distance, eccentricity, area, and the standard equation.
Use the graph to check the shape visually. The plotted ellipse also marks the vertices and foci. Export the result as CSV for spreadsheet use. Export the PDF for reports, homework, study notes, or quick records.
Understanding Ellipse Vertices And Foci
What An Ellipse Represents
An ellipse is a closed curve around two fixed points. These fixed points are called foci. Every point on the ellipse has a special property. The total distance from that point to both foci stays constant. This makes the ellipse useful in geometry, physics, astronomy, and design.
Role Of The Center
The center is the middle point of the ellipse. It is written as (h, k). All main points are measured from this center. When the center changes, the full ellipse shifts on the coordinate plane. The shape stays the same unless the axis values change.
Major And Minor Axes
The major axis is the longest width of the ellipse. Half of that length is the semi-major axis. The minor axis is the shorter width. Half of that length is the semi-minor axis. The vertices sit at the ends of the major axis. The co-vertices sit at the ends of the minor axis.
Why Foci Matter
The foci show how stretched the ellipse is. When the foci are close to the center, the ellipse looks more circular. When they move farther away, the ellipse becomes longer and narrower. This relationship is measured by eccentricity.
Using The Calculator
This calculator saves time by finding all important ellipse points at once. It also creates the equation and graph. Students can use it to check homework. Teachers can use it for examples. Designers can use it to confirm coordinate layouts. It is also helpful for quick graphing work.
Reading The Output
The result gives the vertices, co-vertices, foci, and standard equation. It also gives area and eccentricity. These values help describe both size and shape. Use the download buttons to save the result. The CSV file is best for data use. The PDF file is best for sharing.
Frequently Asked Questions
1. What are ellipse vertices?
Vertices are the two end points of the major axis. They show the longest direction of the ellipse and help define its full length.
2. What are ellipse foci?
Foci are two fixed points inside the ellipse. The sum of distances from any ellipse point to both foci stays constant.
3. How is focal distance calculated?
Focal distance is calculated with c = √(a² - b²). Here, a is the semi-major axis and b is the semi-minor axis.
4. What is the center of an ellipse?
The center is the midpoint of the ellipse. It is written as (h, k) and controls the ellipse position on the graph.
5. What is eccentricity?
Eccentricity measures how stretched an ellipse is. A value near zero looks circular. A higher value means a longer ellipse.
6. Can this calculator handle vertical ellipses?
Yes. Select vertical as the major axis direction. The calculator then places vertices and foci above and below the center.
7. What happens when both radii are equal?
The shape becomes a circle. The focal distance becomes zero, and both foci meet at the center point.
8. Can I download the results?
Yes. You can download the result as a CSV file or a PDF file using the buttons shown after calculation.