Example Data Table
| Vertex 1 |
Vertex 2 |
Eccentricity |
Center |
Semi-Major Axis |
Semi-Minor Axis |
Focus Distance |
| (-6, 2) |
(8, 2) |
0.60 |
(1, 2) |
7 |
5.6 |
4.2 |
| (0, 0) |
(10, 0) |
0.80 |
(5, 0) |
5 |
3 |
4 |
| (2, -3) |
(2, 9) |
0.50 |
(2, 3) |
6 |
5.196 |
3 |
How to Use This Calculator
- Enter the x and y coordinates of the first vertex.
- Enter the x and y coordinates of the second vertex.
- Enter eccentricity between 0 and less than 1.
- Add a unit label if your values use meters, feet, inches, or another unit.
- Choose the number of decimal places for the final answer.
- Press the calculate button to show the result above the form.
- Use the CSV or PDF button to export the result.
Ellipse Calculator Guide
What This Tool Finds
This calculator solves an ellipse from two vertices and eccentricity.
The two vertices define the major axis.
Their midpoint gives the center.
Their distance gives the full major axis length.
The eccentricity controls how stretched the ellipse is.
A value near zero gives a round shape.
A value close to one gives a narrow ellipse.
Why Vertices Matter
Vertices are the endpoints of the longest axis.
They show the direction of the ellipse.
They also show its size along that direction.
When the vertices are horizontal, the equation is simple.
When they are tilted, the equation becomes rotated.
This calculator handles both cases.
It uses vector direction to build accurate results.
Understanding Eccentricity
Eccentricity measures the distance from the center to each focus.
It is written as e.
For an ellipse, e must be less than one.
The focal distance is found by multiplying a by e.
After that, the minor axis is calculated from the relationship between a, b, and e.
This keeps the geometry consistent.
Advanced Output
The output includes center, axes, foci, co-vertices, area, and circumference.
It also gives latus rectum length.
Directrix distance is shown when eccentricity is not zero.
The calculator also builds the rotated standard equation.
It then converts that form into a general equation.
This is helpful for analytic geometry work.
Practical Uses
Students can use this tool to check homework.
Teachers can prepare examples for lessons.
Engineers can inspect elliptical layouts.
Designers can test proportions before drawing.
The export options make the results easy to save.
Use consistent units for every coordinate.
Always check that both vertices are different.
Also make sure eccentricity is valid.
FAQs
1. What inputs are needed?
You need two vertices and the eccentricity. The vertices must be different points. The eccentricity must be at least zero and less than one.
2. What do the vertices represent?
The vertices are endpoints of the major axis. They define the ellipse direction, center, and full major axis length.
3. Can the ellipse be rotated?
Yes. If the two vertices do not form a horizontal or vertical line, the ellipse is rotated. The calculator still builds its equation.
4. What is eccentricity?
Eccentricity shows how stretched the ellipse is. A value near zero is rounder. A value near one is more elongated.
5. Why must eccentricity be less than one?
An ellipse has eccentricity below one. If eccentricity equals one, the conic becomes parabolic. Greater values describe hyperbolas.
6. What is the semi-minor axis?
The semi-minor axis is half of the shortest width of the ellipse. It is calculated from the semi-major axis and eccentricity.
7. What is the latus rectum?
The latus rectum is a chord through a focus. It is perpendicular to the major axis. Its length equals two b squared divided by a.
8. Can I export the results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable summary of the calculated ellipse values.