Ellipse Result
Results appear here after pressing the calculate button.
Equation
Calculator Inputs
Enter opposite major vertices and two foci. Use the same coordinate unit for every field.
Example Data Table
These examples show valid inputs and expected main outputs.
| Vertices | Foci | Center | a | c | b | Equation Type |
|---|---|---|---|---|---|---|
| (-5, 0), (5, 0) | (-3, 0), (3, 0) | (0, 0) | 5 | 3 | 4 | Horizontal |
| (0, -10), (0, 10) | (0, -6), (0, 6) | (0, 0) | 10 | 6 | 8 | Vertical |
| (-4, -3), (4, 3) | (-2.4, -1.8), (2.4, 1.8) | (0, 0) | 5 | 3 | 4 | Rotated |
Formula Used
Let the vertices be V1(x1, y1) and V2(x2, y2). Let the foci be F1(p1, q1) and F2(p2, q2).
- Center:
(h, k) = ((x1 + x2) / 2, (y1 + y2) / 2) - Semi-major axis:
a = distance(V1, V2) / 2 - Focal distance:
c = distance(F1, F2) / 2 - Semi-minor axis:
b = sqrt(a² - c²) - Eccentricity:
e = c / a - Area:
A = πab - Latus rectum length:
L = 2b² / a
For a rotated ellipse, define X = (x - h)cosθ + (y - k)sinθ and Y = -(x - h)sinθ + (y - k)cosθ. The equation becomes X²/a² + Y²/b² = 1.
How To Use This Calculator
- Enter the two opposite major vertices.
- Enter the two foci on the same major axis.
- Choose decimal places and a tolerance level.
- Add a test point if you want inside, outside, or on-curve classification.
- Press calculate to view the equation and measurements.
- Use CSV or PDF export for records and reports.
Complete Ellipse Analysis From Vertices And Foci
Purpose Of This Tool
The Ellipse Calculator Given Vertices And Foci helps you turn four coordinate points into a complete ellipse model. It is designed for geometry, drafting, graphing, and coordinate analysis. You can enter the two major vertices and the two foci. The tool then checks whether those points form a valid ellipse.
Core Geometry
An ellipse has a center, a major axis, a minor axis, and two focus points. The vertices define the longest width. Half of that width is the semi-major axis, called a. The foci define the focal distance, called c. When c is smaller than a, the calculator finds b by using the relation b squared equals a squared minus c squared.
Rotated Ellipse Support
This calculator also supports rotated ellipses. The vertices do not need to be horizontal or vertical. The angle comes from the line joining the vertices. After that, the equation is written in rotated coordinate form. This makes the result useful for graphs that are tilted on the coordinate plane.
Validation And Checks
The consistency checks are important. The midpoint of the vertices should match the midpoint of the foci. The foci should also sit on the major axis line. If the entered points fail these checks, the calculator warns you. It still reports useful measurements when possible, so you can identify the exact issue.
Advanced Outputs
Advanced outputs include eccentricity, area, estimated circumference, focal parameter, latus rectum, and directrix positions. You also get minor co-vertices. These values help with curve sketching, optics examples, orbital paths, and analytic geometry homework.
Point Testing
The point test option is useful when you want to classify a coordinate. Enter any point, and the calculator evaluates the ellipse expression. A value under one is inside. A value near one is on the curve. A value above one is outside.
Export And Accuracy
Use the CSV option for spreadsheets. Use the PDF option for reports, assignments, or client notes. The example table shows common inputs and expected outputs. Always use matching units for all coordinates. The calculator treats coordinates as plain numeric values, so the chosen unit label is only used in the final report.
For best accuracy, use exact coordinates when possible. Round only after finishing the final interpretation and exporting the completed result.
FAQs
1. What inputs are required?
You need two opposite major vertices and two foci. All points should use the same coordinate scale and unit.
2. Can this calculator handle a rotated ellipse?
Yes. The angle is calculated from the line joining the vertices. The final equation uses rotated coordinate variables.
3. What happens if the foci are not aligned?
The calculator shows a warning. The foci should lie on the same major axis formed by the vertices.
4. Why must c be smaller than a?
For a real ellipse, the focal distance must be less than the semi-major axis. Otherwise, the minor axis is not real.
5. What is eccentricity?
Eccentricity is c divided by a. It describes how stretched the ellipse is compared with a circle.
6. What does the point test do?
It substitutes a point into the ellipse equation. The value tells whether the point is inside, on, or outside the ellipse.
7. Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a readable report.
8. Does the unit label affect calculations?
No. The unit label only appears in results. The numeric calculation depends only on the coordinate values entered.