Advanced Ellipse Input Form
Example Data Table
Use these sample values to test the calculator and compare graph shapes.
| Example | h | k | A | B | Rotation | Expected Focus Style |
|---|---|---|---|---|---|---|
| Wide standard ellipse | 0 | 0 | 6 | 3 | 0° | Foci on horizontal axis |
| Tall ellipse | 2 | -1 | 2 | 5 | 0° | Foci on vertical axis |
| Rotated ellipse | -1 | 2 | 5 | 2 | 35° | Foci follow rotated major axis |
| Near circle | 1 | 1 | 4 | 3.8 | 15° | Small focal distance |
Formula Used
The standard centered ellipse uses ((x - h)^2 / A^2) + ((y - k)^2 / B^2) = 1. A rotated ellipse first converts each point into local coordinates. The calculator uses x' = (x - h)cosθ + (y - k)sinθ and y' = -(x - h)sinθ + (y - k)cosθ.
Area is πAB. Focal distance is c = √(major² - minor²). Eccentricity is e = c / major. The circumference is estimated with Ramanujan's formula: π[3(A+B)-√((3A+B)(A+3B))].
Graph points come from the parametric form x = h + A cos(t)cosθ - B sin(t)sinθ and y = k + A cos(t)sinθ + B sin(t)cosθ.
How to Use This Calculator
- Enter the center coordinates h and k.
- Enter horizontal semi-axis A and vertical semi-axis B.
- Add a rotation angle when the ellipse is tilted.
- Choose the number of plotted points for smoother exports.
- Press the graph button to view results above the form.
- Use CSV for spreadsheet work or PDF for reporting.
Understanding Ellipse Graphing
What the graph shows
An ellipse is a closed curve built around a center. It has two main directions. One direction uses the longer semi-axis. The other direction uses the shorter semi-axis. This calculator plots both directions and shows the curve with useful reference points. The center, foci, vertices, co-vertices, and bounding limits help you read the graph faster.
Why rotation matters
Many textbook examples use ellipses aligned with the x-axis or y-axis. Real coordinate problems may be tilted. Rotation changes the visible position of the vertices and foci. The shape stays the same, but the coordinate locations change. This tool applies the rotation angle before drawing each point, so the graph matches the entered angle.
Using the results
The area result is exact when the axes are correct. The circumference is an approximation because an ellipse has no simple elementary perimeter formula. The eccentricity value explains how stretched the ellipse is. A value near zero means the curve is close to a circle. A value closer to one means the shape is long and narrow.
Study and reporting benefits
Graphing an ellipse by hand can be slow. You must calculate points, decide scale, place foci, and verify the equation. This calculator combines those steps in one page. The CSV file stores coordinates for tables and spreadsheets. The PDF file keeps a clean graph and core measurements for assignments, notes, or project records.
FAQs
What is an ellipse?
An ellipse is a closed oval curve. Every point has a constant total distance from two fixed points called foci.
What are A and B in this calculator?
A is the horizontal semi-axis length. B is the vertical semi-axis length before rotation is applied to the graph.
Can this calculator graph a rotated ellipse?
Yes. Enter the rotation angle in degrees. The graph, vertices, foci, and plotted points will rotate around the center.
Why is circumference approximate?
An ellipse does not have a simple exact perimeter formula using elementary functions. This tool uses Ramanujan's strong approximation.
What is eccentricity?
Eccentricity measures how stretched the ellipse is. A value near zero is rounder. A larger value is more elongated.
What happens when A equals B?
The ellipse becomes a circle. The foci meet at the center, and the eccentricity becomes zero.
What does the CSV download include?
The CSV includes summary measurements and plotted coordinate points. You can open it in spreadsheet software for further work.
Is the graph suitable for reports?
Yes. The PDF export includes the main values and the canvas graph, making it useful for notes and assignments.