Calculator Inputs
Example Data Table
| Center | Semi-major | Semi-minor | Direction | Rotation | Equation Type |
|---|---|---|---|---|---|
| (0, 0) | 5 | 3 | Horizontal | 0° | x² / 25 + y² / 9 = 1 |
| (2, -1) | 6 | 4 | Vertical | 0° | (x - 2)² / 16 + (y + 1)² / 36 = 1 |
| (1, 3) | 7 | 2 | Horizontal | 30° | Rotated ellipse with xy term |
Formula Used
Standard center form: (x - h)² / a² + (y - k)² / b² = 1
Rotated coordinates: u = cosθ(x - h) + sinθ(y - k)
Perpendicular coordinate: v = -sinθ(x - h) + cosθ(y - k)
Rotated ellipse: u² / a² + v² / b² = 1
General conic form: Ax² + Bxy + Cy² + Dx + Ey + F = 0
Linear eccentricity: c = √(a² - b²)
Eccentricity: e = c / a
Area: πab
Approximate circumference: π[3(a + b) - √((3a + b)(a + 3b))]
How to Use This Calculator
- Enter the center coordinates h and k.
- Enter the semi-major axis and semi-minor axis.
- Select whether the main axis starts horizontal or vertical.
- Add a rotation angle when the ellipse is tilted.
- Enter a test point if you want location checking.
- Select decimal places for cleaner output.
- Press the calculate button.
- Review the standard equation, general form, features, and graph.
- Use CSV or PDF export for records and sharing.
Understanding an Ellipse Equation
An Ellipse Equation
An ellipse is a stretched circle. It has a center, two axes, two foci, and four main end points. The longer axis is called the major axis. The shorter axis is called the minor axis. When the center is not at the origin, the equation must include h and k.
Why the Standard Form Matters
The standard form shows the shape quickly. It tells you the center, axis lengths, and orientation. For a horizontal ellipse, the larger denominator sits under the x part. For a vertical ellipse, it sits under the y part. A rotated ellipse needs a transformed coordinate system. This calculator handles that by rotating the x and y differences around the center.
Features You Can Read
The vertices sit at the ends of the major axis. The co-vertices sit at the ends of the minor axis. The foci sit inside the ellipse. They control many geometric and optical properties. The eccentricity shows how stretched the ellipse is. A value near zero looks almost circular. A value close to one looks long and narrow.
Using General Form
Many textbooks also use the general conic form. It looks like Ax² + Bxy + Cy² + Dx + Ey + F = 0. This form is helpful when comparing conics or using algebra software. A rotated ellipse usually has an xy term. If there is no rotation, that term becomes zero.
Practical Value
Ellipse equations appear in architecture, physics, astronomy, graphics, and design. They describe orbits, arches, tanks, lenses, paths, and layouts. This tool gives both exact structure and numeric features. It also plots the curve, so mistakes are easier to spot.
Good Input Habits
Use positive axis lengths. Keep the semi-major value greater than or equal to the semi-minor value. Choose the axis direction before adding rotation. Enter a test point when you want to check whether a coordinate lies inside, outside, or on the ellipse. Review the expanded form if your homework asks for a conic equation.
What Results Mean
A clean equation should match the plotted curve. The feature table should also agree with the selected axis direction. Use CSV for records. Use PDF for sharing.
FAQs
1. What does this ellipse calculator find?
It finds the standard equation, rotated equation, general conic form, vertices, co-vertices, foci, eccentricity, area, circumference, and point location status.
2. Can it handle a rotated ellipse?
Yes. Enter the rotation angle in degrees. The calculator transforms coordinates and produces the expanded conic equation with an xy term when needed.
3. Why must a be greater than or equal to b?
The value a represents the semi-major axis. It should be the longest semi-axis. The value b represents the shorter semi-minor axis.
4. What is the general conic form?
The general conic form is Ax² + Bxy + Cy² + Dx + Ey + F = 0. It is useful for algebraic comparison and graphing tools.
5. What are ellipse foci?
Foci are two fixed interior points. For any point on the ellipse, the sum of distances to both foci remains constant.
6. How is the test point checked?
The calculator substitutes the point into the ellipse equation. A value below one is inside. A value above one is outside.
7. Can I download the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a readable report of the calculated values.
8. Does the graph show the same ellipse?
Yes. The graph uses the same center, axes, and rotation angle. It also marks vertices, co-vertices, foci, and test point.