Advanced Ellipse Calculator
Example Data Table
| Center | a | b | Direction | Equation | Area | Eccentricity |
|---|---|---|---|---|---|---|
| (0, 0) | 5 | 3 | Horizontal | x²/25 + y²/9 = 1 | 47.1239 | 0.8 |
| (2, -1) | 6 | 4 | Vertical | (x - 2)²/16 + (y + 1)²/36 = 1 | 75.3982 | 0.745356 |
| (-3, 4) | 8 | 5 | Horizontal | (x + 3)²/64 + (y - 4)²/25 = 1 | 125.6637 | 0.780625 |
Formula Used
The calculator uses the standard axis-aligned ellipse equation. The center is written as (h, k). The semi-major axis is a. The semi-minor axis is b.
The focal distance is calculated by c = √(a² - b²). Eccentricity is e = c / a. Area is πab. The latus rectum length is 2b² / a.
For general equations, the calculator completes the square. It converts Ax² + Cy² + Dx + Ey + F = 0 into standard ellipse form when the coefficients describe a valid axis-aligned ellipse.
How to Use This Calculator
- Select an input method from the dropdown.
- Enter the center and axis values, focal values, or general equation coefficients.
- Choose whether the major axis is horizontal or vertical when needed.
- Press the calculate button.
- Review the equation, foci, vertices, eccentricity, area, and chart points.
- Use the CSV or PDF button to save the result.
Understanding the Equation of an Ellipse
What the Calculator Finds
An ellipse is a smooth closed curve around two focal points. This calculator helps you build its equation from common geometry inputs. You can enter a center, semi-axes, focal distance, or general coefficients. The tool then returns the standard equation, general equation, vertices, co-vertices, foci, area, and eccentricity.
Why the Center Matters
The center controls the translation of the ellipse. When the center is not the origin, the equation uses x minus h and y minus k. These shifts move the curve left, right, up, or down. They do not change the shape unless the axes also change.
Major and Minor Axes
The major axis is the longest diameter. Its half length is called a. The minor axis is the shorter diameter. Its half length is called b. When the major axis is horizontal, a belongs under the x expression. When it is vertical, a belongs under the y expression.
Foci and Shape
The foci show how stretched the ellipse is. The focal distance c comes from a squared minus b squared. Eccentricity equals c divided by a. A value near zero means the ellipse looks more circular. A value closer to one means it is more stretched.
Practical Use
Ellipse equations appear in analytic geometry, astronomy, design, optics, and engineering. The chart points help with graphing. The exports help teachers, students, and content creators keep a clean record. Always check that the semi-major value is not smaller than the semi-minor value.
FAQs
1. What is the standard equation of an ellipse?
The standard equation is based on the center and axis lengths. A horizontal ellipse uses (x - h)²/a² + (y - k)²/b² = 1. A vertical ellipse switches the larger denominator under the y expression.
2. What are h and k in an ellipse?
The values h and k form the center point. The center is written as (h, k). They shift the ellipse away from the origin without changing the basic shape.
3. What is the semi-major axis?
The semi-major axis is half of the longest width of the ellipse. It is usually named a. It controls the longest direction and helps calculate foci, eccentricity, and area.
4. What is the semi-minor axis?
The semi-minor axis is half of the shortest width of the ellipse. It is usually named b. It must be positive and should not be greater than the semi-major axis.
5. How are foci calculated?
The focal distance is c = √(a² - b²). For a horizontal ellipse, foci are left and right of the center. For a vertical ellipse, foci are above and below the center.
6. What does eccentricity mean?
Eccentricity measures how stretched an ellipse is. It equals c divided by a. A smaller value looks more circular. A larger value looks longer and thinner.
7. Can this calculator use general form?
Yes. It can process axis-aligned equations in the form Ax² + Cy² + Dx + Ey + F = 0. It completes the square and converts valid inputs into standard form.
8. Why is my input rejected?
The input may not describe a valid ellipse. Common issues include zero coefficients, opposite signs for A and C, negative denominators, or a focal distance greater than the semi-major axis.