The Ellipse Foci Calculator computes the coordinates of an ellipse’s foci from its semi-axes and centre.
Report an issue
Spotted a wrong result, broken field, or typo? Tell us below and we’ll fix it fast.
About the Ellipse Foci Calculator
An ellipse has two special points called foci. Many design and geometry tasks rely on where those points sit. This Calculator computes the foci once you supply the center and the semi-axes, or the axis lengths. It also reports the distance between foci and the ellipse’s eccentricity.
We assume a standard, axis-aligned ellipse. That means its major and minor axes are parallel to the x- and y-axes. This covers most classroom and engineering cases where rotation is not involved. If you need rotated conics, see the caveats section before you proceed.
How to Use Ellipse Foci (Step by Step)
The process is simple. You provide the center, the semi-major axis a, and the semi-minor axis b. The Calculator returns the focus points and related values. Follow these basic steps to ensure correct inputs and a clean result.
- Decide if the major axis is horizontal or vertical.
- Enter the center coordinates (h, k).
- Enter a and b as positive numbers, with a ≥ b.
- Check units (meters, inches, etc.) for consistency.
- Review the output: foci coordinates, 2c distance, and eccentricity e.
That is it. The Calculator handles the square roots and signs. If a equals b, the ellipse is a circle and both foci collapse to the center. The tool flags that special case so your interpretation is correct.
Equations Used by the Ellipse Foci Calculator
These are the standard equations for an axis-aligned ellipse. We use a for the semi-major axis and b for the semi-minor axis, with a ≥ b. The center is at (h, k). Orientation decides which coordinate gets the “±” for the foci.
- Standard form: (x − h)² / a² + (y − k)² / b² = 1
- Focal distance: c = √(a² − b²)
- Foci (horizontal major axis): (h ± c, k)
- Foci (vertical major axis): (h, k ± c)
- Eccentricity: e = c / a, where 0 ≤ e ≤ 1
- Distance between foci: 2c
If you enter axis lengths instead of semi-axes, the tool divides by two to get a and b. It then applies the same formulas above. All results are rounded sensibly, but you can change precision if needed.
What You Need to Use the Ellipse Foci Calculator
Before you start, gather the core values that define your ellipse. You can provide them in one of two ways: semi-axes directly, or full axis lengths with orientation. Keep in mind that a must be the larger semi-axis.
- Center coordinates (h, k)
- Either semi-major axis a and semi-minor axis b (with a ≥ b)
- Or major axis length (2a) and minor axis length (2b)
- Orientation: horizontal or vertical major axis
- Units (optional label for your values)
All numeric inputs should be real, finite, and positive for the axes. If a is less than b, the Calculator will swap them and note the change. If a or b is zero or negative, it will prompt you to fix the input. Very large numbers are allowed, but rounding may affect the last few decimals.
Step-by-Step: Use the Ellipse Foci Calculator
Here’s a concise overview before we dive into the key points:
- Select the orientation of the major axis (horizontal or vertical).
- Enter the center coordinates h and k.
- Enter a and b (or enter 2a and 2b, and let the tool halve them).
- Confirm that a ≥ b; adjust if needed.
- Press Calculate to generate the foci and related values.
- Review the result: foci coordinates, 2c, and eccentricity e.
These points provide quick orientation—use them alongside the full explanations in this page.
Real-World Examples
Architectural dome layout. A skylight ring is modeled as a horizontally oriented ellipse centered at (0, 0). The semi-major axis is a = 5 m, and the semi-minor axis is b = 3 m. Compute c = √(a² − b²) = √(25 − 9) = √16 = 4 m. The foci are at (±4, 0), the distance between foci is 8 m, and e = 4/5 = 0.8. What this means: To place lighting that emulates whisper-focused acoustics, anchor at x = −4 m and x = +4 m along the centerline.
Satellite ground track planning. A coverage footprint is approximated by a vertical ellipse centered at (120, 45) in map grid units. The semi-major axis is a = 120 km, and the semi-minor axis is b = 110 km. Compute c = √(120² − 110²) = √(14,400 − 12,100) = √2,300 ≈ 47.96 km. The foci are at (120, 45 ± 47.96), the distance between foci is about 95.92 km, and e ≈ 0.3997. What this means: The footprint is only moderately elongated, so focus-driven variations in coverage will be mild along the north–south path.
Assumptions, Caveats & Edge Cases
This Calculator focuses on clean, axis-aligned ellipses in standard form. It assumes accurate inputs and consistent units. The equations are exact, but rounding appears in the display to keep numbers readable.
- Circle case: If a = b, then c = 0 and both foci equal the center.
- Orientation matters: The “±c” applies to x for horizontal, to y for vertical.
- Input correction: If you enter a < b, the tool swaps them and informs you.
- Non-physical values: Negative or zero axes are rejected with a clear message.
- Rotation not supported: Rotated ellipses need a different model (xy term).
For rotated ellipses or data derived from arbitrary conic fits, you need the general conic form and a rotation transform. That is beyond this tool. If your data seems off, revisit measurement steps or convert units first.
Units & Conversions
Units matter when you interpret the distance between foci or the coordinates of the foci. Keep inputs consistent. If your drawing is in inches and your survey in meters, convert before using the Calculator to avoid a mismatched result.
| Unit | To m | To cm | Typical use |
|---|---|---|---|
| m | 1 | 100 | Engineering drawings, site plans |
| cm | 0.01 | 1 | Product design, small models |
| mm | 0.001 | 0.1 | Precision parts, fabrication |
| in | 0.0254 | 2.54 | Furniture, mechanical layouts (Imperial) |
| ft | 0.3048 | 30.48 | Architecture, construction (Imperial) |
Use the “To m” and “To cm” columns to convert any one input or result. Multiply your value by the factor. Convert all related dimensions so the ellipse stays in one unit system.
Troubleshooting
If your outputs look strange, the issue is usually an input mismatch or orientation. Review the following quick checks to get back on track.
- Are a and b swapped? Ensure a ≥ b or let the tool swap them.
- Is the major axis set to the right orientation?
- Do all inputs use the same units?
- Are a and b positive and nonzero?
- Is the center typed correctly, including negative signs?
If a result still seems off, try a simple test ellipse with round numbers. If that works, the problem is likely in your original data, not the Calculator.
FAQ about Ellipse Foci Calculator
What are the foci of an ellipse?
They are two fixed points inside the ellipse such that the sum of distances to any point on the ellipse is constant.
How do I know if my major axis is horizontal or vertical?
If the longer semi-axis aligns with x, it is horizontal; if it aligns with y, it is vertical. Compare a and b and your layout.
What happens if I input a smaller a than b?
The Calculator will swap them so that a ≥ b, then compute the foci and note the corrected ordering in the result.
Can this tool handle rotated ellipses?
No. It handles axis-aligned ellipses only. For rotated cases, you need the general conic equation with a rotation term.
Key Terms in Ellipse Foci
Ellipse
A closed curve where the sum of distances from any point on the curve to two fixed points (foci) is constant.
Focus (plural: Foci)
A special interior point of an ellipse that helps define its shape and geometry.
Semi-major axis (a)
Half the length of the longest diameter of the ellipse; it sets the ellipse’s largest reach from the center.
Semi-minor axis (b)
Half the length of the shortest diameter of the ellipse; it sets the ellipse’s smallest reach from the center.
Eccentricity (e)
A measure of how stretched the ellipse is, defined as e = c / a, where c is the focal distance.
Focal distance (c)
The distance from the center to each focus, computed as c = √(a² − b²) for an axis-aligned ellipse.
Major axis
The longest axis of the ellipse, of length 2a, running through the center and both foci.
Minor axis
The shortest axis of the ellipse, of length 2b, perpendicular to the major axis at the center.
Sources & Further Reading
Here’s a concise overview before we dive into the key points:
- Wikipedia: Ellipse overview, properties, and equations
- Wolfram MathWorld: Ellipse definitions and identities
- Paul’s Online Notes: Parametric equations (context for conics)
- Khan Academy: Conic sections lessons and practice
- Wikipedia: Eccentricity of conic sections
These points provide quick orientation—use them alongside the full explanations in this page.