Foci of Ellipse Calculator - Linear Eccentricity and Focus Points
Use this foci of ellipse calculator to find c, e, focal distance 2c, and the absolute (x, y) coordinates of both foci from a, b, the center, and rotation.
Foci of Ellipse Calculator
Results
What Is Foci of Ellipse Calculator?
The foci of ellipse are the two fixed points on the major axis whose defining property is that the sum of the distances from any point on the curve to those two foci is constant and equal to 2a. A foci of ellipse calculator turns the standard-form ellipse equation x²/a² + y²/b² = 1 into the distance c = sqrt(a² − b²), the eccentricity e = c / a, and the actual (x, y) coordinates of the two foci.
- • Geometry homework: Solve exercises that give you a standard-form ellipse and ask for the coordinates of the two foci, especially when the ellipse is rotated or off-center.
- • Orbit and trajectory analysis: Confirm the perihelion geometry of an elliptical orbit, where the focal distance drives perihelion and aphelion math.
- • Acoustics and optics setups: Lay out an elliptical mirror, whispering gallery, or focal-point lighting rig by computing where the sound or light will concentrate.
- • CAD and drafting checks: Validate that a drawn ellipse, especially a tilted one, has the right focus positions before sending the part to manufacturing.
Every non-circular ellipse has two distinct foci, sitting on the longest axis, symmetric about the center, exactly c units from that center. The single number c is what this calculator is built to produce. Once you know c, you immediately know e, the focal distance 2c, and the (x, y) coordinates of F1 and F2.
If you also need the area of the same ellipse from the same semi-axes, ellipse area calculator gives you area, perimeter, and eccentricity side by side without re-entering inputs.
How Foci of Ellipse Calculator Works
The foci of an ellipse follow directly from the standard-form equation, and the calculator implements that closed-form expression in three short steps.
- a: Semi-major axis (half of the longest diameter). The larger of the two axis values.
- b: Semi-minor axis (half of the shortest diameter). The smaller of the two axis values.
- x_c, y_c: Coordinates of the ellipse center in the plane. Both share the selected linear unit.
- θ (theta): Rotation of the major axis, counter-clockwise from the +x axis, in degrees (0 to 360).
- c: Linear eccentricity — the distance from the center to each focus.
- e: Dimensionless eccentricity, equal to c / a, ranging from 0 (circle) to just below 1 (very flat ellipse).
When the major axis is rotated, the same c is applied as a 2D offset in the direction (cos θ, sin θ) before the center is added, so the foci stay on the major axis even when the ellipse is tilted.
The calculator also accepts full-diameter inputs. In Full diameters mode the values are divided by 2 internally, so a, b, and c stay in the same length unit and the (x, y) coordinates stay consistent with the engineering drawing.
Standard ellipse centered at the origin
a = 5 cm, b = 3 cm, center (0, 0), θ = 0°
c = sqrt(5² − 3²) = sqrt(25 − 9) = sqrt(16) = 4 cm. e = 4 / 5 = 0.8. F1 = (−4, 0), F2 = (4, 0).
c = 4 cm, e = 0.8, focal distance 2c = 8 cm, foci at (−4, 0) and (4, 0).
The textbook 3-4-5 right-triangle ellipse: foci are 8 cm apart along the x-axis, and eccentricity 0.8 shows the ellipse is noticeably elongated.
According to MathWorld (Wolfram Research), the linear eccentricity of an ellipse is c = sqrt(a^2 - b^2) and the dimensionless eccentricity is e = c / a.
According to Wikipedia, Ellipse, the two foci of an ellipse lie on the major axis at distance c = sqrt(a^2 - b^2) from the center.
When the ellipse is given in general form and you need the (x_c, y_c) values used in the focus formulas, center of ellipse calculator extracts them so the rest of the calculation lines up.
Key Concepts Explained
Four definitions keep the calculator honest and the output readable. Each ties a result row back to a property of the ellipse that you can draw or measure.
Linear eccentricity (c)
The straight-line distance from the center of the ellipse to either focus. c is at most a, equals a only in the limit of a perfectly flat ellipse, and is 0 for a circle.
Eccentricity (e)
The ratio c / a, scaled to a unitless number between 0 and 1. It is the standard way to compare the shape of two ellipses without worrying about their absolute size.
Focal distance (2c)
The total distance between the two foci, equal to twice the linear eccentricity. Useful when you need the maximum separation between the two focal points.
Reflective property
A light ray or sound wave from one focus reflects off the ellipse and passes through the other focus. This is the geometric reason satellites and whispering galleries are designed around ellipses.
c and e can be reported side by side because they share the same input unit or are dimensionless. The reflective property is the reason an ellipse-shaped room amplifies a whisper.
When the major axis is rotated, none of these definitions change; only the (x, y) coordinates of the foci move. The eccentricity, c, and 2c values stay identical, which is why the result panel separates 'shape' outputs (c, e, 2c) from 'position' outputs.
An ellipse shares the reflective property with a parabola, and parabola calculator computes the single focus and directrix that the parabolic mirror is built around.
How to Use This Calculator
Five short steps take you from a raw pair of axis numbers to a usable set of focus coordinates, whether the ellipse is axis-aligned or tilted.
- 1 Enter the two axes: Put the longer half-diameter in Axis A and the shorter in Axis B. If you only measured full diameters, switch the Axis Input Mode to Full diameters and enter those values directly.
- 2 Pick the linear unit: Select the unit that matches your drawing. Every output that reports a length — c, 2c, and the focus coordinates — will appear in the same unit.
- 3 Set the ellipse center: Type the (x_c, y_c) coordinates of the center. Leaving both at 0 places the ellipse at the origin; non-zero values move both foci by the same offset.
- 4 Set the major-axis rotation: Enter the angle θ between the major axis and the +x direction, measured counter-clockwise. Use 0° for axis-aligned; use 30°, 45°, or any other value for a tilted ellipse.
- 5 Read the result panel: The result panel reports c, 2c, e, and the (x, y) coordinates of both foci. Each row is recomputed live, so you can adjust any input and see the focal geometry update in real time.
A physics class gives you x²/25 + y²/9 = 1 and asks where the foci are. Enter Axis A = 5, Axis B = 3, Mode = Semi-axes, Unit = cm, Center (0, 0), Rotation = 0. The result returns c = 4 cm, e = 0.8, 2c = 8 cm, F1 = (−4, 0) cm, F2 = (4, 0) cm.
When a equals b, the result is a circle, and circle equation calculator writes out the matching x² + y² = r² form for the same shape in one click.
Benefits of Using This Calculator
Running the focal geometry through this calculator saves algebra time and catches sign errors in rotated ellipses.
- • Closed-form results in real time: Every keystroke updates c, e, 2c, and the focus coordinates. No scratch equation or hand recompute needed.
- • Handles rotated and off-center ellipses: Center coordinates plus rotation angle cover the full standard-form ellipse plus any rigid transform, so no separate rotated-ellipse formula is needed.
- • Unit-aware output: Switch the unit once and every linear output updates in lockstep, so cm, m, in, and ft results stay internally consistent.
- • Auto-corrects swapped axes: If you enter a larger value in Axis B by accident, the calculator swaps them internally and still returns the correct c, e, and foci.
- • Circle and degenerate-limit friendly: When a equals b, the calculator returns c = 0, e = 0, and two coincident foci at the center, the correct circle limit.
These benefits matter most when the ellipse is part of a larger problem: 2c feeds into orbital mechanics, the focus coordinates drop into a CAD sketch, and e lets you compare shapes on the same scale.
Once you know the foci and the ellipse, arc length calculator gives the arc length of any segment of the curve so you can finish a perimeter or design problem.
Factors That Affect Your Results
Shape, orientation, and center all push the focus positions around. Five factors cover the inputs that change the result.
Ratio of b to a
The smaller the ratio b/a, the larger c and the closer e sits to 1. A nearly circular ellipse (b/a near 1) puts both foci near the center; a flat ellipse (b/a near 0) puts them almost at the ends of the major axis.
Major-axis rotation (θ)
Rotating the ellipse does not change c, e, or 2c, but it rotates the line on which both foci sit. The two foci move on a circle of radius c around the center, parameterized by θ.
Center coordinates (x_c, y_c)
Moving the center rigidly shifts both focus coordinates by the same vector. Focal distance and eccentricity are unaffected; only the absolute (x, y) positions change.
Input mode (semi vs diameter)
Full-diameter inputs are divided by 2 internally. Mixing the two modes produces a wrong answer, so pick the mode that matches the numbers on the page or drawing.
Selected linear unit
All linear outputs are reported in the chosen unit. Switch units and every value changes, so always cross-check the displayed unit before copying numbers into a CAD or spreadsheet.
- • The closed-form c = sqrt(a² − b²) assumes the standard-form equation x²/a² + y²/b² = 1 with axes parallel to the coordinate axes before rotation. For a general conic Ax² + Bxy + Cy² + Dx + Ey + F = 0, complete the square or rotate to remove the cross term first.
- • Results round to 4–6 decimal places. For extremely thin ellipses (e above 0.9999) the displayed focus coordinates may differ from a high-precision solver in the last decimal.
According to Paul's Online Math Notes (Lamar University), Algebra - Ellipses, the standard form of an ellipse is (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1, where (h, k) is the center.
When the rotation angle comes in degrees from a drawing but your downstream CAD step expects radians, angle converter converts between the two units without changing the focus positions.
Frequently Asked Questions
Q: What are the foci of an ellipse?
A: The two fixed points on the major axis such that the sum of the distances from any point on the curve to those two points is constant. They sit on the longest axis, are symmetric about the center, and are at distance c = sqrt(a² − b²) from the center.
Q: How do you find the foci of an ellipse from the equation?
A: For x²/a² + y²/b² = 1, take a as the larger semi-axis and b as the smaller. The foci sit at (±c, 0) when the major axis is horizontal, where c = sqrt(a² − b²). If a is vertical, swap x and y in the final coordinates.
Q: What is the formula c = sqrt(a^2 - b^2) for an ellipse?
A: It is the closed-form expression for the linear eccentricity, the distance from the center to either focus. Plug a and b in, take the square root, and that is c. The dimensionless eccentricity is then e = c / a.
Q: Where are the foci of a rotated ellipse?
A: A rotated ellipse keeps the same c value, but the foci now lie on the tilted major axis. Take the unit vector (cos θ, sin θ) along the major axis and place the two foci at (x_c ∓ c·cos θ, y_c ∓ c·sin θ), where (x_c, y_c) is the center.
Q: What is linear eccentricity of an ellipse?
A: The actual length, in your unit of measurement, from the center of the ellipse to either focus. It is usually written as c and computed as sqrt(a² − b²), and is closely related to — but not the same as — the dimensionless eccentricity e = c / a.
Q: Can both foci of an ellipse be at the same point?
A: Yes. When a = b the ellipse is a circle, c = 0, and the two foci coincide at the center. As b approaches a, c approaches 0 and the two focus points pull in toward each other until they merge.