What Are the Foci of an Ellipse?
The foci (plural of focus) are two fixed points that define an ellipse geometrically. They lie along the major axis, positioned symmetrically about the ellipse's center. The defining property of an ellipse is that for any point P on the curve, the sum of distances from P to each focus remains constant and equals 2a, where a is the semi-major axis.
Physically, this property explains why ellipses appear in nature: planetary orbits, satellite trajectories, and acoustic reflectors all exploit the constant-distance property. If you stretch a string between two fixed pins and trace around them with a pencil, you draw an ellipse whose foci are the pin positions.
The distance of each focus from the center depends on the relative sizes of the major and minor axes. A circle (where both axes are equal) has foci that coincide at the center. As an ellipse becomes more elongated, the foci move farther apart.
Calculating Focal Distance and Coordinates
The focal distance F measures how far each focus sits from the ellipse's center along the major axis. Once you have this distance, you can determine the exact coordinates of both foci relative to the center point.
F = √(a² − b²)
Focus₁ = (c₁ − F, c₂) [horizontal ellipse]
Focus₂ = (c₁ + F, c₂) [horizontal ellipse]
Focus₁ = (c₁, c₂ − F) [vertical ellipse]
Focus₂ = (c₁, c₂ + F) [vertical ellipse]
F— Focal distance from center to each focusa— Semi-major axis (longer radius)b— Semi-minor axis (shorter radius)c₁— x-coordinate of the ellipse centerc₂— y-coordinate of the ellipse center
Understanding Orientation and Eccentricity
An ellipse's orientation—whether it stretches horizontally or vertically—determines which axis the foci lie upon. If a (semi-major) is greater than b (semi-minor), the major axis runs horizontally, placing both foci on a horizontal line through the center. If b exceeds a, the major axis is vertical and the foci align vertically.
Eccentricity (e) quantifies how "stretched" the ellipse is and directly relates to focal position:
- Eccentricity = 0: Circle (foci coincide)
- Eccentricity near 1: Highly elongated ellipse (foci far from center)
- Eccentricity formula:
e = √(max(a,b)² − min(a,b)²) / max(a,b)
This metric is crucial in astronomy: Earth's orbital eccentricity is ~0.017 (nearly circular), while Pluto's is ~0.249 (significantly elliptical).
Practical Ellipse Construction Using Foci
Once you know the focal coordinates, you can manually construct an ellipse using the two-pin method. Place pins at both calculated focal points, loop a string of length 2a around them, and draw with tension maintained—the resulting curve is your ellipse.
Example: For an ellipse with semi-major axis a = 5 cm and semi-minor axis b = 3 cm, centered at (0, 0):
- Focal distance: F = √(25 − 9) = √16 = 4 cm
- Focus₁: (−4, 0)
- Focus₂: (4, 0)
- String length: 2 × 5 = 10 cm
This method, rooted in Renaissance geometry, is still used in landscape design and optical instrument fabrication.
Common Mistakes and Practical Considerations
Accurate focal calculation requires attention to several details:
- Confusing axes orientation — Always verify which semi-axis is larger before determining focal position. If <em>b</em> > <em>a</em>, the foci lie on the vertical axis, not horizontal. Swapping these inverts your focal coordinates.
- Forgetting to apply center offset — The formula √(a² − b²) gives focal distance from center, but the actual focus coordinates require adding/subtracting this from the center point (c₁, c₂). Omitting this step places foci at the origin regardless of where the ellipse actually sits.
- Misidentifying major and minor axes — The semi-major axis is always the longer one. Input errors here compound into incorrect focal distance. Double-check axis lengths before calculation, especially when axes values are close.
- Assuming circular symmetry — Only a perfect circle has foci at identical positions (the center). Even a slightly flattened ellipse has two distinct foci. Never assume symmetry without calculating.