Andromeda
Note

Conic Sections: Ellipse

Definition

An ellipse is the collection of all points PP in a plane, the sum of whose distances from two fixed points F1F_1 and F2F_2 (the foci) is a constant. Mathematically, it is defined by the locus of points satisfying d(F1,P)+d(F2,P)=2ad(F_1, P) + d(F_2, P) = 2a, where 2a2a is the length of the major axis.

Why It Matters

Perfect circles are rare in nature; ellipses are the reality of stable interaction. The ellipse matters because it is the geometry of “shared attraction”—it is how planets orbit stars and how sound travels in “whispering galleries.” By understanding that stability comes from the tension between two centers (foci) rather than one, we can design more efficient trajectories for spacecraft and more effective medical tools, aligning our engineering with the “bounded stability” of the physical world.

Core Concepts

  • Fundamental geometric definition (the string property)

    • Sum of distances from any point on the ellipse to the two foci is constant (= 2a, the major axis length).
    • How to read: “The distance between F one and P plus the distance between F two and P equals two a.”
    • Meaning: String property—sum of distances to the two foci is constant and equals the major-axis length 2a2a.

  • Standard equation — Horizontal major axis (major axis left-right) (x−h)2a2+(y−k)2b2=1(a>b)\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \quad (a > b)

    • How to read: “The quantity x minus h squared over a squared, plus the quantity y minus k squared over b squared, equals one.”
    • Meaning: a is semi-major (half the long axis), b is semi-minor. Foci lie on the major axis at distance c from center, where c² = a² − b². Vertices at (h ± a, k).

  • Standard equation — Vertical major axis (major axis up-down) (x−h)2b2+(y−k)2a2=1(a>b)\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \quad (a > b)

    • How to read: “The quantity x minus h squared divided by b squared, plus the quantity y minus k squared divided by a squared, equals one.”
    • Meaning: Vertical major axis—aa under yy (larger denominator). Foci at (h,k±c)(h, k \pm c).

  • Key derived quantities (read the relationships)

    • c² = a² − b² (or b² = a² − c²)
      • How to read: “The value c squared equals a squared minus b squared.”
      • Meaning: Linear eccentricity—distance from center to each focus.
    • Eccentricity e = c / a (0 ≤ e < 1)
      • How to read: “The eccentricity e equals c divided by a.”
      • Meaning: Shape measure—e=0e = 0 is a circle; ee near 1 is very flat.
    • These three numbers (a, b, c) completely determine the shape and location once the center and orientation are known.

  • Decision rule for which form: Look at the larger denominator under which variable. If under x, major axis is horizontal. If under y, major axis is vertical. The larger number is always a².

Connected Concepts

Connected notes