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Conic sections
Slice a double cone with a flat plane — out fall four curves: circle, ellipse, parabola, hyperbola.
Conic sections
Slice a cone four ways and you get these four curves.
Eccentricity
0 < e < 1
Equation
x²/a² + y²/b² = 1
Sum of distances to two foci is constant. Planet orbits trace ellipses.
One family, four shapes
Place two cones tip-to-tip. Slice them with a flat plane. The shape of the slice depends on the angle:
- Horizontal cut — circle.
- Slight tilt — ellipse.
- Parallel to a side — parabola.
- Steep enough to cut both cones — hyperbola.
Eccentricity
A single number, e, places every conic on a continuum. e = 0 is a circle, 0 < e < 1 is an ellipse, e = 1 is a parabola, e > 1 is a hyperbola.
Defining property — focus and directrix
Each conic is the set of points where the ratio of (distance to a fixed point) over (distance to a fixed line) equals e. That ratio is the eccentricity.
Where they show up
- Ellipses — planet orbits (Kepler), satellite paths, whispering galleries.
- Parabolas — projectile paths, satellite-dish reflectors, telescope mirrors.
- Hyperbolas — comet trajectories that escape the sun, Lorentz factor in relativity.