Conic sections are curves formed by the intersection of a plane with a double-napped cone. Depending on the angle of the intersecting plane, you get a circle, ellipse, parabola, or hyperbola. They are fundamental in geometry, physics (orbits, projectile motion), and engineering (reflectors, antennas).

Eccentricity measures how much a conic section deviates from being a perfect circle. Circle: e = 0 (perfectly round). Ellipse: 0 < e < 1 (mildly to highly elongated). Parabola: e = 1 (the boundary case). Hyperbola: e > 1 (open curve, two branches). Eccentricity determines the shape uniquely for each conic family.

Every conic section can be defined as the set of points where the ratio of distance to a focus to the distance to a directrix is constant and equals the eccentricity e. For a parabola (e=1), this means each point is equidistant from the focus and the directrix line. For ellipses and hyperbolas, two foci and two directrices exist.

Conic sections appear everywhere: Ellipses describe planetary orbits (Kepler's laws). Parabolas model projectile trajectories and are used in satellite dishes and car headlights for focusing signals/light. Hyperbolas describe the paths of comets with enough energy to escape the solar system, and are used in LORAN navigation systems. Circles are fundamental in wheel design, gears, and rotary motion.

Look at the general form Ax² + Bxy + Cy² + Dx + Ey + F = 0. If B=0 and A=C (with same sign), it's a circle. If B=0, A≠C but same sign, it's an ellipse. If B=0 and either A=0 or C=0 (but not both), it's a parabola. If B=0 and A and C have opposite signs, it's a hyperbola. The discriminant B²−4AC also classifies them: <0 for ellipse/circle, =0 for parabola, >0 for hyperbola.