Slice a cone and you get ellipses, parabolas, or hyperbolas
A conic section is the curve where a flat plane slices a double cone. Tilt the plane one way and you get a closed ellipse, with the circle as a special case; set it parallel to the cone's side for a parabola; cut through both halves for a hyperbola. Apollonius of Perga systematised them around 200 BC.
The shape depends only on how the plane meets the cone. A closed curve is an ellipse, and it becomes a circle when the plane lies parallel to the cone's generating circle, which for a right cone means square to the axis. A plane parallel to exactly one line on the cone gives an open parabola, and a plane crossing both nappes gives a hyperbola with two separate branches. Planes through the tip produce degenerate cases, a single point, one line or two crossing lines, which some authors refuse to count. Apollonius treated the circle as a fourth type.
Plane geometry offers another route. Fix a point called the focus and a line called the directrix, and collect every point whose distance to the focus is a constant multiple of its distance to the line. That multiple is the eccentricity: below 1 it traces an ellipse, exactly 1 a parabola, above 1 a hyperbola. A circle has eccentricity zero and cannot be built this way in ordinary space, since its directrix would have to be the line at infinity. Dandelin spheres give a neat proof that the two definitions agree.
An ellipse can also be described with two foci, as the points whose distances to them sum to a fixed length, while a hyperbola uses a fixed difference. Algebraically every conic is a curve of degree two, the solutions of a quadratic equation in x and y, and its geometric features can be read from the coefficients.
In the Euclidean plane the three families look unrelated, but adding a line at infinity changes the picture. The two arms of a hyperbola then join at two points at infinity into one closed loop, and a parabola closes up by touching that line. Allowing complex coordinates makes the unification algebraic as well.
Source: Conic section
