Conic Sections Lab
Circles, ellipses, parabolas and hyperbolas are one family. Each is the path of a point whose distance to a focus is e times its distance to a line, and each is a slice through a cone.
Focus, directrix, eccentricity
Pick a point F (the focus) and a line (the directrix). Collect every point P with PF = e · PD, where PD is P's distance to the line. The number e is the eccentricity: e < 1 gives an ellipse, e = 1 a parabola, e > 1 a hyperbola. As e gets close to 0 the ellipse becomes a circle.
Slicing a cone
The Greeks found the same curves by cutting a double cone with a plane. Cut straight across: circle. Tilt the plane: ellipse. Tilt it until it is parallel to the side of the cone: parabola. Tilt it further so it cuts both halves: hyperbola, with two branches. Planets follow ellipses with the Sun at one focus; a ball's path is a parabola.