Conic Sections Explorer

The discriminant Δ = B² - 4AC is the phase boundary for conics: Δ < 0 gives ellipse-family curves, Δ = 0 gives parabola-family curves, and Δ > 0 gives hyperbola-family curves. Linear terms D and E shift the curve in the plane, while F sets the level set value.

Point a flashlight straight at a wall and its beam draws a circle; tilt it and the circle stretches into an ellipse, opens into a parabola, and finally splits into a hyperbola. All four are slices of the same cone of light. This tool does the reverse: from the six coefficients A–F it reads off which slice you have drawn. A circle is simply the special ellipse where A = C and B = 0.

Newton showed that any object moving under inverse-square gravity follows a conic with the Sun at one focus, and the sign of its total energy decides which: negative energy binds it into an ellipse, exactly zero gives a parabolic escape, positive energy opens a hyperbola. That is how astronomers knew ʻOumuamua (2017) and comet 2I/Borisov (2019) came from beyond the solar system — their orbits were measurably hyperbolic (e > 1), so nothing could pull them back.