Conic Parameters Explorer
Adjust center, shape parameters, and orientation to see equation geometry update live.
Quadratic Form Invariants 🖖
Conic sections are geometric manifestations of second-degree polynomials. Their defining parameter, eccentricity, acts as a continuous topological deformation index, smoothly transforming a closed ellipse into an open unbounded hyperbola through the singular parabolic limit.
Where the name comes from 🖖
The word "conic" is literal: slice a double cone with a flat plane and the edge you cut out is one of these curves. Hold the plane level and you get a circle; tilt it gently for an ellipse; tilt until it runs parallel to the cone's side for a parabola; tilt further so it cuts both halves and you get a hyperbola's two branches. Every shape in this tool is just a different cutting angle.
The focus hidden inside the cone 🖖
A conic drawn as a flat equation has a focus, and a conic sliced from a cone has a cutting plane — these seem unrelated. In 1822 Germinal Dandelin showed they are the same point: fit a sphere snugly inside the cone so it touches both the cone and the cutting plane, and the sphere's contact point is exactly the focus. For an ellipse two such spheres fit, pinning down both of its foci.
Example problems
- wide ellipse - Wide ellipse centered at origin with moderate eccentricity.
- tall ellipse - Vertical ellipse showing swapped major/minor orientation.
- parabola up - Upward parabola with visible focus-directrix geometry.
- parabola right - Right-opening parabola emphasizing horizontal orientation.
- hyperbola x - Horizontal hyperbola with two separated branches.
- hyperbola y - Vertical hyperbola centered off origin.