Math101 · Independent Ontario learning library
Math101Quadric Surfaces
A rigorous, example-driven guide to quadric surfaces, including hypotheses, method choice, verification, and practice.
The central idea
A quadric surface in $\mathbb R^3$ is defined by a second-degree equation. Standard forms include ellipsoids, elliptic or hyperbolic paraboloids, one- and two-sheet hyperboloids, cones, and cylinders. Translations locate centers or vertices; signs and the number of squared variables determine type and axis.
Definitions, hypotheses, and notation
A one-sheet hyperboloid has two squared terms with one sign and the axis term with the opposite sign on the side containing one. A two-sheet hyperboloid has one positive term and two negative terms equal to one; sections exist only beyond a gap along the positive term's axis. A cone has a zero right side.
A missing variable produces a cylinder parallel to that coordinate axis. More general quadratic equations with cross terms may require rotation using eigenvectors of the symmetric quadratic-form matrix before matching standard axes.
Conceptual meaning
Coordinate traces reduce a surface to familiar conic sections. Horizontal and vertical slices reveal whether sections expand, contract, split, or change from ellipses to hyperbolas. Degenerate traces can identify vertices, necks, or cones.
A dependable method and decision rule
- Complete squares and move the constant to standard form.
- Normalize the nonzero right side to one when appropriate.
- Count positive, negative, and missing squared terms.
- Identify the distinguished axis from the exceptional sign or unsquared variable.
- Check coordinate-plane and constant-coordinate traces before naming the surface.
