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Math101learn.math101.caQuadric 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.
Fully worked example
Graphical or geometric meaning
At $z=0$ the surface has its smallest elliptical waist. As $|z|$ grows, ellipse semiaxes expand. The negative $z^2$ term does not remove points; moving it to the right increases the ellipse size.
Common mistakes and why they fail
Verification and reasonableness checks
- Substitute coordinate planes to inspect traces.
- Test whether the origin or translated center lies on the surface.
- Compare sign pattern with the normalized standard form.
Traces identify the surface reliably
Put the equation into standard form, then inspect coordinate-plane traces and slices with one coordinate held constant. Ellipsoids have bounded elliptical traces; hyperboloids change between elliptical and hyperbolic behavior according to their signs; cones are homogeneous; paraboloids show a linear variable paired with squared variables. A missing variable often creates a cylinder parallel to that coordinate axis. Sign pattern alone is not enough when constants or degeneracies change the set, so determine intercepts and whether real points exist. Completing squares reveals translated centers or vertices. Sketch several labeled traces before drawing the three-dimensional surface. This method distinguishes one-sheet and two-sheet hyperboloids and prevents treating an empty or degenerate level set as a standard quadric.
Practice
- Classify $x^2+y^2+z^2=4$.
- Classify $z=x^2+y^2$.
- Classify $z=x^2-y^2$.
Answers and brief solutions
- A sphere of radius $2$.
- An elliptic paraboloid opening in $+z$.
- A hyperbolic paraboloid.
Connections and next steps
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What surface is x²+y²−z²=1?
- Horizontal traces are circles x²+y²=1+z².
- They exist for every z and have a minimum waist at z=0.
- Thus the surface is a one-sheet hyperboloid along z.
End of lesson
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