Cylinders and Quadric Surfaces

Planes and spheres are the surfaces a linear or a single squared-distance equation produces. Second-degree equations produce a wider family, and two subfamilies are worth recognizing on sight: cylinders, where one variable is missing, and quadric surfaces, the full second-degree graphs. The tool for sketching and classifying both is the trace, the curve where the surface meets a plane parallel to a coordinate plane.

Cylinders

Consider $z = x^{2}$ . The equation constrains $x$ and $z$ but says nothing about $y$ , so every plane $y = k$ cuts the surface in the same parabola $z = x^{2}$ . Stacking those identical parabolas along the $y$ -axis sweeps out a parabolic cylinder: one curve, translated along the missing axis.

The parabolic cylinder z = x^2. Every plane y = k cuts it in the same parabola, and the rulings run parallel to the y-axis, the missing variable.

The same rule reads circular cylinders. In space $x^{2} + y^{2} = 1$ is not a circle but a surface: the circle $x^{2} + y^{2} = 1$ in each plane $z = k$ , stacked along the $z$ -axis. Its trace in the $x y$ -plane is the circle $x^{2} + y^{2} = 1$ , $z = 0$ , but the surface itself is the full circular cylinder. Likewise $y^{2} + z^{2} = 1$ omits $x$ and is a circular cylinder with axis the $x$ -axis.

Quadric surfaces

Quadrics are the three-dimensional counterparts of the plane conic sections. The translation completes the square in each variable, moving the center or vertex to a chosen point and clearing the linear terms $G x + H y + I z$ ; the rotation removes the cross terms $D x y + E y z + F x z$ , aligning the axes of symmetry with the coordinate axes. What survives is the signs of the squared coefficients and whether one variable remains to the first power, and that data alone fixes the type.

Substituting $z = k$ , $y = k$ , or $x = k$ exposes the traces, and reading them as ellipses, parabolas, or hyperbolas names the surface. The six standard types below are all drawn with the axis of symmetry along $z$ ; a surface symmetric about a different axis has its variables permuted.

Ellipsoid

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} + \frac{z ^{2}}{c ^{2}} = 1.

Every trace is an ellipse. The horizontal trace $z = k$ is $x^{2} / a^{2} + y^{2} / b^{2} = 1 - k^{2} / c^{2}$ , an ellipse that exists only for $∣ k ∣ \leq c$ and shrinks to a point at $k = \pm c$ . The vertical traces are ellipses on the same terms. All even powers make the surface symmetric across each coordinate plane, and $a = b = c$ collapses it to a sphere.

The ellipsoid, a closed surface whose latitude rings and meridian arcs are all ellipses; it is a stretched sphere.

Elliptic paraboloid

\frac{z}{c} = \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} .

Horizontal traces $z = k$ are ellipses (empty when $k / c < 0$ ); vertical traces $x = k$ and $y = k$ are parabolas opening the way $z$ increases. The variable raised to the first power, here $z$ , marks the axis, and its sign sets the opening direction. A bowl.

The elliptic paraboloid z = x^2 + y^2: horizontal traces are ellipses that widen with height, vertical traces are upward parabolas.

Hyperbolic paraboloid

\frac{z}{c} = \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} .

The sign difference bends the two families of parabolas opposite ways: traces $y = k$ open upward, traces $x = k$ open downward, and horizontal traces $z = k$ are hyperbolas. The surface near the origin is a saddle, rising along one axis while falling along the other.

The hyperbolic paraboloid z = x^2 - y^2. Traces with y fixed open upward, the crossing family with x fixed opens down, and the center is a saddle.

Cone

\frac{z ^{2}}{c ^{2}} = \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} .

Horizontal traces $z = k$ are ellipses that grow with $∣ k ∣$ , degenerating to the single point at the origin when $k = 0$ . Vertical traces are hyperbolas away from the axis, and a pair of intersecting lines through the origin when the cutting plane contains the axis. Two nappes meet at the vertex.

The elliptic cone z^2 = x^2 + y^2: two nappes meeting at the vertex, with elliptical rings and straight generating lines.

Hyperboloid of one sheet

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} - \frac{z ^{2}}{c ^{2}} = 1.

Horizontal traces $z = k$ are ellipses for every $k$ , smallest at $z = 0$ (the waist) and widening without bound. Vertical traces are hyperbolas. The one minus sign picks the axis of symmetry, the $z$ -axis here. A single connected surface, pinched at the middle.

The hyperboloid of one sheet: one connected surface with a waist at z = 0, elliptical rings flaring above and below, meridians that are hyperbolas.

Hyperboloid of two sheets

- \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} + \frac{z ^{2}}{c ^{2}} = 1.

Two minus signs, two sheets. A horizontal trace $z = k$ is an ellipse only when $∣ k ∣ > c$ , empty in the gap $∣ k ∣ < c$ ; the sheets are separated caps opening away from the origin along the axis of the positive term. Vertical traces are hyperbolas.

The hyperboloid of two sheets: two disconnected caps, one above z = c and one below z = -c, with an empty band between them.

The trace signatures

Two horizontal and two vertical trace shapes fix the surface. The sign pattern of the squared terms, together with which variable (if any) appears to the first power, is enough to classify a quadric by inspection.

Surface	Standard form	Horizontal traces $z = k$	Vertical traces
Ellipsoid	$\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} + \frac{z ^{2}}{c ^{2}} = 1$	ellipses ( $∣ k ∣ \leq c$ )	ellipses
Elliptic paraboloid	$\frac{z}{c} = \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}}$	ellipses	parabolas
Hyperbolic paraboloid	$\frac{z}{c} = \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}}$	hyperbolas	parabolas
Cone	$\frac{z ^{2}}{c ^{2}} = \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}}$	ellipses (point at $k = 0$ )	hyperbolas; lines through the axis
Hyperboloid of one sheet	$\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} - \frac{z ^{2}}{c ^{2}} = 1$	ellipses (all $k$ )	hyperbolas
Hyperboloid of two sheets	$- \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} + \frac{z ^{2}}{c ^{2}} = 1$	ellipses ( $∣ k ∣ > c$ )	hyperbolas

The count of minus signs among the three squared terms distinguishes the constant-term family: zero gives the ellipsoid, one the hyperboloid of one sheet, two the hyperboloid of two sheets. A missing squared term with a first-power variable gives a paraboloid, elliptic or hyperbolic according to the sign of the two surviving squares.

The cone ties the two hyperboloids together. Replacing the $1$ on the right of either hyperboloid with $0$ gives $x^{2} / a^{2} + y^{2} / b^{2} - z^{2} / c^{2} = 0$ , exactly the cone. As $∣ z ∣$ grows, both hyperboloids approach this cone: the one-sheet surface flares out toward it from outside, the two-sheet surface opens toward it from within each cap. The cone is the shared asymptotic surface, the boundary case in which the waist of the one-sheet surface pinches to the vertex and the gap of the two-sheet surface closes. Reading a general second-degree equation then reduces to three checks: clear the linear terms by completing the square, count the signs of the squared terms, and note whether any variable survives to the first power.

Classifying by completing the square

A quadric that is shifted or scaled arrives in general form; dividing to make the right side $1$ and completing the square in each variable returns it to standard form.

Identifying a surface from its traces

Running the classification backward, a surface handed over as three trace families is named by matching the pattern in the table.

Quadrics model physical shapes directly. A rotating planet flattens into an ellipsoid rather than a sphere; a circular paraboloid focuses parallel rays of light or radio to a single point, the geometry of a satellite dish and a radio telescope; a hyperboloid of one sheet gives a nuclear cooling tower its structural stability, and paired hyperboloids transmit rotation between skew axes through their straight generating lines.