---
title: Cylinders and Quadric Surfaces
module: Vectors and the Geometry of Space
moduleNumber: 10
lessonNumber: 3
order: 1003
summary: >
  A surface whose equation omits one variable is a cylinder: the graph of a
  plane curve swept along the missing axis. A second-degree equation in three
  variables is a quadric, and translation and rotation reduce every one to a
  short standard list. Traces — the curves cut by planes parallel to the
  coordinate planes — sort the six quadrics into ellipsoid, the two paraboloids,
  the cone, and the two hyperboloids.
topics: [Vectors and the Geometry of Space]
sources:
  - book: Stewart
    ref: "Ch. 12 — Vectors and the Geometry of Space; §12.6 Cylinders and Quadric Surfaces"
draft: false
---

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.

> **Definition (Trace).** A trace of a surface is its curve of intersection with
> a plane parallel to one of the coordinate planes: a horizontal trace sets
> $z = k$, a vertical trace sets $x = k$ or $y = k$. The traces are the
> cross-sections a slicing plane reveals.

## Cylinders

> **Definition (Cylinder).** A cylinder is a surface made of all lines
> (**rulings**) parallel to a fixed line, each passing through a fixed plane
> curve. If one of $x, y, z$ is absent from the equation of a surface, the
> surface is a cylinder with rulings parallel to the axis of the missing
> variable.

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.

$$
% caption: The parabolic cylinder z = x^2. Every plane y = k cuts it in the same
% parabola, and the rulings run parallel to the y-axis, the missing variable.
\tdplotsetmaincoords{70}{115}
\begin{tikzpicture}[tdplot_main_coords, scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0,0) -- (2.1,0,0) node[anchor=north east] {$x$};
  \draw[->, black] (0,0,0) -- (0,3.4,0) node[anchor=north west] {$y$};
  \draw[->, black] (0,0,0) -- (0,0,2.4) node[anchor=south] {$z$};
  % parabola ribs z = x^2 at several y
  \foreach \Y in {0,1,2,3} {
    \draw[acc, thick] plot[domain=-1.35:1.35, samples=30, variable=\t] (\t, \Y, {\t*\t});
  }
  % rulings parallel to the y-axis
  \foreach \X in {-1.35,-0.7,0,0.7,1.35} {
    \draw[black, thin] (\X,0,{\X*\X}) -- (\X,3,{\X*\X});
  }
  \node[black, anchor=west] at (1.05,3.0,1.9) {rulings};
\end{tikzpicture}
$$

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 $xy$-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.

> **Worked example.** Identify the surface $y^2 + z^2 = 1$ in $\mathbb{R}^3$.
>
> The variable $x$ is absent, so the surface is a cylinder with rulings parallel
> to the $x$-axis. Setting $x = k$ gives the circle $y^2 + z^2 = 1$ in the plane
> $x = k$, radius $1$, centered on the $x$-axis. The surface is the circular
> cylinder obtained by sliding the circle $y^2 + z^2 = 1$, $x = 0$ (in the
> $yz$-plane) along the $x$-axis.

## Quadric surfaces

> **Definition (Quadric surface).** A quadric surface is the graph of a
> second-degree equation in $x, y, z$. The general form
>
> $$
> Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0
> $$
>
> reduces, by translation and rotation, to one of two standard forms,
>
> $$
> Ax^2 + By^2 + Cz^2 + J = 0
> \qquad\text{or}\qquad
> Ax^2 + By^2 + Iz = 0.
> $$

Quadrics are the three-dimensional counterparts of the plane
[conic sections](/calculus/parametric-and-polar/conic-sections). The translation
completes the square in each variable, moving the center or vertex to a chosen
point and clearing the linear terms $Gx + Hy + Iz$; the rotation removes the
cross terms $Dxy + Eyz + Fxz$, 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
$\abs{k} \le c$ 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.

> **Worked example.** Sketch $x^2 + \dfrac{y^2}{9} + \dfrac{z^2}{4} = 1$ from its
> traces.
>
> The surface is an ellipsoid with $a = 1$, $b = 3$, $c = 2$, reaching to
> $(\pm 1, 0, 0)$, $(0, \pm 3, 0)$, $(0, 0, \pm 2)$ on the axes. The horizontal
> trace $z = k$ is
>
> $$
> x^2 + \frac{y^2}{9} = 1 - \frac{k^2}{4},
> $$
>
> an ellipse for $-2 < k < 2$, largest at $k = 0$ and shrinking to the poles at
> $k = \pm 2$. The vertical traces are ellipses too:
> $y^2/9 + z^2/4 = 1 - k^2$ in the plane $x = k$ (for $-1 < k < 1$), and
> $x^2 + z^2/4 = 1 - k^2/9$ in the plane $y = k$ (for $-3 < k < 3$). Three
> families of nested ellipses assemble the closed surface.

$$
% caption: The ellipsoid, a closed surface whose latitude rings and meridian
% arcs are all ellipses; it is a stretched sphere.
\tdplotsetmaincoords{70}{110}
\begin{tikzpicture}[tdplot_main_coords, scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % latitude rings: x = a sinφ cosθ, y = b sinφ sinθ, z = c cosφ  (a=b=1.6, c=2.2)
  \foreach \ph in {30,60,90,120,150} {
    \draw[black, thin] plot[domain=0:360, samples=44, variable=\t]
      ({1.6*sin(\ph)*cos(\t)}, {1.6*sin(\ph)*sin(\t)}, {2.2*cos(\ph)});
  }
  % meridian half-arcs
  \foreach \th in {0,45,90,135,180,225,270,315} {
    \draw[acc, thin] plot[domain=0:180, samples=34, variable=\t]
      ({1.6*sin(\t)*cos(\th)}, {1.6*sin(\t)*sin(\th)}, {2.2*cos(\t)});
  }
\end{tikzpicture}
$$

### 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.

$$
% caption: The elliptic paraboloid z = x^2 + y^2: horizontal traces are ellipses
% that widen with height, vertical traces are upward parabolas.
\tdplotsetmaincoords{70}{110}
\begin{tikzpicture}[tdplot_main_coords, scale=1.05, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % elliptical rings at height r^2
  \foreach \r in {0.5,0.85,1.2,1.5} {
    \draw[black, thin] plot[domain=0:360, samples=44, variable=\t]
      ({\r*cos(\t)}, {\r*sin(\t)}, {\r*\r});
  }
  % parabolic ribs from the vertex
  \foreach \th in {0,45,90,135,180,225,270,315} {
    \draw[acc, thin] plot[domain=0:1.5, samples=20, variable=\r]
      ({\r*cos(\th)}, {\r*sin(\th)}, {\r*\r});
  }
  \fill[black] (0,0,0) circle (1pt);
\end{tikzpicture}
$$

### 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.

$$
% caption: The hyperbolic paraboloid z = x^2 - y^2. Traces with y fixed open
% upward, the crossing family with x fixed opens down, and the center is a saddle.
\tdplotsetmaincoords{70}{110}
\begin{tikzpicture}[tdplot_main_coords, scale=1.25, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % parabolas with y fixed (open upward in z)
  \foreach \Y in {-1.2,-0.6,0,0.6,1.2} {
    \draw[acc, thin] plot[domain=-1.2:1.2, samples=24, variable=\t]
      (\t, \Y, {\t*\t - \Y*\Y});
  }
  % parabolas with x fixed (open downward in z)
  \foreach \X in {-1.2,-0.6,0,0.6,1.2} {
    \draw[black, thin] plot[domain=-1.2:1.2, samples=24, variable=\t]
      (\X, \t, {\X*\X - \t*\t});
  }
  \fill[black] (0,0,0) circle (1pt);
\end{tikzpicture}
$$

### 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 $\abs{k}$, 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.

$$
% caption: The elliptic cone z^2 = x^2 + y^2: two nappes meeting at the vertex,
% with elliptical rings and straight generating lines.
\tdplotsetmaincoords{70}{110}
\begin{tikzpicture}[tdplot_main_coords, scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % rings at z = +/- level, radius = level
  \foreach \zz in {0.6,1.2,1.8} {
    \draw[black, thin] plot[domain=0:360, samples=44, variable=\t]
      ({\zz*cos(\t)}, {\zz*sin(\t)}, {\zz});
    \draw[black, thin] plot[domain=0:360, samples=44, variable=\t]
      ({\zz*cos(\t)}, {\zz*sin(\t)}, {-\zz});
  }
  % generating lines through the vertex
  \foreach \th in {0,45,90,135,180,225,270,315} {
    \draw[acc, thin] ({1.8*cos(\th)},{1.8*sin(\th)},1.8) -- (0,0,0)
       -- ({1.8*cos(\th)},{1.8*sin(\th)},-1.8);
  }
  \fill[black] (0,0,0) circle (1pt);
\end{tikzpicture}
$$

### 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.

$$
% caption: The hyperboloid of one sheet: one connected surface with a waist at
% z = 0, elliptical rings flaring above and below, meridians that are hyperbolas.
\tdplotsetmaincoords{70}{110}
\begin{tikzpicture}[tdplot_main_coords, scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % rings, radius sqrt(1 + z^2)
  \foreach \zz/\rr in {-1.6/1.89,-0.8/1.28,0/1,0.8/1.28,1.6/1.89} {
    \draw[black, thin] plot[domain=0:360, samples=44, variable=\t]
      ({\rr*cos(\t)}, {\rr*sin(\t)}, {\zz});
  }
  % hyperbolic meridians
  \foreach \th in {0,45,90,135,180,225,270,315} {
    \draw[acc, thin] plot[domain=-1.7:1.7, samples=28, variable=\z]
      ({sqrt(1+\z*\z)*cos(\th)}, {sqrt(1+\z*\z)*sin(\th)}, {\z});
  }
  \node[black, anchor=west] at (1.05,0,0) {waist};
\end{tikzpicture}
$$

### 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
$\abs{k} > c$, empty in the gap $\abs{k} < c$; the sheets are separated caps
opening away from the origin along the axis of the positive term. Vertical
traces are hyperbolas.

$$
% caption: The hyperboloid of two sheets: two disconnected caps, one above
% z = c and one below z = -c, with an empty band between them.
\tdplotsetmaincoords{70}{110}
\begin{tikzpicture}[tdplot_main_coords, scale=1.0, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % rings on each sheet, radius sqrt(z^2 - 1)
  \foreach \zz/\rr in {1.2/0.66,1.6/1.25,2.1/1.85} {
    \draw[black, thin] plot[domain=0:360, samples=44, variable=\t]
      ({\rr*cos(\t)}, {\rr*sin(\t)}, {\zz});
    \draw[black, thin] plot[domain=0:360, samples=44, variable=\t]
      ({\rr*cos(\t)}, {\rr*sin(\t)}, {-\zz});
  }
  % hyperbolic meridians on each sheet
  \foreach \th in {0,60,120,180,240,300} {
    \draw[acc, thin] plot[domain=1:2.1, samples=18, variable=\z]
      ({sqrt(\z*\z-1)*cos(\th)}, {sqrt(\z*\z-1)*sin(\th)}, {\z});
    \draw[acc, thin] plot[domain=1:2.1, samples=18, variable=\z]
      ({sqrt(\z*\z-1)*cos(\th)}, {sqrt(\z*\z-1)*sin(\th)}, {-\z});
  }
  \fill[black] (0,0,1) circle (1pt);
  \fill[black] (0,0,-1) circle (1pt);
\end{tikzpicture}
$$

## 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 ($\abs{k}\le c$) | 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 ($\abs{k}>c$) | 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 $\abs{z}$ 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.

> **Worked example.** Identify and sketch $4x^2 - y^2 + 2z^2 + 4 = 0$.
>
> The equation has no linear terms, so no translation is needed; only the
> constant must be cleared. Divide by $-4$ to make the right side $1$:
>
> $$
> -x^2 + \frac{y^2}{4} - \frac{z^2}{2} = 1.
> $$
>
> Two of the three squared terms are negative, so this is a **hyperboloid of two
> sheets**, with the axis along the one positive term, the $y$-axis. The trace
> $y = 0$ is empty (it would need $-x^2 - z^2/2 = 1$), confirming the gap; for
> $\abs{k} > 2$ the trace $y = k$ is the ellipse
> $x^2 + z^2/2 = k^2/4 - 1$. The two sheets cap off at $(0, \pm 2, 0)$.

> **Worked example.** Classify $x^2 + 2z^2 - 6x - y + 10 = 0$.
>
> The variable $y$ appears only to the first power, which points to a paraboloid;
> complete the square in $x$ to locate the vertex:
>
> $$
> x^2 - 6x = (x - 3)^2 - 9,
> $$
>
> so the equation becomes $(x - 3)^2 - 9 + 2z^2 - y + 10 = 0$, that is
>
> $$
> y - 1 = (x - 3)^2 + 2z^2.
> $$
>
> Both squared terms are positive and $y$ is linear, so this is an **elliptic
> paraboloid** with axis parallel to the $y$-axis and vertex $(3, 1, 0)$. Traces
> $y = k$ for $k > 1$ are the ellipses $(x - 3)^2 + 2z^2 = k - 1$; the trace in
> the $xy$-plane is the parabola $y = 1 + (x - 3)^2$, $z = 0$.

## 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.

> **Worked example.** A surface has vertical traces $x = 0$ and $y = 0$ that are
> both upward parabolas, and horizontal traces $z = k$ (for $k > 0$) that are
> ellipses. Name it.
>
> Two parabolic vertical families and an elliptical horizontal family match one
> row of the table: the **elliptic paraboloid**. Concretely $z = 4x^2 + y^2$ fits.
> Its trace $x = 0$ is $z = y^2$ and its trace $y = 0$ is $z = 4x^2$, both upward
> parabolas, while $z = k$ gives $4x^2 + y^2 = k$, an ellipse for $k > 0$. The
> first-power variable $z$ names the axis, and the positive coefficients confirm
> the surface opens upward rather than saddling.

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.