Cylinders

Definition

A cylinder is a surface that consists of all lines (called rulings) that

1. pass through a given plane curve $C$ (called the directrix), and
2. are parallel to a fixed direction $\vec{v}$.

The simplest case: when $\vec{v}$ is one of the coordinate axes, the equation of the cylinder involves only the other two coordinates — the missing variable can be anything.

Key rule: If an equation in 3D involves only two of the three variables, its graph is a cylinder whose rulings are parallel to the axis of the missing variable.

Example: Parabolic Cylinder $y = x^2$

The directrix is the parabola $y = x^2$ lying in the $xy$-plane. Every vertical line through this curve (parallel to the $z$-axis) lies on the surface. The result is the parabolic cylinder — a parabola swept straight up and down.

Notice how the cross-section at any fixed $z = z_0$ is the same parabola $y = x^2$. The surface is completely determined by the 2D curve; the $z$-direction adds no new constraint.

Example: Circular Cylinder $x^2 + y^2 = 1$

The directrix is the unit circle in the $xy$-plane. Each ruling is a vertical line through a point on that circle. The result is the familiar right circular cylinder of radius 1.

Example: Circular Cylinder $y^2 + z^2 = 1$

Now the variable $x$ is missing. The directrix is the unit circle in the $yz$-plane; rulings run parallel to the $x$-axis.

This is the same shape as the previous cylinder, just tilted 90°. Comparing the two pictures builds intuition for how the "missing variable" determines the orientation.

Quadratic Surfaces

Definition

A quadratic surface (also called a quadric surface) is the set of all points $(x, y, z)$ satisfying a second-degree polynomial equation in three variables:

This is the 3D analogue of a conic section (ellipse, parabola, hyperbola) in 2D.

Reducing to Standard Form via Completing the Square

A general quadratic equation can always be simplified by:

1. Rotating axes to eliminate the cross terms $Dxy$, $Exz$, $Fyz$.
2. Translating axes (completing the square) to eliminate the linear terms $Gx$, $Hy$, $Iz$.

After these steps we arrive at one of the standard forms below.

How completing the square works (one variable)

Given a term like $Ax^2 + Gx$, write:

Setting $X = x + \tfrac{G}{2A}$ shifts the center to the origin. Repeat for $y$ and $z$.

1. Ellipsoid

Every cross-section parallel to a coordinate plane is an ellipse (or a circle when two semi-axes are equal). The surface is bounded — it fits inside the box $|x| \le a$, $|y| \le b$, $|z| \le c$.

When $a = b = c$ the ellipsoid is a sphere of radius $a$.

Drag the sliders to change $a$, $b$, $c$. Notice:

• Making all three equal gives a sphere.
• Shrinking one axis compresses the surface into a disc (oblate spheroid when $c \ll a = b$).
• Elongating one axis stretches it into a football shape (prolate spheroid when $a \gg b = c$).

2. Elliptic Paraboloid

Cross-sections at constant $z = k > 0$ are ellipses; cross-sections at constant $x$ or $y$ are parabolas. The surface opens upward (toward $+z$) and has its vertex at the origin.

When $a = b$ this is a circular paraboloid (a bowl with circular cross-sections), which appears in satellite dishes and telescope mirrors.

3. Hyperbolic Paraboloid

This is the saddle surface. Cross-sections at constant $z = k$:

• $k > 0$: hyperbola opening in the $x$-direction.
• $k < 0$: hyperbola opening in the $y$-direction.
• $k = 0$: two lines through the origin (the "saddle point").

Cross-sections at constant $x$ are downward parabolas; at constant $y$ are upward parabolas.

The saddle point at the origin is a minimax: a minimum along the $x$-direction and a maximum along the $y$-direction. This shape appears in architecture (hyperbolic paraboloid shells) and in multivariable calculus when classifying critical points.

4. Elliptic Cone

Cross-sections at constant $z = k \ne 0$ are ellipses; the vertex is at the origin. The surface consists of two nappes (upper $z > 0$ and lower $z < 0$) joined at the origin.

The cone is the "boundary case" between the one-sheet and two-sheet hyperboloids (see below). Setting the right-hand side to $+1$ or $-1$ moves the surface off the cone.

When $a = b$ the cross-sections are circles and we get the familiar right circular cone. Changing $c$ controls how steep the cone is.

5. Hyperboloid of One Sheet

The surface is connected (one piece). Cross-sections at constant $z$ are ellipses; the smallest is at $z = 0$ (the waist) with semi-axes $a$ and $b$. Cross-sections at constant $x$ or $y$ are hyperbolas.

The hyperboloid of one sheet is a ruled surface — through every point on it pass two straight lines that lie entirely within the surface. This makes it structurally rigid and is why cooling towers and skyscrapers use this shape.

Notice: as $c \to \infty$ (very flat hyperbolas) the surface approaches a cylinder; as you reduce $c$ the waist narrows and the surface flares out faster.

6. Hyperboloid of Two Sheets

The surface has two separate sheets — one with $z \ge c$ and one with $z \le -c$. There are no points between $-c < z < c$. Cross-sections at constant $|z| = k > c$ are ellipses; cross-sections at constant $x$ or $y$ are hyperbolas.

Compare this with the one-sheet hyperboloid: both have the same equation form, differing only in the sign of the constant ($+1$ vs $-1$). Increasing $c$ pushes the two sheets further apart; decreasing $c$ brings them closer together until, at the limit $c \to 0$, the two sheets merge at the origin and become the cone $x^2/a^2 + y^2/b^2 = z^2/c^2$.

Summary Table

| Surface | Standard equation | Shape at $z = k$ | Connected? | | ---------------------- | --------------------------------------------------- | -------------------- | ----------------- | ------- | --------------- | | Ellipsoid | $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ | Ellipse (for $| k | \lt c$) | Yes (bounded) | | Elliptic paraboloid | $z=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ | Ellipse | Yes (unbounded) | | Hyperbolic paraboloid | $z=\frac{x^2}{a^2}-\frac{y^2}{b^2}$ | Hyperbola | Yes (saddle) | | Elliptic cone | $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2}$ | Ellipse (two nappes) | Yes (vertex only) | | Hyperboloid (1 sheet) | $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ | Ellipse (all $k$) | Yes | | Hyperboloid (2 sheets) | $\frac{z^2}{c^2}-\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ | Ellipse ($| k | >c$) | No (two pieces) |

Exercises