Hyperbolic Cylinder: Two Walls from One Hyperbola

The idea: a hyperbola slid along z

The hyperbolic cylinder is the surface

x²/a² − y²/b² = 1

in 3D. The letter z is missing. So a point (x, y, z) is on the surface exactly when (x, y) is on the hyperbola x²/a² − y²/b² = 1, whatever z is. Take the hyperbola in the floor z = 0 and slide it straight up and down: that is the whole surface.

Every point of the hyperbola gives a whole vertical line on the surface, called a ruling. The surface is made of these straight lines, which is why it is called a cylinder, even though it is not round. (Demo 1: slide the hyperbola along z)

Two walls, nothing between

Write the equation as x²/a² = 1 + y²/b². The right side is at least 1, so x² ≥ a²: every point has x ≥ a or x ≤ −a. The surface is two separate curved walls, one on each side, and the slab −a < x < a is empty. Each wall comes closest to the z-axis along the vertical line x = ±a, y = 0. (Demo 5: a and b reshape the walls)

The cut z = h: the same hyperbola every time

Put z = h into the equation. Nothing happens, because there is no z to replace. Every level cut is the same hyperbola x²/a² − y²/b² = 1, just at a different height. (Demo 2: slide the cut z = h, nothing changes)

The asymptotic planes y = ±(b/a)x

Far from the origin the 1 on the right hardly matters, and the equation is almost x²/a² = y²/b², that is y = ±(b/a)x. In 2D those are the asymptotes of the hyperbola. In 3D, with z free, they become two vertical asymptotic planes through the z-axis. The walls get closer and closer to them but never touch. The slope b/a decides how fast the walls open: b/a > 1 gives steep planes and wide walls, b/a < 1 flat planes and narrow walls, and a = b gives planes at right angles. (Demo 4: the asymptotic planes y = ±(b/a)x)

Upright cuts x = h and y = h: lines

The planes x = h and y = h stand upright, parallel to the z-axis like the rulings. So they can only meet the walls in vertical lines.

• x = h: y²/b² = h²/a² − 1. For |h| < a the right side is negative: nothing, the plane passes between the walls. At |h| = a it is 0: one line y = 0, the edge of a wall. For |h| > a: two parallel lines y = ±b√(h²/a² − 1). (Demo 3: cut x = h: none → one line → two lines)
• y = h: x²/a² = 1 + h²/b², always positive: always two lines x = ±a√(1 + h²/b²), one on each wall.

cut	equation left	what you get
z = h	x2/a2 − y2/b2 = 1	the same hyperbola for every h
x = h, |h| < a	y2/b2 = negative	nothing
x = h, |h| = a	y2 = 0	one line y = 0
x = h, |h| > a	y2/b2 = h2/a2 − 1	two lines y = ±b√(h2/a2 − 1)
y = h	x2/a2 = 1 + h2/b2	two lines x = ±a√(1 + h2/b2)

Swapping the signs turns the walls

In −x²/a² + y²/b² = 1 the minus sign sits on x instead. Now y² ≥ b², so the walls are y ≥ b and y ≤ −b: the same kind of surface turned 90° about the z-axis. The asymptotic planes y = ±(b/a)x do not change, and the roles of the two upright cuts swap: y = h can miss, x = h always gives two lines. The variable with the plus sign names the axis the walls open along. (Demo 6: swap the signs, the walls turn 90°)

Facts

property	x2/a2 − y2/b2 = 1
pieces	two walls, x ≥ a and x ≤ −a
closest lines to the axis	x = ±a, y = 0 (any z)
asymptotic planes	y = ±(b/a)x
bounded	no
ruled	yes: vertical lines
general form	(1/a2)x2 − (1/b2)y2 − 1 = 0, no z at all

See also Quadric Surfaces (all nine shapes on one page), Elliptic Cylinder (the same idea with an ellipse: one round tube) and Hyperbolic Paraboloid (the saddle, whose level cuts are these hyperbolas, but changing with h).

What the page leaves out

Tilted cuts (a slanted plane cuts a hyperbola or two lines), a hyperbolic cylinder along another axis (x missing or y missing), shifted or rotated versions with xy and linear terms, and the parabolic cylinder y = ax², the third kind of quadric cylinder.

References

OpenStax Calculus Volume 3, 2.6 Quadric Surfaces

Paul's Online Math Notes — Quadric Surfaces

Wolfram MathWorld — Hyperbolic Cylinder