Cylinders and Surfaces in 3D

We’ve explored vectors, lines, planes, and spheres in three-dimensional space. Now we’ll investigate a broader variety of surfaces that can be graphed in 3D coordinate systems, starting with cylinders.

Identifying Cylinders

When most people hear “cylinder,” they think of a hollow pipe or soda can. But the mathematical definition is much broader—a rectangular heating duct is a cylinder, as is a rolled-up yoga mat with spiral cross-section.

Consider the equation [latex]x^2 + y^2 = 9[/latex]. In 2D, this describes a circle centered at the origin with radius 3. In 3D, this same equation represents a surface. Imagine stacking copies of this circle on top of each other, all centered on the [latex]z[/latex]-axis. This creates a hollow tube extending infinitely in both directions along the [latex]z[/latex]-axis.

cylindrical surface

A set of parallel lines (called rulings) passing through a given curve forms a cylindrical surface, or cylinder. The rulings can be parallel to any line, and the original curve can have any shape.

This definition shows that cylinders don’t need circular cross-sections. Any curve can form a cylinder, and the parallel lines can point in any direction.

Identifying Cylinders: An equation describes a cylinder if one variable is completely missing. The missing variable corresponds to the direction of the rulings:

• Missing [latex]z[/latex]: rulings parallel to [latex]z[/latex]-axis
• Missing [latex]y[/latex]: rulings parallel to [latex]y[/latex]-axis
• Missing [latex]x[/latex]: rulings parallel to [latex]x[/latex]-axis

Sketch the graphs of the following cylindrical surfaces.

1. [latex]x^{2}+z^{2}=25[/latex]
2. [latex]z=2x^{2}-y[/latex]
3. [latex]y=\sin x[/latex]

Show Solution

1. The variable [latex]y[/latex] can take on any value without limit. Therefore, the lines ruling this surface are parallel to the [latex]y[/latex]-axis. The intersection of this surface with the [latex]xz[/latex]-plane forms a circle centered at the origin with radius 5 (see the following figure).
   

   

2. In this case, the equation contains all three variables–[latex]x[/latex], [latex]y[/latex], and [latex]z[/latex]–so none of the variables can vary arbitrarily. The easiest way to visualize this surface is to use a computer graphing utility (see the following figure).
   

   

3. In this equation, the variable [latex]z[/latex] can take on any value without limit. Therefore, the lines composing this surface are parallel to the [latex]z[/latex]-axis. The intersection of this surface with the [latex]xy[/latex]-plane outlines curve [latex]y=\sin x[/latex] (see the following figure).
   

   

Traces: Cross-Sections of Surfaces

When sketching surfaces, it helps to examine where the surface intersects planes parallel to the coordinate planes. These intersections are called traces.

traces

The traces of a surface are the cross-sections created when the surface intersects a plane parallel to one of the coordinate planes ([latex]xy[/latex], [latex]xz[/latex], or [latex]yz[/latex] planes).

For cylindrical surfaces, only one set of traces provides useful information: The trace perpendicular to the rulings shows the shape of the generating curve. Traces parallel to the rulings just show parallel lines.

For example, with [latex]z = \sin x[/latex]:

• [latex]xz[/latex]-plane trace ([latex]y = 0[/latex]): Shows the sine wave that generates the cylinder
• [latex]xy[/latex]-plane trace ([latex]z = 0[/latex]): Shows parallel lines (not very informative)
• [latex]yz[/latex]-plane trace ([latex]x = 0[/latex]): Shows a single line (also not informative)

Traces become an essential tool as we explore more complex surfaces beyond cylinders, where the geometry isn’t as straightforward as parallel lines through a curve.

 Cylindrical surfaces are the simplest type of 3D surface—they’re formed by parallel lines. More complex surfaces require examining multiple traces to understand their shape fully.