Cylinders And Vertical Surfaces In Cylindrical Coordinates

Reading Cylindrical Equations as Surfaces

Cylindrical coordinates describe a point by its distance from the \(z\)-axis, its angle around that axis, and its height. The coordinates are \((r,\theta,z)\), with \(r\geq 0\), and their Cartesian relationships are

x=r\cos\theta,\qquad y=r\sin\theta,\qquad z=z.

A surface equation in these coordinates often has a simple geometric interpretation. An equation that fixes \(r\) but leaves \(\theta\) and \(z\) free gives one kind of surface. An equation that fixes \(\theta\) but leaves \(r\) and \(z\) free gives another. The key is to identify what may vary, not just to recognize the symbols.

Definition: A vertical surface is a surface that extends parallel to the \(z\)-axis. In particular, if membership in a surface is determined by a condition on \(r\) and \(\theta\) alone, then each permitted horizontal point \((x,y)\) occurs at every height \(z\).

Fixed Radius: \(r=c\)

Suppose \(c>0\). The equation \(r=c\) fixes the distance from the \(z\)-axis while allowing the angle and height to vary. At every height, the horizontal cross-section is a circle of radius \(c\) centered at the origin of the \(xy\)-plane. Extending that circle through all heights makes a circular cylinder whose axis is the \(z\)-axis.

This interpretation can also be checked in Cartesian coordinates. Since \(r^2=x^2+y^2\), the equation \(r=c\) is equivalent to \(x^2+y^2=c^2\). The equation does not constrain \(z\), so every point on the corresponding circle occurs at every height.

r=c,\quad c>0 \qquad\Longleftrightarrow\qquad x^2+y^2=c^2.

The case \(c=0\) is different: \(r=0\) means \(x=0\) and \(y=0\), with \(z\) unrestricted. This is the \(z\)-axis, not a cylinder of positive radius. If \(c<0\), the equation has no solutions under the convention \(r\geq0\).

Worked Example: The Surface \(r=3\)

Interpret \(r=3\) geometrically and check the interpretation in Cartesian coordinates.

Because \(r\) is the distance to the \(z\)-axis, the equation fixes that distance at \(3\). The angle \(\theta\) can take any value, and \(z\) can take any value. Thus the surface is the circular cylinder of radius \(3\) centered on the \(z\)-axis.

To convert the equation, square both sides: \(r^2=9\). Substitute \(r^2=x^2+y^2\), obtaining \(x^2+y^2=9\), with \(z\) unrestricted. For example, at \(\theta=\pi/2\) and \(z=4\), the coordinates are \(x=3\cos(\pi/2)=0\), \(y=3\sin(\pi/2)=3\), and \(z=4\). The Cartesian equation checks: \(0^2+3^2=9\). More generally, every point on \(x^2+y^2=9\) has \(r=3\), so the conversion gives exactly the same cylinder.

Fixed Angle: \(\theta=c\)

Now fix an angle \(c\), while allowing \(r\geq0\) and \(z\) to vary. In each horizontal plane, the points lie on a ray starting at the origin and pointing in direction \((\cos c,\sin c)\). Extending that ray through all heights produces a vertical half-plane containing the \(z\)-axis.

The word half matters. The restriction \(r\geq0\) allows points in the direction of angle \(c\), but not points on the opposite ray. A parameterization makes this explicit:

(x,y,z)=(r\cos c,\;r\sin c,\;z), \qquad r\geq0,\quad z\in\mathbb R.

Every point on this half-plane has a horizontal direction given by \(c\), except points on the \(z\)-axis, where \(r=0\) and the angle is not uniquely determined. Angles differing by a multiple of \(2\pi\) describe the same direction, so \(\theta=c\) and \(\theta=c+2\pi k\) describe the same half-plane for any integer \(k\).

Worked Example: The Surface \(\theta=\pi/3\)

Describe the surface and find a Cartesian condition that distinguishes it from the full plane containing it.

Here \(\cos(\pi/3)=1/2\) and \(\sin(\pi/3)=\sqrt{3}/2\). Its points therefore have the form

(x,y,z)=\left(\frac r2,\frac{\sqrt{3}r}{2},z\right), \qquad r\geq0.

Eliminating \(r\) gives \(y=\sqrt{3}x\). Since \(x=r/2\) and \(r\geq0\), we also have \(x\geq0\). Thus the surface is the half of the vertical plane \(y=\sqrt{3}x\) with \(x\geq0\), including its boundary, the \(z\)-axis.

For a check, choose \(r=2\) and \(z=-1\). Then \(x=1\), \(y=\sqrt{3}\), and \(z=-1\). These coordinates satisfy \(y=\sqrt{3}x\) and \(x\geq0\). Conversely, if \(y=\sqrt{3}x\) and \(x\geq0\), then \(r=\sqrt{x^2+y^2}=2x\), so \(r\geq0\), and the point’s direction is \(\pi/3\) when \(r>0\). When \(r=0\), it lies on the \(z\)-axis and is included as well.

Radial Curves Extended Vertically: \(r=f(\theta)\)

An equation \(r=f(\theta)\) specifies a distance from the \(z\)-axis for each allowed angle. Since it contains no \(z\), every permitted horizontal point extends through all heights. Geometrically, first consider the polar curve \(r=f(\theta)\) in the \(xy\)-plane; then extend that curve parallel to the \(z\)-axis to obtain a surface.

This description has conditions. The formula must be considered only at angles in its domain, and the resulting radius must satisfy \(f(\theta)\geq0\). If a formula gives a negative value, it cannot be the radius under the convention \(r\geq0\). Also, a point with \(r=0\) lies on the \(z\)-axis, where different angle values represent the same point. A radial curve need not make a circular cylinder: its shape can vary with angle.

Worked Example: The Surface \(r=2\cos\theta\)

Identify the surface and verify its Cartesian equation.

Because \(r\geq0\), the equation \(r=2\cos\theta\) is possible only where \(\cos\theta\geq0\). Multiply both sides by \(r\):

r^2=2r\cos\theta.

Using \(r^2=x^2+y^2\) and \(r\cos\theta=x\), this becomes \(x^2+y^2=2x\). Completing the square gives

$$ (x-1)^2+y^2=1. $$

This is a circle of radius \(1\) centered at \((1,0)\) in each horizontal cross-section. Since \(z\) is unrestricted, the surface is a circular cylinder parallel to the \(z\)-axis, but it is not centered on the \(z\)-axis.

For a direct check, take \(\theta=\pi/3\). Then \(r=2\cos(\pi/3)=1\), giving \(x=1/2\) and \(y=\sqrt{3}/2\). The Cartesian equation holds because \(x^2+y^2=1=2x\), and \(z\) may be any real number. Conversely, a point on \((x-1)^2+y^2=1\) satisfies \(x^2+y^2=2x\), so \(r^2=2x=2r\cos\theta\). If \(r>0\), division by \(r\) gives \(r=2\cos\theta\); if \(r=0\), the point is the origin in the cross-section and also satisfies the original equation for an angle with \(\cos\theta=0\). Thus the two descriptions represent the same surface.

Why These Surfaces Extend Vertically

The extension in the \(z\)-direction follows directly from how cylindrical coordinates encode a point. A condition involving only \(r\) and \(\theta\) is unchanged when the height coordinate is replaced by another value. If \((r,\theta,z_1)\) satisfies that condition, then \((r,\theta,z_2)\) satisfies it for every real \(z_2\). In Cartesian space, the horizontal coordinates \(x=r\cos\theta\) and \(y=r\sin\theta\) stay fixed while \(z\) varies.

Takeaway: An equation involving \(r\) and \(\theta\) but not \(z\) describes a surface made by extending its allowed horizontal points parallel to the \(z\)-axis. The resulting surface may be a cylinder, a half-plane, or another vertical surface; its exact shape depends on the horizontal condition.

A Common Misinterpretation

A frequent error is to treat \(\theta=c\) as the entire vertical plane through the ray at angle \(c\). For example, from \(\theta=\pi/4\), one may correctly obtain \(y=x\), but incorrectly conclude that the whole plane \(y=x\) is described. The original equation also requires \(r\geq0\), which at this angle gives \(x\geq0\). It therefore describes only one half of that plane, together with the \(z\)-axis.

The full plane through the \(z\)-axis contains both this half-plane and the opposite half-plane. The opposite half-plane has angle \(\pi/4+\pi\), so the union is represented by the two angular conditions \(\theta=\pi/4\) or \(\theta=5\pi/4\), with angles understood modulo \(2\pi\). Keeping the radial restriction visible prevents the half-plane from being mistaken for a plane.

Another useful check is to ask what is free. In \(r=c\), the angle and height vary. In \(\theta=c\), the radius and height vary. In \(r=f(\theta)\), the angle determines the radius, while height remains free. This simple accounting often reveals whether an interpretation has accidentally added points or omitted part of the surface.

Check Your Understanding

For each question, identify the allowed ranges and explain what geometric set is represented.

What set is described by \(r=0\), and why is it not a positive-radius cylinder?
Describe the surface \(\theta=0\). Which restriction distinguishes it from the entire plane \(y=0\)?
For \(r=4\), write the Cartesian equation and state which coordinate is unrestricted.
Why does an equation \(r=f(\theta)\) with no \(z\) describe a vertical surface?
What geometric surface is represented by \(r=2\cos\theta\), and where is its circular cross-section centered?