How a Curve Moves on a Cylinder
A vertical circular cylinder fixes the distance of each point from its central axis, while allowing the point to move around that axis and up or down. For a cylinder whose axis is the \(z\)-axis, the circular coordinates \(x\) and \(y\) therefore satisfy one fixed equation. The height \(z\), by contrast, can vary freely.
This separation gives a direct way to build curves on a cylinder: use circular functions for the first two coordinates, and choose a height function for the third. The resulting parametrization can describe a horizontal circle, a helix, or many other paths that wind around the cylinder as their height changes.
The Basic Parametrization
At angle \(\theta\), the point on a circle of radius \(R\) centered at the origin has coordinates \(x=R\cos\theta\) and \(y=R\sin\theta\). A curve can use one angle function and a separate height function:
Here \(I\) is the parameter interval, \(\theta(s)\) determines the position around the cylinder, and \(h(s)\) determines the height. If the curve is to be differentiable, choose differentiable functions \(\theta\) and \(h\). A common convenient choice is to use the angle itself as the parameter: set \(\theta(t)=t\). Then the formula becomes
The parameter need not be the angle. For example, \(\theta(s)=2s\) makes the point go around the cylinder twice as fast as \(\theta(s)=s\). The height can change at a different rate from the angle.
Why the Formula Stays on the Cylinder
The construction works because the fundamental trigonometric identity \(\cos^2\theta+\sin^2\theta=1\) fixes the distance of the first two coordinates from the \(z\)-axis. Substituting the parametrization into the cylinder equation gives a complete verification.
Let \(x(s)=R\cos\theta(s)\) and \(y(s)=R\sin\theta(s)\).
Then \(x(s)^2+y(s)^2=R^2\cos^2\theta(s)+R^2\sin^2\theta(s)\).
Factoring out \(R^2\) gives \(R^2(\cos^2\theta(s)+\sin^2\theta(s))=R^2\), as required.
This check applies for every \(s\in I\), not just at a starting point or at a few sample values. It establishes that the entire parametrized curve lies on the cylinder.
Constant Height Gives a Circle
If \(h(s)\) is constant, say \(h(s)=c\), the curve remains at height \(c\) while its circular coordinates vary. Taking the angle through a full interval of length \(2\pi\) gives one complete horizontal circle. A shorter interval gives only an arc, while a longer interval traces the circle repeatedly.
Worked Example: A Circle at Height 2
Parametrize one complete circle of radius \(3\) on the cylinder \(x^2+y^2=9\) at height \(z=2\), and check the parametrization.
Use the angle \(t\) on the interval \(0\leq t\leq 2\pi\). The radius is \(R=3\), and the constant height function is \(h(t)=2\). Therefore
At the start, \(t=0\), so \(\cos 0=1\) and \(\sin 0=0\). Thus \(\mathbf{r}(0)=(3,0,2)\). At \(t=\pi/2\), \(\cos(\pi/2)=0\) and \(\sin(\pi/2)=1\), so \(\mathbf{r}(\pi/2)=(0,3,2)\). At the end, \(t=2\pi\), \(\cos(2\pi)=1\) and \(\sin(2\pi)=0\), so \(\mathbf{r}(2\pi)=(3,0,2)\). The endpoint agrees with the starting point, completing one turn.
For every \(t\), the cylinder check is \(x(t)^2+y(t)^2=(3\cos t)^2+(3\sin t)^2=9\cos^2t+9\sin^2t=9\). The height is \(z(t)=2\) throughout, so the parametrization is the requested circle on the cylinder.
A Prescribed Height Function
When the height is not constant, the point still travels on the same circular cross-section pattern, but rises or falls as the angle changes. In the angle-parameter form, the height is simply inserted as the third coordinate. Its formula does not alter the fixed radius.
Worked Example: A Curve with Quadratically Increasing Height
Parametrize a curve on the cylinder of radius \(2\) with height \(z=t^2\), using \(0\leq t\leq\pi\). Then check its location and evaluate three sample points.
Here \(R=2\) and \(h(t)=t^2\), so substitution into the standard form gives
At \(t=0\), the coordinates are \(x=2\cos 0=2\), \(y=2\sin 0=0\), and \(z=0^2=0\), so \(\mathbf{r}(0)=(2,0,0)\).
At \(t=\pi/2\), \(x=2\cos(\pi/2)=0\), \(y=2\sin(\pi/2)=2\), and \(z=(\pi/2)^2=\pi^2/4\), so \(\mathbf{r}(\pi/2)=(0,2,\pi^2/4)\).
At \(t=\pi\), \(x=2\cos\pi=-2\), \(y=2\sin\pi=0\), and \(z=\pi^2\), so \(\mathbf{r}(\pi)=(-2,0,\pi^2)\).
For every \(t\), the circular-coordinate check gives \(x(t)^2+y(t)^2=4\cos^2t+4\sin^2t=4\). Thus every point lies on the cylinder \(x^2+y^2=4\), while the height increases from \(0\) to \(\pi^2\) on this interval.
Helices as a Special Case
A helix on a circular cylinder results when height changes linearly with the angle. If \(h(t)=at+b\), then each increase of \(2\pi\) in \(t\) makes one complete revolution, while the height changes by \(2\pi a\). The constant \(a\) determines the signed vertical change per turn; a positive value rises as the angle increases, and a negative value descends.
Worked Example: A Helix Rising by 3π per Turn
Construct a helix of radius \(2\) that rises by \(3\pi\) in one full revolution, starting at height \(1\). Parametrize two complete turns and check the endpoints and cylinder equation.
Use \(t\) as the angle. A full turn increases \(t\) by \(2\pi\), so the linear height rate \(a\) must satisfy \(2\pi a=3\pi\). Dividing by \(2\pi\) gives \(a=3/2\). Since the starting height is \(1\), choose \(h(t)=1+(3/2)t\). Two turns correspond to \(0\leq t\leq4\pi\). Therefore
At \(t=0\), the point is \((2,0,1)\). After one turn, at \(t=2\pi\), the coordinates are \(x=2\), \(y=0\), and \(z=1+(3/2)(2\pi)=1+3\pi\), so the point is \((2,0,1+3\pi)\). After two turns, at \(t=4\pi\), \(x=2\), \(y=0\), and \(z=1+(3/2)(4\pi)=1+6\pi\), so the endpoint is \((2,0,1+6\pi)\).
The vertical change per turn is \((1+3\pi)-1=3\pi\), as specified. For every \(t\), \(x(t)^2+y(t)^2=4\cos^2t+4\sin^2t=4\), so the whole helix lies on the cylinder of radius \(2\).
Parameter Interval and Interpretation
The formula describes both a path and the way the path is traced as the parameter increases. The parameter interval determines how much of the path is included. In the angle-parameter form, an interval of length \(2\pi\) covers one full turn around the axis. An interval of length \(4\pi\) covers two turns when the angle increases at unit rate. The height may make the endpoints different even when the circular coordinates return to their starting values.
More generally, if the angle is \(\theta(s)\), the number of turns depends on how much \(\theta(s)\) changes over the parameter interval. Increasing angle moves around the cylinder in the direction specified by \((\cos\theta,\sin\theta)\); decreasing angle reverses that direction. The function \(h(s)\) independently determines whether the curve rises, falls, or changes height in some other way.
A Common Mistake: Including the Height in the Radius Check
The cylinder equation involves only \(x\) and \(y\). A frequent error is to add the square of \(z\) when checking whether a curve lies on a vertical cylinder. For instance, consider the curve \(\mathbf{r}(t)=(2\cos t,2\sin t,t)\). A mistaken check might claim that \(x^2+y^2+z^2=4\) is required. At \(t=1\), this expression equals
not \(4\). That does not show that the point is off the cylinder. The correct check is \(x^2+y^2=4\cos^2t+4\sin^2t=4\) for every \(t\). The height \(z=t\) is unrestricted by the cylinder equation.
Why This Representation Is Useful
Separating the circular coordinates from the height makes it easy to design a path with a desired vertical behavior without changing its distance from the axis. A constant height produces a circle; a linear height produces a helix; and a nonlinear height function produces a curve whose vertical motion changes as it winds around the cylinder.
The same separation also makes verification straightforward: the trigonometric identity checks the radial constraint, and the third coordinate records the height directly. Keeping these roles distinct helps prevent mistakes when constructing or interpreting parametrizations.
Check Your Understanding
Use the cylinder equation and the angle-height representation to answer these questions.
- Write a parametrization of a circle of radius \(5\) at height \(-2\), with one complete turn.
- For \(\mathbf{r}(t)=(4\cos t,4\sin t,2t)\), what is the height change in one full turn?
- Give a parametrization on \(0\leq t\leq\pi\) of a curve on the cylinder of radius \(3\) with height \(h(t)=\sin t\).
- Why does \(x(t)^2+y(t)^2=R^2\) hold when \(x(t)=R\cos\theta(t)\) and \(y(t)=R\sin\theta(t)\)?
- For a parametrization using \(t\) as the angle, how many turns are traced over \(0\leq t\leq6\pi\)?