From a Circle to a Helix
A circle in the horizontal plane can be described by a pair of sine and cosine components. To make a space curve that circles while steadily rising or falling, keep those circular components and let the height vary linearly with the parameter.
The result is a circular helix with constant radius and constant pitch. Its projection onto the horizontal plane follows a circle, while each complete revolution produces the same vertical rise. The parameter interval determines how much of this curve is traced.
The Standard Parametrization
Let \(R>0\) be the radius, let \((a,b)\) be the horizontal center of the circular motion, let \(z_0\) be the height at parameter value \(0\), and let \(p\) be the signed vertical rise per complete revolution. Use the angle \(\theta\), measured in radians, as parameter. A standard parametrization is
The horizontal components trace a circle centered at \((a,b)\). The height component is linear in \(\theta\). If \(p>0\), the helix rises as \(\theta\) increases; if \(p<0\), it descends. If \(p=0\), the parametrization traces a circle at constant height rather than a rising or falling helix.
The factor \(1/(2\pi)\) converts angular change to vertical rise: an increase of \(2\pi\) in \(\theta\) is one full revolution. In particular, \(p\) is the vertical rise per revolution, not the vertical change per unit of angle.
Why the Radius and Pitch Stay Constant
The radius and pitch can be read directly from the components. Subtract the horizontal center from the first two coordinates and square the results:
Since \(R>0\), the horizontal distance from \((x,y)\) to \((a,b)\) is always \(R\). The point therefore circles at a fixed distance from the vertical line through \((a,b)\).
To check the rise, compare heights at parameters \(\theta\) and \(\theta+2\pi\):
This establishes the stated pitch. More generally, over any parameter change \(\Delta\theta\), the height change is \(\frac{p}{2\pi}\Delta\theta\), because the height component is linear.
Building a Helix from Its Radius and Rise
When the parameter is the angle itself, constructing the helix takes three choices: select the circular radius and center, set the height at \(\theta=0\), and divide the desired rise per full revolution by \(2\pi\) to obtain the coefficient of \(\theta\) in the height component.
Use \(a+R\cos\theta\) and \(b+R\sin\theta\) for a circle of radius \(R\) centered at \((a,b)\).
For signed rise \(p\) per revolution, use \(z_0+\frac{p}{2\pi}\theta\).
An interval of length \(2\pi\) gives one revolution; shorter or longer intervals give the corresponding fraction or multiple of a turn.
Evaluate the position at the parameter values bounding the interval to confirm its start and finish.
Worked Example: Two Upward Turns
Worked Example: Two Upward Turns
Construct a helix of radius \(3\), centered on the vertical axis, starting at height \(1\). It must rise \(8\) units per revolution and make two complete turns, beginning at \(\theta=0\).
The center is \((a,b)=(0,0)\), the radius is \(R=3\), the initial height is \(z_0=1\), and the signed pitch is \(p=8\). Two turns require a parameter interval of length \(2(2\pi)=4\pi\), so choose \(0\leq\theta\leq4\pi\). The parametrization is
At the start, \(\cos 0=1\) and \(\sin 0=0\), so \(\mathbf{r}(0)=\langle3,0,1\rangle\). At the end, \(\cos(4\pi)=1\), \(\sin(4\pi)=0\), and \(1+\frac{4}{\pi}(4\pi)=17\). Thus \(\mathbf{r}(4\pi)=\langle3,0,17\rangle\).
The horizontal distance from the axis is \(3\), since \(x^2+y^2=9(\cos^2\theta+\sin^2\theta)=9\). The total rise is \(17-1=16\), which is \(2\cdot8\), exactly the rise for two turns.
Parameter Intervals and Partial Turns
The angle parameter makes the connection between interval length and number of turns direct. For an interval \([\alpha,\beta]\), the angular change is \(\beta-\alpha\), so the number of turns traced is \((\beta-\alpha)/(2\pi)\). For example, an interval of length \(\pi\) traces half a turn. The vertical change on that interval is \(\frac{p}{2\pi}(\beta-\alpha)\).
The interval’s starting value also matters. The point at \(\theta=\alpha\) is the starting point of the parametrized portion, even if \(\alpha\) is not zero. The initial height \(z_0\) in the standard formula is the height at \(\theta=0\); at a general starting value \(\alpha\), the height is \(z_0+\frac{p}{2\pi}\alpha\).
Worked Example: A Descending Half-Turn
Consider a helix centered at \((2,-1)\), of radius \(4\), with height \(z=5\) at \(\theta=0\) and signed pitch \(-6\). Find its parametrization on \(\pi/2\leq\theta\leq3\pi/2\), and determine its starting and ending positions.
Here \(a=2\), \(b=-1\), \(R=4\), \(z_0=5\), and \(p=-6\). The height coefficient is \(\frac{p}{2\pi}=-\frac{3}{\pi}\). Therefore,
The starting point is found using \(\cos(\pi/2)=0\) and \(\sin(\pi/2)=1\):
At the endpoint, \(\cos(3\pi/2)=0\) and \(\sin(3\pi/2)=-1\):
The parameter interval has length \(3\pi/2-\pi/2=\pi\), or half of \(2\pi\), so it traces half a turn. The height change is \(\frac12-\frac72=-3\), also equal to \(\frac{-6}{2\pi}\cdot\pi=-3\). The negative pitch gives the expected downward motion.
Reading a Helix from Its Formula
When a parametrization is already given, inspect its components in order. The first two components reveal the horizontal center and radius; the coefficient of the angle in the height component determines pitch; and the interval determines how much of the curve is included. An interval endpoint calculation can confirm the total height change.
Worked Example: Identifying the Radius, Pitch, and Rise
For
identify the radius, vertical center line, signed pitch, number of turns, and total height change.
The horizontal components are \(-1+2\cos t\) and \(3+2\sin t\), so the circle is centered at \((-1,3)\) and has radius \(2\). The corresponding vertical center line is \(x=-1,\ y=3\). To find pitch, compare the height coefficient \(\frac{5}{3\pi}\) with \(\frac{p}{2\pi}\):
The parameter interval has length \(5\pi-(-\pi)=6\pi\), so it spans \(6\pi/(2\pi)=3\) turns. The total height change, calculated from the linear component, is
This agrees with three turns each rising \(\frac{10}{3}\): \(3\cdot\frac{10}{3}=10\). As a direct endpoint check, the heights are \(4-\frac{5}{3}=\frac{7}{3}\) and \(4+\frac{25}{3}=\frac{37}{3}\), whose difference is \(\frac{30}{3}=10\).
A Common Pitch Mistake
A frequent error is to use \(z=z_0+p\theta\) when \(p\) is supposed to mean rise per revolution. That expression rises by \(2\pi p\), not by \(p\), during a complete turn. For instance, if the desired rise is \(8\), setting \(z=1+8\theta\) produces a height change of \(8(2\pi)=16\pi\) over one revolution.
To make the height change equal to \(8\) when the angle increases by \(2\pi\), the coefficient must satisfy \(2\pi c=8\). Thus \(c=8/(2\pi)=4/\pi\), and the correct height component is \(z=1+\frac{4}{\pi}\theta\). The general rule is to divide the desired rise per revolution by \(2\pi\).
It is also important to keep the meaning of pitch signed: a negative value describes a downward rise as the parameter increases. Reversing the parameter direction changes which way the curve is traversed, so the direction should always be interpreted together with the specified parameter interval and increasing parameter values.
Check Your Understanding
Use the angle parameter convention and show how you identify each quantity.
- Write a parametrization for a helix of radius \(5\), centered at \((1,2)\), with initial height \(-1\) and signed pitch \(4\), over one complete turn starting at \(\theta=0\).
- For \(\mathbf{r}(\theta)=\langle 3\cos\theta,3\sin\theta,2-\theta/\pi\rangle\), find the radius, signed pitch, and vertical change over one complete turn.
- How many turns are traced on the interval \([-\pi,3\pi]\)?
- A helix has signed pitch \(-10\). What height change occurs when the angle increases by \(\pi/2\)?
- For a height component \(z=7+ct\), what value of \(c\) gives a rise of \(12\) per complete revolution when \(t\) is the angle in radians?