Archimedean Spirals

What Makes a Spiral Archimedean?

An Archimedean spiral is traced by a point whose signed radius changes linearly with its angle. As the angle advances, the point winds around the pole while moving closer to or farther from it. The amount of radial change in each full turn is constant, provided the corresponding parts of the spiral remain on the same ray.

Two common equations are \(r=a\theta\) and \(r=a+b\theta\). In both, the angle interval matters: it determines which part of the curve is traced, how many turns appear, and whether the radius passes through zero. The signs of the radius and of its rate of change also matter, because a negative radius plots on the ray opposite the one named by the angle.

Definition: An Archimedean spiral is a polar curve whose signed radius is a linear function of the angle. The forms \(r=a\theta\) and \(r=a+b\theta\) are linear in \(\theta\). If the coefficient of \(\theta\) is zero, the radius is constant instead; if \(r=a\theta\) has \(a=0\), the curve is just the pole.

Radius Change and Winding Direction

For \(r=a\theta\), the signed radius changes at the constant rate \(a\): increasing \(\theta\) by \(\Delta\theta\) changes \(r\) by \(a\Delta\theta\). For \(r=a+b\theta\), that change is \(b\Delta\theta\). Thus the coefficient of \(\theta\) controls whether the signed radius increases or decreases as the angle increases. The constant term \(a\) in the second form sets the radius at \(\theta=0\).

When \(\theta\) increases, the point is traced counterclockwise on any interval where the radius has one fixed, nonzero sign. For positive radius, the plotted ray has angle \(\theta\); for negative radius, it has angle \(\theta+\pi\) (modulo \(2\pi\)). That angle still increases as \(\theta\) increases, so negative radius does not reverse the winding direction. At \(r=0\), the point is at the pole and has no defined direction. If the interval is traversed with \(\theta\) decreasing, the tracing direction reverses.

The distance from the pole is \(|r|\), not \(r\). Consequently, a positive signed-radius rate does not always mean that the point is moving farther from the pole: on an interval where \(r<0\), increasing \(r\) toward zero makes \(|r|\) smaller. To decide whether the plotted point moves inward or outward, track the absolute value of the radius as well as its sign.

Why Successive Turns Have Constant Radial Change

A full revolution changes the angle by \(2\pi\). If the equation is \(r=a+b\theta\), compare the radii at angles \(\theta\) and \(\theta+2\pi\):

r(\theta+2\pi)-r(\theta) = \bigl(a+b(\theta+2\pi)\bigr)-\bigl(a+b\theta\bigr) = 2\pi b.

This proves that the signed radius changes by the same amount, \(2\pi b\), on every full turn. For \(r=a\theta\), the same calculation gives \(2\pi a\). When the two compared radii have the same sign, these points lie on the same ray, so the distance between them along that ray is \(2\pi|b|\) for \(r=a+b\theta\), or \(2\pi|a|\) for \(r=a\theta\). If the radii have opposite signs, the points are on opposite rays; the signed-radius difference alone is then not a gap between successive points on one ray.

This comparison applies only when both angles lie in the specified interval. An interval shorter than \(2\pi\) traces less than one full revolution, and a short interval can show only a fragment of the spiral. The graph’s appearance on that interval should not be mistaken for the entire curve.

Key takeaway: Over a full turn, \(r=a\theta\) changes its signed radius by \(2\pi a\); \(r=a+b\theta\) changes it by \(2\pi b\). This is a same-ray spacing only when the compared radii have the same sign.

Worked Examples

Worked Example: Tracing \(r=2\theta\) Through Two Turns

Sketch the curve \(r=2\theta\) for \(0\leq\theta\leq4\pi\), and find the radial change in each full turn.

First, the signed-radius rate is \(2\). The interval length is \(4\pi\), so the angle advances through two full turns. At successive quarter-turn angles, substitution gives:

\begin{aligned} \theta&=0: &r&=2(0)=0,\\ \theta&=\frac{\pi}{2}: &r&=2\left(\frac{\pi}{2}\right)=\pi,\\ \theta&=\pi: &r&=2\pi,\\ \theta&=\frac{3\pi}{2}: &r&=3\pi,\\ \theta&=2\pi: &r&=4\pi,\\ \theta&=\frac{5\pi}{2}: &r&=5\pi,\\ \theta&=3\pi: &r&=6\pi,\\ \theta&=\frac{7\pi}{2}: &r&=7\pi,\\ \theta&=4\pi: &r&=8\pi. \end{aligned}

All listed radii are nonnegative. As \(\theta\) increases, the spiral winds counterclockwise and moves outward from the pole. For example, the points at \(\theta=0\) and \(\theta=2\pi\) lie on the positive horizontal ray, with radii \(0\) and \(4\pi\). The points at \(\theta=2\pi\) and \(\theta=4\pi\) are also on that ray, with radii \(4\pi\) and \(8\pi\). Each full turn therefore adds \(4\pi\) to the radius.

Check the result directly: for any \(\theta\), \(r(\theta+2\pi)=2(\theta+2\pi)=2\theta+4\pi=r(\theta)+4\pi\). This agrees with the values in the table and verifies the full-turn change.

Worked Example: Reading the Turn Spacing of \(r=1+\frac{\theta}{2}\)

For \(r=1+\frac{\theta}{2}\) on \(0\leq\theta\leq4\pi\), find the radii at each full turn and describe the graph.

The interval covers two full turns. Substituting the angles \(0,2\pi,4\pi\) gives:

\begin{aligned} r(0)&=1+\frac{0}{2}=1,\\ r(2\pi)&=1+\frac{2\pi}{2}=1+\pi,\\ r(4\pi)&=1+\frac{4\pi}{2}=1+2\pi. \end{aligned}

At these three angles, the plotted points lie on the positive horizontal ray because each angle is an integer multiple of \(2\pi\) and each radius is positive. Each turn increases the radius by \(\pi\): \((1+\pi)-1=\pi\), and \((1+2\pi)-(1+\pi)=\pi\). More generally, \(b=\frac12\), so \(2\pi b=\pi\), as expected. Since the radius stays positive on this interval and increases with \(\theta\), the spiral winds counterclockwise and outward.

Check the full-turn calculation by substitution: \(r(\theta+2\pi)=1+\frac{\theta+2\pi}{2}=1+\frac{\theta}{2}+\pi=r(\theta)+\pi\). This confirms the same radial increment for both turns in the specified interval.

Worked Example: Interpreting Negative Radius in \(r=1-\frac{\theta}{\pi}\)

Trace \(r=1-\frac{\theta}{\pi}\) for \(0\leq\theta\leq2\pi\), including the point where the radius changes sign.

Set the radius equal to zero to find where the curve reaches the pole: \(1-\frac{\theta}{\pi}=0\), so \(\theta=\pi\). Substituting several angles gives:

\begin{aligned} r(0)&=1,\\ r\left(\frac{\pi}{2}\right)&=1-\frac{1}{2}=\frac12,\\ r(\pi)&=1-1=0,\\ r\left(\frac{3\pi}{2}\right)&=1-\frac32=-\frac12,\\ r(2\pi)&=1-2=-1. \end{aligned}

Before \(\theta=\pi\), the radius is positive and decreases from \(1\) to \(0\), so the point approaches the pole while its angle increases counterclockwise. At \(\theta=\pi\), the point is at the pole. After that, the radius is negative. For instance, at \(\theta=\frac{3\pi}{2}\), the plotted point has distance \(\frac12\) from the pole and lies on the ray with angle \(\frac{3\pi}{2}+\pi=\frac{5\pi}{2}\), equivalent to \(\frac{\pi}{2}\). At \(\theta=2\pi\), the radius is \(-1\), so the point is one unit from the pole on the negative horizontal ray.

The negative radius does not mean a negative distance. The distance at \(\theta=2\pi\) is \(|-1|=1\). Check the zero and sign change by substituting \(\theta=\pi\) into the equation: \(r=1-\frac{\pi}{\pi}=0\); for \(\theta=\frac{3\pi}{2}\), substitution gives \(r=1-\frac32=-\frac12\), as used in locating the point.

A Common Plotting Mistake

A common error is to treat a negative radius as a negative distance along the ray \(\theta\). For example, at \(\theta=\frac{3\pi}{2}\), a radius of \(-\frac12\) is sometimes incorrectly plotted below the pole on the ray \(\frac{3\pi}{2}\). That would place the point at Cartesian coordinates \((0,-\frac12)\), but those coordinates correspond to positive radius \(\frac12\) at angle \(\frac{3\pi}{2}\), not negative radius.

The correct plotting rule is to reverse the ray and use the positive distance \(|r|\). Thus \(\left(-\frac12,\frac{3\pi}{2}\right)\) is plotted at radius \(\frac12\) and angle \(\frac{3\pi}{2}+\pi=\frac{5\pi}{2}\), which is equivalent to \(\frac{\pi}{2}\). Its Cartesian coordinates are \((0,\frac12)\), as the conversion formulas confirm:

x=r\cos\theta =\left(-\frac12\right)\cos\left(\frac{3\pi}{2}\right)=0, \qquad y=r\sin\theta =\left(-\frac12\right)\sin\left(\frac{3\pi}{2}\right)=\frac12.

Reading a Spiral on a Specified Interval

A reliable sketch starts with the interval, not with an assumption about what the whole curve looks like. The following checks organize the task:

Count the angular sweep.
Compare the interval length with \(2\pi\) to see how many full turns, or partial turns, are traced.

Find the radius at the endpoints.
Substitute both endpoint angles into the equation to identify the signed radii at the start and finish.

Locate any passage through the pole.
Solve \(r=0\) and check whether each solution lies in the interval.

Plot key angles and interpret signs.
Evaluate the radius at convenient angles, then plot a negative radius on the opposite ray using distance \(|r|\).

Check turn-to-turn change.
Use \(r(\theta+2\pi)-r(\theta)\), and call its magnitude same-ray spacing only when both radii have the same sign.

These steps help distinguish three related but separate questions: which way the point is traced, whether its distance from the pole is increasing or decreasing, and how far apart corresponding parts of successive turns lie. A sketch is more reliable when each question is answered from the parameter values rather than from the spiral’s appearance alone.

Check Your Understanding

Use substitution and the sign of the radius to support each answer.

For \(r=3\theta\), what is the signed-radius change over one full turn?
For \(r=2-\theta\), at what angle does the curve pass through the pole?
For \(r=1+\frac{\theta}{4}\), compare the radii at \(\theta=\pi\) and \(\theta=3\pi\). What is their difference?
Where should the point \(\left(-2,\frac{\pi}{3}\right)\) be plotted using a nonnegative radius?
Why does \(r(\theta+2\pi)-r(\theta)\) not always give the distance between points on the same ray?

What You'll Learn