Finding Fundamental Periods of Polar Curves

Two Different Questions About Period

A polar equation \(r=r(\theta)\) assigns a signed radius to each angle. If the radius function repeats after an angle change \(T\), the equation has a period \(T\). But the plotted curve is a set of points, and it can have geometric symmetries or repeated parametrized points that do not match the periods of \(r\).

These distinctions matter because polar coordinates can represent the same point in more than one way. For example, \((r,\theta)\) and \((-r,\theta+\pi)\) represent the same Cartesian point. We will use this fact to separate the algebraic repetition of the equation from the geometry of its trace.

For a polar equation involving sine and cosine of integer multiples of \(\theta\), the fundamental period can be found from the frequencies that remain after like terms are combined. A constant term does not affect the period. Cancellations do matter: if the combined sine and cosine coefficients at some frequency are both zero, that frequency is absent.

Finding the Fundamental Period

Consider a finite trigonometric expression with positive integer frequencies. For each frequency, combine every cosine term and every sine term having that frequency. A frequency is active precisely when at least one of its combined coefficients is nonzero.

For instance, frequencies \(8\) and \(12\) have greatest common divisor \(4\), so a function containing those as its only active frequencies has fundamental period \(2\pi/4=\pi/2\). A constant term, such as \(5\), changes the values of the function but not its periods.

Why the Greatest Common Divisor Gives the Answer

Write the combined nonconstant part using complex exponentials. At each active positive frequency \(n\), the cosine and sine terms together have the form \(c_n e^{in\theta}+\overline{c_n}e^{-in\theta}\), where \(c_n\ne0\). This representation packages the two real coefficients at that frequency into one complex coefficient.

If \(T\) is a period, then subtracting \(r(\theta)\) from \(r(\theta+T)\) gives an identically zero trigonometric polynomial. In that difference, the coefficient of \(e^{in\theta}\) is \(c_n(e^{inT}-1)\). Distinct exponential frequencies are linearly independent: multiplying the zero expression by \(e^{-ik\theta}\) and integrating from \(0\) to \(2\pi\) isolates its coefficient at frequency \(k\), because the integral of \(e^{i(n-k)\theta}\) over this interval is zero for \(n\ne k\), and is \(2\pi\) for \(n=k\). Therefore each active frequency must satisfy \(e^{inT}=1\), or equivalently \(nT\) is an integer multiple of \(2\pi\).

Now let \(d\) be the greatest common divisor of the active frequencies. Since \(d\) divides every active \(n\), choosing \(T=2\pi/d\) makes \(nT=2\pi(n/d)\), an integer multiple of \(2\pi\). Thus \(2\pi/d\) is a period.

To see that no smaller positive period is possible, suppose \(T\) is any period and write \(nT=2\pi k_n\) for integers \(k_n\). The greatest common divisor \(d\) can be expressed as an integer combination of the active frequencies: \(d=\sum_n q_n n\) for integers \(q_n\). Multiplying by \(T/(2\pi)\) gives \(dT/(2\pi)=\sum_n q_n k_n\), an integer. Hence \(T\) is a positive integer multiple of \(2\pi/d\). The smallest positive choice is exactly \(2\pi/d\), proving the theorem.

Worked Examples

Equation Periods and the Plotted Curve

The fundamental period just computed belongs to the scalar function \(r\). It does not automatically give the smallest angle of rotational symmetry of the plotted point set, nor does it tell us when the parametrization returns to exactly the same point. These are related but different questions.

The point traced at parameter \(\theta\) has Cartesian coordinates

If \(r\) has period \(T\), then \(r(\theta+T)=r(\theta)\), but the direction angle has also changed by \(T\). Thus the point at \(\theta+T\) is generally a rotation of the point at \(\theta\), not the same point. In particular, an equation period often indicates a rotational symmetry of the plotted set, rather than a return of each parameter value to its original point.

A different distinction concerns the parametrized point itself. For \(r(\theta)=\cos\theta\), the equation has fundamental period \(2\pi\), since its sole active frequency is \(1\). But

Here the parametrization returns to the same point after a shift of \(\pi\), even though the radius function does not repeat after \(\pi\). The curve is the circle \(x^2+y^2=x\), as follows from \(x=\cos^2\theta\), \(y=\cos\theta\sin\theta\), and \(x^2+y^2=\cos^2\theta\). That circle is not invariant under rotation about the pole: its center is \((1/2,0)\), not the pole. So even a repeated parametrized point and a rotational symmetry of the point set are not interchangeable ideas.

A Common Mistake: Counting Terms Instead of Frequencies

A tempting but incorrect procedure is to take the greatest common divisor of every frequency that appears in the written expression, without first combining repeated terms. For example, the written expression \(\cos\theta-\cos\theta\) seems to contain frequency \(1\), and counting it would suggest a fundamental period \(2\pi\). But the terms cancel exactly, leaving \(r(\theta)=0\). Every positive number is a period, so no fundamental period exists.

The correction is straightforward: collect the sine and cosine coefficients at each frequency, add them, and only then identify the active frequencies. A frequency is included if and only if the combined coefficient pair is not \((0,0)\). The same check prevents a constant function from being assigned a false fundamental period.

Using the Result Reliably

This procedure answers a precise algebraic question about the radius function. To study a plotted curve, also ask what kind of repetition is meant: an equation period, an exact return of the parametrized point, or a rotation that carries the plotted point set onto itself. These statements need not have the same smallest positive angle.