Angle Periodicity and Coordinate Aliases

One Angle, Several Coordinate Pairs

A point in the plane can be described using many different polar coordinates. The simplest source of multiple descriptions is that a full turn brings the direction back to itself: angles that differ by \(2\pi\) point in the same direction. A second source is the use of a negative radius, which reverses the direction by half a turn.

These facts concern coordinate representations of a point. A polar equation has a separate notion of periodicity: its radial value may repeat after an angle change that is not \(2\pi\). Keeping those ideas separate helps prevent a repeated value of \(r\) from being mistaken for a repeated point.

Angle Aliases for a Point

In polar coordinates, a pair \((r,\theta)\) represents the Cartesian point \((x,y)=(r\cos\theta,r\sin\theta)\). Here \(r\) is a signed radius and \(\theta\) is an angle. The identities \(\cos(\theta+2\pi k)=\cos\theta\) and \(\sin(\theta+2\pi k)=\sin\theta\), for any integer \(k\), show why adding full turns leaves the point unchanged.

Definition: For a nonzero point represented by \((r,\theta)\), the pairs \((r,\theta+2\pi k)\), where \(k\in\mathbb Z\), are angle aliases: they represent the same point. The same point can also be represented by \((-r,\theta+(2k+1)\pi)\). The origin is a special case: it is represented by \((0,\theta)\) for every angle \(\theta\).

The negative-radius rule follows because changing the sign of \(r\) reverses both Cartesian coordinates, while adding \(\pi\) to the angle also reverses both cosine and sine. Thus the two changes cancel each other. More generally, for a nonzero point, its signed radius and angle may be changed together in these ways without changing its location.

(-r)\cos(\theta+\pi)=(-r)(-\cos\theta)=r\cos\theta, \qquad (-r)\sin(\theta+\pi)=(-r)(-\sin\theta)=r\sin\theta.

Worked Example: Listing Aliases of a Point

Find two other polar coordinate pairs for the point \((3,\pi/6)\), and verify that all three pairs represent the same Cartesian point.

First convert the given pair to Cartesian coordinates. Since \(\cos(\pi/6)=\sqrt{3}/2\) and \(\sin(\pi/6)=1/2\),

x=3\cos(\pi/6)=\frac{3\sqrt{3}}{2}, \qquad y=3\sin(\pi/6)=\frac{3}{2}.

Adding \(2\pi\) to the angle gives the alias \((3,\pi/6+2\pi)=(3,13\pi/6)\). Its Cartesian coordinates are

x=3\cos(13\pi/6)=\frac{3\sqrt{3}}{2}, \qquad y=3\sin(13\pi/6)=\frac{3}{2},

because \(13\pi/6=2\pi+\pi/6\). For an alias with negative radius, change the sign of the radius and add \(\pi\) to the angle. This gives \((-3,7\pi/6)\). Since \(7\pi/6=\pi+\pi/6\), its Cartesian coordinates are

x=-3\cos(7\pi/6)=-3\left(-\frac{\sqrt{3}}{2}\right)=\frac{3\sqrt{3}}{2}, \qquad y=-3\sin(7\pi/6)=-3\left(-\frac{1}{2}\right)=\frac{3}{2}.

All three coordinate pairs give the same Cartesian point, \(\left(3\sqrt{3}/2,3/2\right)\), as required.

Period of a Polar Equation

A polar equation often gives the radius as a function of the angle, written \(r=f(\theta)\). For this type of equation, the period refers to repetition in the radial value: it asks how far the angle can increase while leaving \(f(\theta)\) unchanged.

Definition: A positive number \(T\) is a period of \(f\) if \(f(\theta+T)=f(\theta)\) for every \(\theta\) in the domain (here, the real numbers). If a least positive period exists, it is called the least positive period. A period describes repetition of function values; by itself, it does not assert that the corresponding points in the plane are the same.

Angle aliases and function periods are different statements. The angle-alias rule applies to every point: the same signed radius at angles separated by \(2\pi\) gives the same point. A period \(T\) of \(f\) says that the equation assigns the same radius at angles separated by \(T\). If \(T\) is not a whole number of full turns, those equal radii will generally be placed in different directions.

Worked Example: A Polar Equation with Period π

Find the least positive period of \(r=1+\cos(2\theta)\), then compare the points produced at \(\theta=0\) and \(\theta=\pi\).

Using the \(2\pi\)-periodicity of cosine,

f(\theta+\pi) =1+\cos(2(\theta+\pi)) =1+\cos(2\theta+2\pi) =1+\cos(2\theta) =f(\theta).

Thus \(\pi\) is a period. To see it is the least positive period, suppose \(T>0\) is any period. Then \(\cos(2\theta+2T)=\cos(2\theta)\) for every \(\theta\). A shift \(s\) leaves \(\cos u\) unchanged for every \(u\) only when \(s\) is an integer multiple of \(2\pi\). Therefore \(2T=2\pi k\) for some positive integer \(k\), so \(T=k\pi\). The smallest possibility is \(T=\pi\).

Now evaluate the equation at the two angles. At \(\theta=0\),

r=1+\cos 0=2, \qquad (x,y)=(2\cos 0,2\sin 0)=(2,0).

At \(\theta=\pi\),

r=1+\cos(2\pi)=2, \qquad (x,y)=(2\cos\pi,2\sin\pi)=(-2,0).

The radius repeats after \(\pi\), but the point does not: the two Cartesian points are \((2,0)\) and \((-2,0)\). This verifies that the equation’s radial period is not automatically a period of the point’s position.

Coordinate Aliases Can Also Repeat a Point on an Equation

The reverse distinction matters too. A point may appear more than once as an equation is traced, because two different coordinate pairs satisfy the equation and represent the same point. In particular, a negative-radius alias shifts the angle by an odd multiple of \(\pi\), not by a full turn.

Worked Example: The Same Point at Two Angles

Consider \(r=\sin\theta\). First, its least positive period as a radial function is \(2\pi\): the sine function repeats after \(2\pi\), and no smaller positive shift leaves its values unchanged for every angle.

At \(\theta=\pi/2\), the equation gives

r=\sin(\pi/2)=1, \qquad (x,y)=(1\cos(\pi/2),1\sin(\pi/2))=(0,1).

At \(\theta=3\pi/2\), it gives

r=\sin(3\pi/2)=-1, \qquad (x,y)=(-1\cos(3\pi/2),-1\sin(3\pi/2))=(0,1).

Both parameter values represent \((0,1)\). Indeed, the coordinate alias of \((1,\pi/2)\) with negative radius is \((-1,3\pi/2)\), and both pairs satisfy the equation. The radial values \(1\) and \(-1\) are not equal, so this repetition of the point is not explained by a period of \(f\). It is explained by the negative-radius alias rule. Substitution into the equation and conversion to Cartesian coordinates confirm the result in both cases.

Why the Two Ideas Must Stay Separate

For a radial function \(r=f(\theta)\), a period \(T\) guarantees \(f(\theta+T)=f(\theta)\). The Cartesian coordinates at the two angles are nevertheless

\bigl(f(\theta)\cos(\theta+T),f(\theta)\sin(\theta+T)\bigr) \quad\text{and}\quad \bigl(f(\theta)\cos\theta,f(\theta)\sin\theta\bigr).

These coordinates need not agree. If \(T=2\pi\), they do agree because the direction has made a full turn. If \(T=\pi\), they are opposites when the radius is nonzero. For other periods, the direction changes by the corresponding angle. Separately, the negative-radius alias can identify two parameter values even when the radial values have opposite signs.

This distinction matters when interpreting a polar equation, checking whether an interval repeats radial values, or deciding whether two coordinate pairs mark one point or two. A function’s period is a claim about its outputs. A coordinate alias is a claim about the location represented by a pair. Neither claim should be substituted for the other.

A Common Mistake

A tempting but incorrect argument is: “Angles repeat after \(2\pi\), so every polar equation has period \(2\pi\).” The angle-alias rule does not force the radial function to repeat. For example, take \(r=\theta\). At \(\theta=0\), the equation gives \(r=0\); at \(\theta=2\pi\), it gives \(r=2\pi\). The radial values differ, so \(2\pi\) is not a period of this function.

Correction: The angle of a direction is unchanged by adding \(2\pi\), but a polar equation may assign different radii at those two angle inputs. Check the equation itself by testing whether \(f(\theta+T)=f(\theta)\) for every \(\theta\); do not infer a function period from the coordinate alias rule.

Summary

For a nonzero point, adding \(2\pi k\) to its angle while keeping its signed radius unchanged gives an angle alias. Reversing the radius sign and adding an odd multiple of \(\pi\) also gives an alias. By contrast, a period of \(r=f(\theta)\) is an angle shift that leaves the radial function value unchanged for every input. Equal radial values at different angles need not represent the same point, and different radial values can still represent the same point through a negative-radius alias.

Check Your Understanding

Use the distinction between point coordinates and radial-function values to answer these questions.

Give two angle aliases of \((2,\pi/3)\), one with positive radius and one with negative radius.
For \(r=\cos(2\theta)\), what is the least positive period of the radial function?
For \(r=\cos(2\theta)\), compare the Cartesian points obtained at \(\theta=0\) and \(\theta=\pi\). Are they the same?
In \(r=\sin\theta\), why do \(\theta=\pi/2\) and \(\theta=3\pi/2\) represent the same point even though their radial values differ?
Does the angle-alias rule imply that every polar radial function has period \(2\pi\)? Explain.