Circles Centered at the Pole

What Does a Constant Polar Radius Trace?

A polar equation gives a condition on the coordinates \(r\) and \(\theta\). When the equation is \(r=a\), the radius is fixed at the constant \(a\), while the angle \(\theta\) is free to vary. As the angle varies, the point moves around the pole. The resulting curve is a circle centered at the pole.

The circle’s radius is not always the signed value written in the equation. A geometric radius is nonnegative, so the circle described by \(r=a\) has radius \(|a|\). In particular, \(r=-a\) also describes a circle of radius \(a\) when \(a>0\).

Why the Locus Is a Circle

Polar coordinates and Cartesian coordinates are related by \(x=r\cos\theta\) and \(y=r\sin\theta\). If \(r=a\), then every point on the locus satisfies

The identity \(\cos^2\theta+\sin^2\theta=1\) shows that these points are all at distance \(\sqrt{a^2}=|a|\) from the origin. That is the defining distance property of a circle centered at the pole.

For \(a\ne 0\), the equation describes the entire circle, not just some of its points. To see this, take any point on the circle \(x^2+y^2=a^2\). It has distance \(|a|\) from the pole, so it can be represented with polar radius \(|a|\) and an angle pointing toward it. If \(a>0\), that representation has \(r=a\). If \(a<0\), use the opposite angle and \(r=a\): a negative radius places the point in the direction opposite to \(\theta\). Thus every point on the circle is represented by \(r=a\).

Sketching a Circle from \(r=a\)

To sketch a constant-radius polar equation, first identify the circle’s center and radius. The center is the pole, and the radius is the absolute value of the constant. Then plot a few points at familiar angles. For \(r=a\), the angles \(0\), \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\) give points on the positive horizontal, positive vertical, negative horizontal, and negative vertical axes when \(a>0\). Join them with a circle.

When the constant is negative, the points associated with those same angles appear on the opposite sides of the pole. But as \(\theta\) varies through a full turn, the point still goes all the way around the same circle. The negative sign changes which angle represents a given point; it does not make the geometric radius negative.

What Changes When the Radius Is Negative?

A negative polar radius means that the point is placed opposite the direction indicated by its angle. The coordinate rule makes this precise: \((r,\theta)\) and \((-r,\theta+\pi)\) represent the same point, because changing the sign of \(r\) and turning the angle by \(\pi\) cancel each other’s effects.

For example, \(r=-a\) with \(a>0\) gives the same circle as \(r=a\). As \(\theta\) ranges over all angles, the negative-radius point at angle \(\theta\) is the positive-radius point at angle \(\theta+\pi\). Since the angle is unrestricted, both equations describe the same full set of points.

This equivalence is about the whole locus. At a particular fixed angle, the points described by \(r=-a\) and \(r=a\) need not be the same. For instance, at \(\theta=0\), \(r=-a\) gives the point \((-a,0)\), whereas \(r=a\) gives \((a,0)\). The curves coincide because their angles vary freely, not because the two equations give identical points at every matching angle.

Repeated Representations of Points

Polar coordinates are not unique. A point can be represented by angles that differ by whole turns, since adding \(2\pi\) does not change the direction. A point can also be represented using the opposite signed radius and an angle shifted by \(\pi\). Therefore, points on these circles generally have many polar-coordinate representations.

For a circle with positive radius \(a\), a point in direction \(\phi\) can be represented as \((a,\phi+2k\pi)\) for any integer \(k\). The same point can also be represented as \((-a,\phi+\pi+2k\pi)\). These are different coordinate pairs for the same geometric point.

There is a related distinction between a curve and a tracing of the curve. If \(\theta\) runs from \(0\) to \(2\pi\), the circle is traced once, with the starting point repeated at the endpoint. If \(\theta\) is allowed to range over all real numbers, the equation provides infinitely many representations of points on the same circle. Repeated coordinates do not create additional circles.

A Common Misinterpretation

A common mistake is to interpret a negative \(r\) as an impossible radius, or to change its sign without changing the angle. For example, at \(\theta=0\), the equation \(r=-3\) gives

It does not give \((3,0)\). The equivalent positive-radius representation is \((3,\pi)\), since \((3\cos\pi,3\sin\pi)=(-3,0)\). The correction is to remember that changing the sign of the radius requires shifting the angle by \(\pi\). For the complete curve, the negative constant still gives a circle of radius \(3\), because the angle varies through all directions.

When This Recognition Is Useful

Recognizing \(r=a\) immediately identifies a simple polar graph: its center is the pole and its radius is the magnitude of the constant. This avoids treating each angle as a separate point or mistaking a negative radius for a negative geometric length. It also helps distinguish the curve itself from the many coordinate pairs that can represent points on it.

For a quick sketch, identify \(|a|\), mark the four axis points, and draw the circle. If the constant is zero, mark only the pole. If the constant is negative, use the same radius \(|a|\); the resulting locus is still the full circle because the angle is free to vary.