What a Negative Radius Means
A polar coordinate \((r,\theta)\) gives a direction, \(\theta\), and a signed distance, \(r\), from the pole. When \(r\) is positive, the point lies \(r\) units from the pole in the direction of angle \(\theta\). When \(r\) is negative, the point lies \(|r|\) units from the pole in the opposite direction.
Thus a negative radius does not mean “move left” regardless of the angle. It means reverse the direction specified by the angle. The same point can consequently be described either with a negative radius or with a positive radius and an angle shifted by \(\pi\).
Why Reversing the Angle Works
The coordinate formulas \(x=r\cos\theta\) and \(y=r\sin\theta\) show exactly how the sign change works. If \(r=-\rho\), where \(\rho>0\), then
The trigonometric identities \(\cos(\theta+\pi)=-\cos\theta\) and \(\sin(\theta+\pi)=-\sin\theta\) give
These are the Cartesian coordinates of \((\rho,\theta+\pi)\). Since both polar coordinates produce the same \(x\) and \(y\), they locate the same point. This proves the conversion rule, rather than relying only on a sketch.
Plotting a Negative-Radius Point
A reliable sketching method is to first mark the ray at the given angle, then reverse direction along the same line and measure the positive distance \(|r|\). Another method is to convert to a positive radius by adding \(\pi\) to the angle. Both methods should produce the same location.
Worked Example: Plotting a Negative Radius in the First Quadrant Direction
Plot \((-4,\pi/3)\). The angle \(\pi/3\) is \(60^\circ\), so the specified ray points into the first quadrant. Because the radius is negative, the point is four units in the opposite direction. Convert to a positive radius:
The direction \(4\pi/3\) is in the third quadrant. To check the plotted location, calculate its Cartesian coordinates from the original representation:
The converted representation gives the same result:
Both checks place the point at \((-2,-2\sqrt3)\), in the third quadrant, four units from the pole.
Finding All Representations in an Angle Range
To find every representation of a nonzero point, first identify its direction using a positive radius. Call that direction \(\alpha\), and let the distance from the pole be \(\rho>0\). The equivalent-coordinate rules give two families: positive-radius representations have angle \(\alpha+2\pi k\), while negative-radius representations have angle \(\alpha+(2k+1)\pi\), where \(k\) is an integer. These are the equivalent-coordinate rules applied to the point’s direction and its opposite.
For a specified interval, keep exactly the angles from these families that lie in the interval. This amounts to testing the interval bounds; it does not require guessing how many representations there will be. If the interval includes both endpoints of a full turn, be attentive to the fact that those endpoints describe the same direction.
Convert the given coordinate to a point with radius \(\rho>0\), and call its direction \(\alpha\).
Use angles \(\alpha+2\pi k\) for radius \(\rho\), and \(\alpha+(2k+1)\pi\) for radius \(-\rho\).
Retain each integer \(k\) for which the resulting angle is in the required interval.
Confirm that each candidate gives the same Cartesian coordinates as the original point.
Worked Example: All Representations with Angles from Zero to Two Pi
Find all representations of \((-5,7\pi/6)\) whose angles lie in \([0,2\pi)\). First convert to positive radius. Adding \(\pi\) to the angle gives \(13\pi/6\), which differs from \(\pi/6\) by a full turn. Thus the point has positive-radius direction \(\alpha=\pi/6\) and distance \(5\).
The positive-radius family is \((5,\pi/6+2\pi k)\). The angle is in \([0,2\pi)\) only for \(k=0\), giving \((5,\pi/6)\). The negative-radius family is \((-5,\pi/6+(2k+1)\pi)\). For \(k=0\), this gives angle \(7\pi/6\), which is in the interval. The neighboring choices give \(-5\pi/6\) and \(19\pi/6\), which are outside it. Therefore the complete list is
Check the Cartesian coordinates. The first representation gives
The second gives
The coordinates agree, and both listed angles satisfy the specified range.
Worked Example: A Different Angle Interval
Find all representations of \((2,\pi/4)\) whose angles lie in \([-\pi,\pi]\). Here \(\rho=2\) and \(\alpha=\pi/4\). For positive radius, the angles are \(\pi/4+2\pi k\). Only \(k=0\) falls in the interval. For negative radius, the angles are \(\pi/4+(2k+1)\pi\). The choice \(k=-1\) gives \(-3\pi/4\); the neighboring choices lie outside the interval. The full list is
For \((2,\pi/4)\), the Cartesian coordinates are
For \((-2,-3\pi/4)\), they are
The representations agree, and both angles lie in \([-\pi,\pi]\).
The Pole Is a Special Case
The families above assume \(\rho>0\). At the pole, the radius is zero, so the angle does not change the location. In fact, \((0,\theta)\) represents the pole for every angle \(\theta\). A restricted angle interval can limit which coordinates are allowed, but a continuous interval still gives infinitely many representations.
Worked Example: Representations of the Pole
List the representations of the pole whose angles lie in \([0,2\pi)\). For every allowed angle, the radius must be zero, so the list is
For any such \(\theta\), the Cartesian check gives
Thus every listed coordinate describes the pole. There is no finite list containing all of them, even though the angle is restricted to one full turn.
A Common Plotting Mistake
A frequent error is to treat a negative radius as a positive distance along the stated angle. For instance, this incorrect calculation plots \((-3,\pi/6)\) as though it were \((3,\pi/6)\), giving \((3\sqrt3/2,3/2)\). That point lies in the first quadrant and is not the requested point.
The correction is to reverse the direction, or to retain the negative sign in the coordinate calculation:
The point is in the third quadrant. Equivalently, \((-3,\pi/6)=(3,7\pi/6)\), whose positive-radius direction also lies in the third quadrant. The sign of the radius reverses both coordinate components; it does not change just one of them.
Check Your Understanding
Use the direction-reversal rule and check your answers by reasoning about the corresponding point.
- Rewrite \((-6,\pi/4)\) using a positive radius and an angle between \(0\) and \(2\pi\).
- Plot \((-2,5\pi/6)\). In which quadrant does the point lie?
- Find all representations of \((3,\pi/3)\) whose angles are in \([0,2\pi)\).
- Find all representations of \((4,5\pi/4)\) whose angles are in \([-\pi,\pi]\).
- Why can the pole not be described by a finite list of all its polar coordinates when every angle in \([0,2\pi)\) is allowed?