Polar Coordinates

Introduction to Polar Coordinates

In analytic geometry, you first meet the Cartesian (rectangular) coordinate system, where a point in the plane is described by an ordered pair $(x,y)$. Polar coordinates provide a different way to locate points: instead of horizontal and vertical distances, we use a distance from the origin and an angle.

Polar coordinates are especially useful for describing circles, spirals, and many curves that look complicated in Cartesian coordinates but simple in polar form.

The Polar Coordinate System

A point in the plane can be described by polar coordinates $(r,\theta)$, where:

• $r$ is the radial coordinate (or radius): the distance from the point to the origin.
• $\theta$ is the angular coordinate (or argument): the angle, measured from the positive $x$-axis (the “polar axis”) to the line segment connecting the origin to the point.

Unless otherwise specified, $\theta$ is measured in radians and:

• Positive $\theta$ corresponds to counterclockwise rotation.
• Negative $\theta$ corresponds to clockwise rotation.

You can imagine starting at the origin, facing along the positive $x$-axis:

1. Rotate by $\theta$.
2. Move straight out from the origin a distance $r$.

The point where you arrive is the point with polar coordinates $(r,\theta)$.

Converting Between Polar and Cartesian Coordinates

Because polar and Cartesian coordinates describe the same plane, there are standard formulas for converting between them.

From Polar to Cartesian

Given a point in polar form $(r,\theta)$, its Cartesian coordinates $(x,y)$ are
$$
x = r\cos\theta, \qquad y = r\sin\theta.
$$

This comes directly from viewing $x$ and $y$ as the legs of a right triangle with hypotenuse $r$ and angle $\theta$ from the $x$-axis.

Example: Convert $(r,\theta) = (4,\tfrac{\pi}{3})$ to Cartesian.
$$
x = 4\cos\frac{\pi}{3} = 4\cdot\frac{1}{2} = 2, \qquad
y = 4\sin\frac{\pi}{3} = 4\cdot\frac{\sqrt{3}}{2} = 2\sqrt{3}.
$$
So $(4,\tfrac{\pi}{3})$ in polar form is $(2,\,2\sqrt{3})$ in Cartesian form.

From Cartesian to Polar

Given $(x,y)$ with $(x,y)\neq (0,0)$, the radial coordinate $r$ and angle $\theta$ satisfy:
$$
r = \sqrt{x^2 + y^2},
$$
and $\theta$ is an angle whose cosine and sine match $x/r$ and $y/r$:
$$
\cos\theta = \frac{x}{r}, \qquad \sin\theta = \frac{y}{r}.
$$

In practice, you often determine $\theta$ using an inverse tangent, but you must be careful about the quadrant:

• A common formula is
  $$
  \theta = \arctan\left(\frac{y}{x}\right),
  $$
  but this only gives an angle in the correct quadrant if you adjust based on the signs of $x$ and $y$.
• In many computing contexts, a special function like atan2(y,x) is used; it automatically chooses the correct quadrant.

Example: Convert $(x,y) = (-3,3)$ to polar coordinates.

First find $r$:
$$
r = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}.
$$
Next, find $\theta$.
We know
$$
\frac{y}{x} = \frac{3}{-3} = -1.
$$
An angle with tangent $-1$ is $-\tfrac{\pi}{4}$ or $\tfrac{3\pi}{4}$, etc. Since $(x,y)=(-3,3)$ is in the second quadrant, we take
$$
\theta = \frac{3\pi}{4}.
$$
So one polar description is $(3\sqrt{2},\,\tfrac{3\pi}{4})$.

Non-Uniqueness of Polar Coordinates

Unlike Cartesian coordinates, polar coordinates for a given point are not unique.

Multiple Angles

If $(r,\theta)$ represents a point, then so does $(r,\theta + 2k\pi)$ for any integer $k$, because adding $2\pi$ corresponds to a full revolution.

Example: The points
$$
(5,\tfrac{\pi}{6}),\ (5,\tfrac{\pi}{6} + 2\pi),\ (5,\tfrac{\pi}{6} - 4\pi)
$$
all represent the same physical point in the plane.

Negative Radius

You can also use a negative value of $r$ to describe the same point with a different angle. The pair $(-r,\theta)$ corresponds to the same point as $(r,\theta + \pi)$. Moving a distance $-r$ in the direction of $\theta$ is equivalent to moving a distance $r$ in the opposite direction, which is $\theta+\pi$.

In symbols:
$$
(r,\theta) \equiv (-r,\theta+\pi)
$$
for $r>0$.

Example: The point $(2,\tfrac{\pi}{3})$ is the same as $(-2,\tfrac{\pi}{3}+\pi) = (-2,\tfrac{4\pi}{3})$.

Because of this non-uniqueness, when working with polar coordinates you often restrict to a “principal” range for $\theta$ (such as $0 \le \theta < 2\pi$ or $-\pi < \theta \le \pi$) and require $r\ge 0$. This gives each point a more or less standard representation, except for the origin.

The Origin in Polar Coordinates

At the origin, $r=0$. The angle $\theta$ becomes irrelevant, because all angles with $r=0$ represent the same point. Thus
$$
(0,\theta_1) \equiv (0,\theta_2)
$$
for any angles $\theta_1,\theta_2$.

Often we simply write the origin as $(0,\,\text{any }\theta)$ or just refer to it as the origin without specifying $\theta$.

Interpreting Polar Equations

A key use of polar coordinates is describing curves by equations relating $r$ and $\theta$.

A polar equation is an equation of the form
$$
r = f(\theta)
$$
or more generally some relation $F(r,\theta)=0$. The graph is the set of all points $(r,\theta)$ satisfying the equation, interpreted in the plane.

To sketch the graph of $r = f(\theta)$:

1. Choose values of $\theta$ in a suitable interval (often $0$ to $2\pi$).
2. Compute the corresponding $r$ values.
3. Plot the points described by $(r,\theta)$, remembering that $r$ can be negative.
4. Connect them smoothly when appropriate.

Converting Polar Equations to Cartesian Form

Sometimes you want to rewrite a polar equation using $x$ and $y$. The basic relationships are
$$
x = r\cos\theta,\quad
y = r\sin\theta,\quad
r^2 = x^2 + y^2,\quad
\tan\theta = \frac{y}{x}\ (\text{when }x\neq 0).
$$

You use these to eliminate $r$ and $\theta$.

Example: Convert $r = 4\cos\theta$ to Cartesian form.

Multiply both sides by $r$:
$$
r^2 = 4r\cos\theta.
$$
Now use $r^2 = x^2 + y^2$ and $r\cos\theta = x$:
$$
x^2 + y^2 = 4x.
$$
This is the Cartesian equation of a circle.

Example: Convert $r = 2$ to Cartesian form.

Again, multiply by $r$:
$$
r^2 = 2r \quad\Rightarrow\quad x^2 + y^2 = 2r.
$$
We still have $r$ on the right. But note that $r = \sqrt{x^2 + y^2}$, so:
$$
x^2 + y^2 = 2\sqrt{x^2 + y^2}.
$$
If $x^2 + y^2 \neq 0$, divide both sides by $\sqrt{x^2 + y^2}$:
$$
\sqrt{x^2 + y^2} = 2.
$$
Squaring:
$$
x^2 + y^2 = 4.
$$
This is a circle of radius $2$ centered at the origin. (The origin itself $x^2+y^2=0$ also satisfies $r=0$, which does not satisfy $r=2$, so the origin is not included.)

Converting Cartesian Equations to Polar Form

To go the other way, you substitute $x = r\cos\theta$, $y = r\sin\theta$, and $r^2=x^2+y^2$ into a Cartesian equation and solve for $r$ (or sometimes for $\theta$).

Example: Convert $x^2 + y^2 = 9$ to polar form.

Use $x^2 + y^2 = r^2$:
$$
r^2 = 9 \quad\Rightarrow\quad r = 3\ \text{or}\ r=-3.
$$
With the convention $r\ge 0$, we typically write simply $r = 3$.

Example: Convert $y = x$ to polar form.

Substitute $x = r\cos\theta$ and $y = r\sin\theta$:
$$
r\sin\theta = r\cos\theta.
$$
If $r\neq 0$, divide both sides by $r$:
$$
\sin\theta = \cos\theta \quad\Rightarrow\quad \tan\theta = 1.
$$
So $\theta = \tfrac{\pi}{4} + k\pi$ for any integer $k$. This represents a family of lines through the origin, which together form the same geometric line $y=x$.

Typical Polar Curves

Many curves have simple polar equations and distinctive shapes. Here are a few common types, focusing on how to recognize the form.

Circles Centered on the Origin

The circle of radius $a$ centered at the origin has polar equation
$$
r = a.
$$

Any point on this circle is at distance $a$ from the origin, regardless of $\theta$.

Circles Offset from the Origin

Circles not centered at the origin often have equations of the form
$$
r = a\cos\theta \quad\text{or}\quad r = a\sin\theta
$$
or more generally
$$
r = a\cos(\theta - \theta_0),
$$
with $a>0$. These represent circles whose centers lie on a ray from the origin at angle $\theta_0$.

For instance, $r = 4\cos\theta$ describes a circle of radius $2$ centered at $(2,0)$ in Cartesian coordinates.

Lines Through the Origin

Any line through the origin can be written as
$$
\theta = \theta_0
$$
for some fixed angle $\theta_0$. The set of points is all $(r,\theta_0)$ with $r$ any real number.

In Cartesian form, this corresponds to a line of the form $y = (\tan\theta_0)\,x$ (except when $\theta_0=\tfrac{\pi}{2} + k\pi$, in which case the line is vertical: $x=0$).

Roses (Rhodonea Curves)

Curves of the form
$$
r = a\cos(n\theta)\quad\text{or}\quad r = a\sin(n\theta)
$$
often look like flowers with “petals” and are called rose curves.

• If $n$ is an odd positive integer, the curve has $n$ petals.
• If $n$ is even, it has $2n$ petals.
• The parameter $a$ controls the length of each petal.

Example: $r = 3\cos(2\theta)$ has $4$ petals of length $3$.

Spirals

The simplest spiral in polar coordinates is the Archimedean spiral:
$$
r = a\theta,
$$
with $a>0$. As $\theta$ increases, $r$ increases linearly, so the curve winds outward from the origin.

Other spirals have different functional relationships between $r$ and $\theta$, but the idea is always that $r$ changes as $\theta$ increases, causing the curve to wind around the origin.

Plotting Polar Curves by Hand

To sketch a polar curve:

1. Choose an interval for $\theta$. For many curves, $0\le\theta\le 2\pi$ is sufficient, but some periodicity may allow a smaller interval.
2. Make a table of sample $\theta$ values and corresponding $r$ values.
3. For each $(r,\theta)$:
• If $r>0$, plot the point at distance $r$ along angle $\theta$.
• If $r<0$, plot the point at distance $|r|$ along angle $\theta+\pi$.
4. Connect the plotted points smoothly, following the order of increasing $\theta$.

Pay particular attention to:

• Where $r=0$ (intersections with the origin).
• Symmetries (discussed briefly below), which can reduce the amount of plotting needed.

Symmetry in Polar Curves

Simple checks can reveal symmetry, making graphs easier:

• Symmetry about the polar axis (the $x$-axis in Cartesian terms):
  If replacing $\theta$ by $-\theta$ leaves the equation unchanged, the curve is symmetric about the $x$-axis.
• Symmetry about the vertical line $\theta=\tfrac{\pi}{2}$ (the $y$-axis):
  If replacing $\theta$ by $\pi - \theta$ leaves the equation unchanged, the curve is symmetric about the $y$-axis.
• Symmetry about the origin:
  If replacing $(r,\theta)$ by $(-r,\theta)$ or equivalently $(r,\theta)$ by $(r,\theta+\pi)$ leaves the equation unchanged, the curve is symmetric about the origin.

These tests are not exhaustive, but they are often helpful for common polar equations.

Distance and Angle Between Points in Polar Form

When points are given in polar form, you can still compute geometric quantities like distance and angle between them, either by converting to Cartesian or by using formulas.

Suppose you have points $P_1=(r_1,\theta_1)$ and $P_2=(r_2,\theta_2)$.

Distance Between Two Polar Points

The distance $d$ between $P_1$ and $P_2$ satisfies
$$
d^2 = r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_1-\theta_2),
$$
so
$$
d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_1-\theta_2)}.
$$

This follows from the law of cosines applied to the triangle connecting the origin and the two points.

Angle Between Radius Vectors

The angle formed at the origin between the lines to $P_1$ and $P_2$ is simply
$$

$$
(modulo $2\pi$), reduced to an appropriate range if needed (for example, $0$ to $\pi$ for the smallest angle).

When Polar Coordinates Are Useful

Polar coordinates are particularly natural in problems where:

• There is rotational symmetry around a point (commonly the origin).
• Distances from the origin and angles are more important than horizontal and vertical components.
• Curves like circles, spirals, and roses appear.

In more advanced topics, polar coordinates also play a role in integration, differential equations, and complex numbers, but those uses are handled in their own chapters.