KAHIBARO
Table of Contents
Seeing Position by Distance and Angle
In Cartesian coordinates, a point is described by how far it is along perpendicular axes. In polar coordinates, a point in a plane is described in a different way, by its distance from a chosen origin and by its direction relative to a chosen reference line.
A point is written as $(r,\theta)$. Here, $r$ is the radial distance from the origin, and $\theta$ is the angle measured from a reference direction, usually the positive $x$ axis. This system is especially useful when the geometry of a problem involves circles, rotation, or radial symmetry.
For example, a point 5 units from the origin at an angle of $30^\circ$ is written as $(5,30^\circ)$, or in radians as $(5,\pi/6)$.
The Meaning of $r$ and $\theta$
The quantity $r$ tells you how far the point is from the origin. The quantity $\theta$ tells you the direction. If $r$ increases while $\theta$ stays fixed, the point moves straight outward from the origin. If $\theta$ changes while $r$ stays fixed, the point moves along a circle centered at the origin.
In many physics problems, angles are measured counterclockwise from the positive $x$ axis. Angles may be given in degrees or radians, though radians are often preferred in physics and mathematics.
A point can sometimes have more than one polar description. For example, adding a full turn to the angle gives the same direction:
$$
(r,\theta) = (r,\theta + 2\pi)
$$
Also, a negative radius can be used if the angle is shifted by $\pi$:
$$
(r,\theta) = (-r,\theta + \pi)
$$
This means polar coordinates are not unique in the same way Cartesian coordinates are.
Important facts about polar coordinates:
$$
(r,\theta) = (r,\theta + 2\pi n), \quad n \in \mathbb{Z}
$$
and also
$$
(r,\theta) = (-r,\theta + \pi)
$$
So the same physical point can have several polar coordinate representations.
Visualizing Polar Coordinates
A polar grid is made of circles centered at the origin and rays extending outward at different angles. Instead of moving left and right, up and down, you think in terms of outward distance and direction.
In this picture, the point is located by first choosing the angle $\theta$, then moving outward a distance $r$.
Relation to Cartesian Coordinates
Polar coordinates and Cartesian coordinates describe the same plane. The connection comes from basic trigonometry. If a point has polar coordinates $(r,\theta)$, then its Cartesian coordinates $(x,y)$ are
$$
x = r\cos\theta
$$
$$
y = r\sin\theta
$$
These formulas come from the right triangle formed by the radius line and its horizontal and vertical projections.
Conversely, if $(x,y)$ is known, then
$$
r = \sqrt{x^2 + y^2}
$$
and the angle is found from
$$
\tan\theta = \frac{y}{x}
$$
but care is needed because the tangent alone does not determine the correct quadrant. In practice, one uses the signs of $x$ and $y$, or a function such as $\operatorname{atan2}(y,x)$.
Key conversion formulas:
$$
x = r\cos\theta, \quad y = r\sin\theta
$$
$$
r = \sqrt{x^2 + y^2}
$$
$$
\tan\theta = \frac{y}{x}
$$
Always check the quadrant when finding $\theta$ from $x$ and $y$.
A Simple Example
Suppose a point has polar coordinates
$$
(r,\theta) = (4,60^\circ)
$$
Then
$$
x = 4\cos 60^\circ = 4\cdot \frac{1}{2} = 2
$$
$$
y = 4\sin 60^\circ = 4\cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}
$$
So the Cartesian coordinates are
$$
(x,y) = (2,2\sqrt{3})
$$
Now go the other way. If a point has Cartesian coordinates $(3,4)$, then
$$
r = \sqrt{3^2 + 4^2} = 5
$$
and
$$
\theta = \tan^{-1}\left(\frac{4}{3}\right)
$$
So the point is approximately
$$
(r,\theta) = \left(5,53.1^\circ\right)
$$
Common Angle Conventions
In physics, the angle is often measured from the positive $x$ axis in the counterclockwise direction. This is the standard mathematical convention. However, some applications may choose a different reference direction, such as north in navigation. When using polar coordinates, always state clearly where the angle starts and in which direction it increases.
Radians are often more convenient than degrees because many physical formulas involving rotation and oscillation naturally use radians. For example, one full turn is
$$
2\pi \text{ radians} = 360^\circ
$$
The following table gives some common angle values.
| Degrees | Radians |
|---|---|
| $0^\circ$ | $0$ |
| $30^\circ$ | $\pi/6$ |
| $45^\circ$ | $\pi/4$ |
| $60^\circ$ | $\pi/3$ |
| $90^\circ$ | $\pi/2$ |
| $180^\circ$ | $\pi$ |
| $360^\circ$ | $2\pi$ |
Why Polar Coordinates Are Useful in Physics
Polar coordinates are very natural when motion or forces depend on distance from a center or direction from a center. Circular motion, central forces, wave fronts, and rotational systems often become easier to describe in polar form.
For example, if an object moves around a circle of fixed radius, then in polar coordinates its position is especially simple because $r$ stays constant while only $\theta$ changes.
Likewise, curves such as circles and spirals are often simpler in polar coordinates than in Cartesian coordinates. A circle centered at the origin is just
$$
r = \text{constant}
$$
This is much simpler than the Cartesian equation $x^2 + y^2 = R^2$.
Polar Curves
In polar coordinates, a curve is often written as $r$ as a function of $\theta$:
$$
r = f(\theta)
$$
This means that for each angle $\theta$, the distance from the origin is given by the function value. This can produce many shapes, including circles, spirals, and flower-like curves.
Some simple examples are shown below.
| Polar equation | Shape |
|---|---|
| $r = R$ | Circle centered at origin |
| $r = a\theta$ | Spiral |
| $r = a(1+\cos\theta)$ | Cardioid-like curve |
Important Practical Points
Because polar coordinates are not unique, it is common to restrict the angle to a standard interval such as
$$
0 \le \theta < 2\pi
$$
or sometimes
$$
-\pi < \theta \le \pi
$$
This makes descriptions more consistent.
Also, the origin is a special point. When $r=0$, the angle becomes irrelevant because all angles correspond to the same point. So the origin does not have a unique angle.
Special case at the origin:
If
$$
r=0
$$
then the point is the origin, and $\theta$ is not uniquely defined.
Summary
Polar coordinates describe points in a plane using distance from the origin and direction from a reference axis. A point is written as $(r,\theta)$, where $r$ is the radius and $\theta$ is the angle. They are especially useful for circular and rotational situations.
The essential relations are
$$
x = r\cos\theta, \quad y = r\sin\theta
$$
and
$$
r = \sqrt{x^2+y^2}
$$
Polar coordinates give a different and often simpler way to represent position when symmetry around a center is important.