Overcoming the Prejudice: Cartesian vs. Polar Coordinates

Written by tutor Steve C.

In his 1962 book, Toward a Psychology of Being, Abraham Maslow define the typical human tendency of “over-reliance upon a familiar tool” with the “Law of
the Instrument,” or “Maslow’s Hammer”, which is commonly accepted today as the simple proverb:

“If all you have is a hammer, everything looks like a nail.”

Basically, if you know how to use a hammer but don’t know how to use a screwdriver, and you need to insert a screw into wood, you will probably use the
hammer—even though it is not best tool for the task. The implication of this proverb is that most problems have more than one solution, but people tend
to use the familiar one, even when it’s not the best one.

Getting back to mathematics, most students learn about the XY Cartesian Coordinate Plane around age 8 or 9, but don’t learn about the Polar Coordinate
Plane until around age 15 or 16. Naturally, when presented a problem at age 16, they prefer solving it with their familiar XY hammer to their unfamiliar
Polar screwdriver.

Doing so is human nature. We are not all narrow-minded, but rather just inexperienced. The goal of this discussion: demonstrate the polar coordinates
are not that scary after all. Simply put, when you use the tool that makes the most sense, math actually becomes easier to understand.

Here are two blank systems: the XY Cartesian Coordinate Plane, and the Polar Coordinate Plane.

Cartesian Plane Polar Coordinate Plane

Polar coordinate plane image source is found here, used with permission.

Discussions of these two systems often include methods of converting points from one to the other. In this example, the
red point P is defined as (x, y) and by (r cos θ, r sin θ).

Image source is found here, used with permission.

As such, most points (and by extension, most equations) described in one system can be converted into the other system. However, in many cases, there is no
benefit from doing so.

Here is the same line (green) in each system, and the same circle (purple) in each system.

Comparing lines in Cartesian and Polar Coordinates

The basic rule of thumb for which system to use depends upon which system grid lines follows the plot. Although this may seem like an over simplification,
there is some validity to it.

The straight horizontal line on the XY plane is: y = 2
That same line in the Polar Coordinate plane is: r = 1 / cos θ

The circle on the XY plane is: x2 + y2 = 32
That same circle in the Polar Coordinate plane is: r =3

Notice that the simplicity of the line in Cartesian space is no longer simple in Polar space.
Likewise, the simplicity of a circle in Polar space is far less simple in Cartesian space.

The same holds for lines that are not parallel to the X axis.

Here is the same line (blue) in each system.

The straight 45° line on the XY plane is: y = x+1
That same line in the Polar Coordinate plane is: r = 1/ (sin θ – cos θ)

Lines on a polar coordinate and cartesian plane

We could also extend this pattern with a straight line parallel to the y axis, and with straight lines at any angle in
the Cartesian coordinate plane. In every case, the XY plane is simpler equation.

While much has been written about plotting curves on the XY Cartesian Coordinate plane, many of them are done with far simpler equations when reproduced on the Polar Coordinate plane. There are countless examples of curves in the Cartesian, involving polynomials,
exponents and logarithms. While these are often discussed in the XY plane, the Polar plans appears to be better suited for the curved equations.

The most impressive example of this is Archimedes’ Spiral, which is graphed as follows:

In Polar Coordinate r = θ
In XY Cartesian coordinates (x2 + y2)1/2 = arctan (y/x)

Source found here, used with permission.

The bottom line here is this:

Don’t fear the Polar Coordinate System.
It can make life easier more often than you think.

Learn how to use a screwdriver.

It is usually much easier to use than a hammer.

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