Polar Coordinates
Polar coordinates are most useful when it is convenient to know distance and angle. In many such cases using Cartesian coordinates is cumbersome.
Polar coordinates use the distance from the origin, called the radius, and the angle from the positive horizontal axis in the counterclockwise direction to locate a point.
A point a distance r from the origin and an angle θ counterclockwise from the positive axis is identified (r, θ) in the polar coordinate system.
If starting from the Cartesian coordinate system to convert to the polar coordinate system use the Pythagorean theorem, x2 + y2 = r2. The angle is found by using the arctangent of the angle at the origin, tan-1(y/x).
The above leads to the equations r = (x2 + y2)1/2 and θ = tan-1(y/x). Points where x = 0 must be handled with thought since the arctangent is not defined. In such cases θ = π/2 when y is positive, and θ = 3π/2 when y is negative. Also, the signs of both x and y need be considered, and adjustments made if the point (x, y) is not in the first quadrant.
Example 1: Find the polar coordinate representation of the point (3, 4).
r = (x2 + y2)1/2 = (32 + 42)1/2 = (9 + 16)1/2 = 5
θ = tan-1(4/3)
(5, tan-1(4/3))
Example 2: Find the polar coordinate representation of the point (-3, -3).
In this example there are two ways to approach the transformation into polar coordinates.
In one case the radius is -3, taking the negative root of ((-3)2 + (-3)2)1/2 = -3 and θ = tan-1(1) = π/4.
Another possibility is ((-3)2 + (-3)2)1/2 = 3 and θ = tan-1(1) = π/4, adjusted to 5π/4 since both x and y are negative, placing the point in the third quadrant.
(-3, π/4) or equivalently (3, 5π/4)
Example 3: Find the polar coordinate representation of the point (0, 0).
The radius is 0, and the angle is arbitrary.
Example 4: Find the polar coordinate representation of the point (0, 3).
The radius is (02 + 32)1/2 = 3, and θ = π/2, since y is positive. The arctangent does not exist for any angle terminating of the vertical axis.
Basic trigonometry gives the definitions of cos(θ) as x/r and sin(θ) as y/r. Solving cos(θ) = x/r for x gives x = r cos(θ). Solving sin(θ) = y/r for x gives y = r sin(θ).
Example 1: Find the Cartesian coordinate representation of (r, θ) = (2, π/3).
x = r cosθ = 2(1/2) = 1
y = r sinθ = 2(21/2/2) = 21/2
(1, 21/2)
It is possible to convert entire equations from Cartesian coordinates to polar coordinates. Simply replace x with r cosθ and y with r sinθ.
Example 1: Transform x2 + y2 = 3x into polar coordinates.
(r cosθ)2 + (r sinθ)2 = 3r cosθ
r2 cos2θ + r2 sin2θ = 3r cosθ
r2 = 3r cosθ
r = 3 cosθ
To convert an equation in polar coordinates to Cartesian coordinates replace r with (x2 + y2)1/2 and θ with arctan(y/x), which can be awkward. It may be better to eliminate θ using x = r cosθ or y = r sinθ. Notice any of the above expressions for θ can be used in the same problem. In fact, it is allowed to express any trigonometry function in terms of x, y, and r = (x2 + y2)1/2.
Example 1: Express r2 = cotθ in Cartesian coordinates.
r2 = x2 + y2 and cotθ = x/y.
r2 = x/y
The graph of θ = a is a straight line.
The graph of r = aθ is a spiral.
The graph of r = a is a circle with a radius of a centered at the origin.
The graph of r = a cosθ is a circle of diameter a that goes from (0, 0) to (0, a).
The graph of r = a sinθ is a circle of diameter a that goes from (0, 0) to (a, 0).
A limaçon has a graph symmetric about the horizontal axis when an equation is of the form r = b + a cosθ and a graph symmetric about θ = π/2 for an equation of the form r = b + a sinθ.
Notice negative values can occur for b. If a is negative factor our -1. If b is 0 the graph is a circle, as handled above.
There are four cases for limaçons.
A rose figure results from the graph of an equation of the form r = a cosnθ or of the form a sinnθ. The figure had n petals if n is odd and 2n petals if n is even. The reason for the difference is that when n is odd the graph repeats itself.
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Using a graphing calculator for polar coordinates is tricky. If Δθ is selected too large the graph can be unrecognizable. The reason is graphing calculators determine points, then connect those points with line segments. Graphing a rose, for example, can have the graph jump from one petal to another with too large of an increment for θ.
Both cylindrical and spherical coordinate systems use polar coordinates with an added dimension. For cylindrical coordinates the added dimension is z. For spherical coordinates the third dimension is an angle..
One excellent place to find more on polar coordinates is in Chapter 7 of Calculus 2 by OpenStax, the open resource from Rice University.
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Comments
Thank you for your comment in answer to my previous observation and question.
The fifth subheading, Polar graphs, contains, with its information about limaçons, the statement that "If the magnitude of a is between the magnitude of b and of 2b there is a dimple."
What is a dimple in the afore-indicated context?
Since there is no need it could run into opposition. Scientific article space is inadequate, so something less significant could find difficulty in being included.
Thank you for your comment in answer to my previous observation and question.
If someone invented an alternative to limaçons, is it enough to insert that alternative in a scientific-journal article?
Or is there some agency/body/group/institution or process to insert it as an alternative or even as a replacement in polar-coordinate terminology?
I have never heard of an alternative, but I might have missed it. I rarely concern myself with what we call something as long as I understand it.
The fifth subheading, polar graphs, alerts us to limaçons.
Is there another term or is it always the Gilles de Roberval inventive invention?