Polar Coordinates: two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction (Merriam-Webster Dictionary,2022)
Cartesian Coordinates: a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length (Merriam-Webster Dictionary,2022)
Polar Axis: positive x-axis; the fixed line, usually horizontal, from which the angle made by the radius vector is measured in a polar coordinate system (Merriam-Webster Dictionary,2022)
Coordinate Plane: gird or graph; two-dimensional surface formed by two number lines where coordiantes are plotted
To understand the equations behind spirals, a clear understanding of Polar Coordinates is necessary.
Many people are taught how to graph equations in what is called the Cartesian Coordinate system.
This is where a coordinate is described in terms of X and Y with the location of the points labeled (X,Y) as seen below.
Manually type in various number into the A section set up as (x,y). Then notice how that translates onto the graph and what those numbers you entered mean. Do this until you understand how the Cartesian Coordinate system operates.
Lets start exploring Polar Coordinates! The video below provides a quick overview if you want to skip the play by play:
Graph
Notice that the Cartesian Coordinate system operates on a grid coordinate plane.
Many don't know that there are differing grid systems in mathematics. Below are three examples, one of which we will use.
On the left is the now known Cartesian gird system, in the middle is the Polar grid system and on the right is a Parallel gird system where the x coordinates are on the left most line and the y coordinates are on the right most line.
As you probably guessed, Polar Coordinates are mapped and plotted, most commonly, on a Polar Coordinate system.
Rather than using the locations of x and y to identify where the location of a point is, Polar Coordinates measures the distance from the origin (r) a point is and what angle away from the Polar Axis (θ) it is.
Thus leading to the point (r,θ) where θ starts at the Polar Axis, recall defineitons, and moves counterclockwise.
Now this may sound very confusing but it will all get explained just bear with me.
Just to get you used to the grid itself below is an activity created by Tim Brzezinski that allows you to input any radius (r) and angle (θ) value.
As you move the slider, its demonstrates how the point is discovered.
Remeber that this is just to get you familiar with the grid, next will be an explanantion of the math and more consepts that go with it.
Radius - r
The first coordiante in this system is the radius, denoted as r.
The radius is the distance the point will be from the origin, (0,0).
In order to find this we will start in the familiar Cartesian Coordinate system.
As you move the point A, notice that the blue line measures the length of y while the pink line measures the length of x.
When you draw a line from the origin to the point call this r, a right triangle is formed with the x and y lengths known and r unknown.
Recall what you learned in geometry/trigonometry about the Pythagorean theorem; when there is a right triangle with base lengths a and b and a missing side length of c.
In order to find that third side length the equation: $$ c^2 = a^2 + b^2 $$ is employed.
Now apply this to our case and let r = c x = a y = b
Notice that when you make those substitutions you are left with the equation $$r^2 = x^2 + y^2$$ which can be rearranged to $$r = \sqrt{(x^2 + y^2)}$$.
If you want to try this before moving on to finding θ , visit the Practice page for assistance.
Angle - θ
Returning to our knowledge of triangles in trigonometry, if you have a right triangle and are attempting to find θ, you can use: $$tan(θ) = \frac{oppositie}{adjacent}$$ to do so.
When it comes to Polar Coordinates, the angle that we use to describe the location of a point is the angle from the Polar Axis to r as seen below.
In order to apply the formula for tan(θ) we can rearrange our trangle to look like this.
In this case we will make y = opposite x = adjacent Thus $$tan(θ) = \frac{y}{x}$$
Using algebra we isolate θ to get $$ θ = arctan(\frac{y}{x})$$
Notice in the applet that when the triangle moves into the second, third, and fourth quadrant that calculating θ isn't enough.
Quadrant 2
$$\pi-arctan(\frac{y}{x})$$
Quadrant 3
$$\pi + arctan(\frac{y}{x})$$
Quadrant 4
$$-arctan(\frac{y}{x})$$
But why is that?
Notice that the angle in these quadrants is formed between r and the X-axis.
However, the angle we need is the angle starting at the Polar Axis.
Thus, if we subtract or add this angle from π, depending on which quadrant, we get the angle we desire.
There we have it, (r,θ)!
Formulas
Calculating r and θ from x and y:
$$r = \sqrt{(x^2 + y^2)}$$
$$θ = arctan(\frac{y}{x})$$
Calculation x and y from r and θ
$$x = r*cos(θ)$$
$$ y = r*sin(θ)$$
