Make conversions between rectangular (Cartesian) and polar plane coordinates.
This calculator is used to convert rectangular to polar coordinates and vice versa for a 2D system. It shows the step-by-step calculation for both conversions.
Polar coordinates represent a point that is positioned with respect to the origin (0,0) and an angle from the origin in the reference direction. They are written as (r, θ), such that:
Cartesian coordinates indicate a point in an XY-plane by a pair of numerical values. These coordinates are signed distances from a point to two perpendicular directed lines. They are usually written as (x, y), such that:
There are two methods of converting between the coordinates:
2.1.1 To Convert from Cartesian to Polar
For rectangular coordinates (x, y) given in the above picture, you can find polar coordinates (r, θ) as follows:
\(r = \sqrt{x^{2} + y^{2}}\)
\(θ = arctan (\dfrac{y}{x}\)
2.1.2 To Convert from Polar to Cartesian
Use the following polar equations to cartesian equations for converting: \(x = r cos θ\) \(y = r sin θ\)
Solution:
Data Given:
Calculations:
Step 1: Determine ‘r’
\(r = \sqrt{x^{2} + y^{2}}\)
\(r = \sqrt{6^{2} + 2^{2}}\)
\(r = \sqrt{36 + 4}\)
\(r = \sqrt{40}\)
\(r = 6.324555320336759\)
Step 1: Determine ‘θ’
\(θ = arctan (\dfrac{y}{x}\)
\(θ = arctan (\dfrac{2}{6}\)
\(θ = arctan (\dfrac{1}{3}\)
Solution:
Data Given:
Calculations:
Step 1: Determine Horizontal Rectangular Component ‘x’
\(x = r cos θ\)
\(x = 2 * 0.707\)
\(x = 1.414\)
Step 2: Determine Vertical Rectangular Component ‘y’
\(y = r sin θ\)
\(y = 2 * sin\left(45^\text{o}\right)\)
No, the polar coordinates of a point are not unique. The reason is that each point can be represented by infinite polar coordinates in infinite ways.
In polar coordinates, z represents the complex number in polar form, such that: z = x + iy (where i (iota) = \(\sqrt{-1}\)
Where
The polar coordinate representation of 1 is given as:
\(1 = 1e^{i} 0\)
The same equation holds true for \(1 = 1e^{i} 2π\) due to the fact that sine and cosine are both periodic with the angle 2π.
Cartesian Coordinates (x, y) | Polar Coordinates (r, θ) |
---|---|
(3, 4) | √(25) , arctan(4/3) ≈ 53.13° |
(-2, 1) | √(5) , arctan(-1/2) ≈ -26.57° |
(0, 5) | 5, 90° |
(-4, 4) | √(32) , arctan(1/1) ≈ 45° (or 315°) |
This polar coordinates calculator can handle all the conversions between coordinates including all ones listed in the above table, with the complete solution shown.
From the source of Wikipedia: Polar coordinate system, Conventions, Uniqueness of polar coordinates, Converting between polar and Cartesian coordinates.
From the source of Math Insight: Conversion formulas, the plane of Cartesian coordinates, coordinates r and θ.
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