Converting Cartesian to Polar and Sorting

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In this article, we will explore how to convert a set of points from the Cartesian coordinate system to polar coordinates and then sort them based on their angles. We’ll use R as our programming language for this example.

Introduction

The Cartesian coordinate system is a two-dimensional system where each point in space is represented by an ordered pair of numbers, (x, y). On the other hand, the polar coordinate system represents points using a distance from a reference point and the angle between the line connecting that point to the origin and the positive x-axis.

Converting Cartesian to Polar Coordinates

To convert points from Cartesian coordinates (x, y) to polar coordinates (r, θ), we use two formulas:

1. The radial distance ( r ) is calculated using the formula: [ r = \sqrt{x^2 + y^2} ]
2. The angle ( \theta ) can be calculated using the arctangent function and considering the quadrant of the point to ensure it falls within the range (0, 2π). Here’s how you can do it in R:

library(dplyr)

coords <- as.data.frame(X)
mutate(coords, theta = ifelse(x < 0, atan(y / x) + pi,
                              ifelse(y < 0 , atan(y / x) + 2*pi, atan(y / x)))) %>%
arrange(theta)

Understanding the Code

The code snippet above uses the dplyr library in R to perform data manipulation tasks.

1. The first line imports the necessary library.
2. It then converts the input data frame X into a new data frame coords.
3. In the mutate function, it applies transformations to each row of the coords data frame:
   • If ( x ) is less than 0, it calculates the arctangent of ( y/x ) and adds π to ensure the angle falls within the range (0, 2π).
   • Similarly, if ( y ) is less than 0, it calculates the arctangent of ( y/x ) and adds 2π to bring the angle into the correct range.
   • If neither x nor y is negative, it simply calculates the arctangent of y divided by x without any adjustments.
4. Finally, the arrange function sorts the data frame based on the calculated angles in ascending order.

Sorting Points Based on Angle

Now that we have converted our points to polar coordinates and ensured they fall within the correct range, we can sort them based on their angles.

However, the approach taken above results in duplicate rows for points with the same angle but opposite signs. If you want to remove such duplicates or perform further analysis based on these angles, you would need a different approach.

Example Use Case

Suppose we have a set of points representing locations on a map and we want to determine the order of these locations based on their distances from a central point.

Here’s how we could use the polar coordinates calculated above:

# Calculate distances from the origin (central point)
distances <- sqrt(coords$X^2 + coords$Y^2)

# Merge the original data with the distances
merged_data <- merge(coords, distances, by.x = "index", by.y = "value")

# Sort the data based on the angle
sorted_data <- merged_data[order(merged_data$theta), ]

In this example, we calculate the distances from the origin for each point using the radial distance formula. We then merge this information with the original coordinates and sort the resulting data frame based on the calculated angles.

Conclusion

Converting points between Cartesian and polar coordinate systems is a fundamental task in various fields, including geography, physics, and engineering. By understanding how to perform these conversions, we can gain valuable insights into the properties of our data and make more informed decisions. In this article, we explored how to convert Cartesian coordinates to polar coordinates using R and demonstrated its applications.

Note: The examples provided here are simplified for illustrative purposes. Depending on your specific requirements or use case, you might need to modify the code or consider additional factors such as handling edge cases or ensuring data consistency across different variables.

Last modified on 2023-06-13