Coordinate systems in R Programming - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of coordinate transformation using base R

What is the output of this R code that converts Cartesian coordinates (x, y) to polar coordinates (r, theta)?

R Programming

cartesian_to_polar <- function(x, y) {
  r <- sqrt(x^2 + y^2)
  theta <- atan2(y, x)
  return(c(r = r, theta = theta))
}

result <- cartesian_to_polar(3, 4)
print(round(result, 2))

A r 7.00, theta 0.93

B r 5.00, theta 0.93

C r 5.00, theta 0.64

D r 7.00, theta 0.64

💡 Hint

Recall that r is the distance from origin and theta is the angle in radians from the x-axis.

The distance r is sqrt(3^2 + 4^2) = 5. The angle theta is atan2(4, 3) ≈ 0.93 radians.

🧠 Conceptual

intermediate

Understanding coordinate system origin shifts

If you shift the origin of a 2D Cartesian coordinate system by vector (2, -3), what are the new coordinates of point (5, 7)?

A (3, 10)

B (7, 4)

C (3, 4)

D (7, 10)

💡 Hint

Subtract the shift vector from the original point coordinates.

Shifting the origin by (2, -3) places the new origin at old (2, -3). New coordinates = (5 - 2, 7 - (-3)) = (3, 10).

🔧 Debug

advanced

Identify the error in coordinate rotation code

What error does this R code produce when rotating a point (x, y) by angle theta (in radians)?

R Programming

rotate_point <- function(x, y, theta) {
  x_new <- x * cos(theta) - y * sin(theta)
  y_new <- x * sin(theta) + y * cos(theta)
  return(c(x_new, y_new))
}

rotate_point(1, 0, pi / 2)

A Returns numeric vector c(1, 0) without rotation

B Error: object 'theta' not found

C Returns numeric vector c(0, 1) as expected

D Error: non-numeric argument to binary operator

💡 Hint

Check if all variables are defined and used correctly.

The function correctly computes the rotated coordinates. Rotating (1,0) by 90 degrees (pi/2 radians) results in (0,1).

📝 Syntax

advanced

Syntax error in coordinate transformation function

Which option contains the correct syntax for defining a function that converts polar to Cartesian coordinates in R?

A

polar_to_cartesian <- function(r, theta) {
  x <- r * cos(theta)
  y <- r * sin(theta)
  return(c(x = x, y = y))
}

B

polar_to_cartesian <- function(r, theta) {
  x <- r * cos(theta)
  y <- r * sin(theta)
  c(x = x, y = y)
}

C

polar_to_cartesian <- function(r, theta) {
  x <- r * cos(theta)
  y <- r * sin(theta)
  return(x, y)
}

D

polar_to_cartesian <- function(r, theta) {
  x = r * cos(theta)
  y = r * sin(theta)
  return(c(x, y))
}

💡 Hint

Check the return statement syntax and how multiple values are returned.

Option A uses return() with a named vector correctly. Option A returns unnamed vector but is syntactically valid. Option A tries to return two values separately which is invalid. Option A returns vector without return() but is valid in R as last expression is returned.

🚀 Application

expert

Calculate distance between two points in different coordinate systems

Given point A in Cartesian coordinates (3, 4) and point B in polar coordinates (r = 5, theta = pi/3), what is the Euclidean distance between A and B?

A 3.0

B 4.0

C 5.0

D 0.6

💡 Hint

Convert point B to Cartesian coordinates first, then use distance formula.

Point B in Cartesian: x = 5*cos(pi/3) = 2.5, y = 5*sin(pi/3) ≈ 4.33. Distance = sqrt((3-2.5)^2 + (4-4.33)^2) ≈ sqrt(0.25 + 0.11) ≈ 0.6.

Practice

1. Which function in R's ggplot2 package is used to flip the x and y axes on a plot?

easy

A. coord_flip()

B. coord_cartesian()

C. coord_polar()

D. coord_fixed()

Solution

1. Step 1: Understand coordinate functions in ggplot2

   The ggplot2 package provides several coordinate functions to control plot appearance.

2. Step 2: Identify the function that flips axes

   coord_flip() swaps the x and y axes, effectively flipping the plot.

3. Final Answer:

   coord_flip() -> Option A

4. Quick Check:

   Flip axes = coord_flip() [OK]

Hint: Flip axes with coord_flip() in ggplot2 [OK]

Common Mistakes:

• Confusing coord_flip() with coord_cartesian()
• Using coord_polar() which changes to polar coordinates
• Thinking coord_fixed() flips axes

2. Which of the following is the correct syntax to set fixed aspect ratio in ggplot2?

easy

A. coord_fixed_ratio(1)

B. coord_fixed(scale = 1)

C. coord_fixed(aspect = 1)

D. coord_fixed(ratio = 1)

Solution

1. Step 1: Recall the function for fixed aspect ratio

   coord_fixed() sets a fixed aspect ratio between x and y axes.

2. Step 2: Check correct argument name

   The argument to specify ratio is ratio, so coord_fixed(ratio = 1) is correct syntax.

3. Final Answer:

   coord_fixed(ratio = 1) -> Option D

4. Quick Check:

   Fixed ratio uses ratio= in coord_fixed() [OK]

Hint: Use ratio= argument in coord_fixed() for aspect ratio [OK]

Common Mistakes:

• Omitting argument name and just passing 1
• Using aspect= instead of ratio=
• Using a non-existent function coord_fixed_ratio()

3. What will be the effect of this code snippet in R using ggplot2?

ggplot(mtcars, aes(wt, mpg)) + geom_point() + coord_cartesian(xlim = c(2, 4), ylim = c(15, 30))

medium

A. Zooms the plot to show only points with weight between 2 and 4 and mpg between 15 and 30

B. Removes points outside the specified limits from the data

C. Flips the x and y axes

D. Changes the plot to polar coordinates

Solution

1. Step 1: Understand coord_cartesian() behavior

   coord_cartesian() zooms the plot to specified limits without removing data points.

2. Step 2: Analyze the limits given

   The x-axis is limited to 2 to 4 (weight), and y-axis to 15 to 30 (mpg), so plot zooms to this area.

3. Final Answer:

   Zooms the plot to show only points with weight between 2 and 4 and mpg between 15 and 30 -> Option A

4. Quick Check:

   coord_cartesian() zooms plot, does not remove data [OK]

Hint: coord_cartesian() zooms plot, keeps all data points [OK]

Common Mistakes:

• Thinking coord_cartesian() removes points outside limits
• Confusing with coord_flip() which flips axes
• Assuming coord_polar() is used here

4. Identify the error in this ggplot2 code snippet:

ggplot(mtcars, aes(wt, mpg)) + geom_point() + coord_flip(ratio = c(1, 5))

medium

A. geom_point() is missing required argument

B. coord_flip() does not accept ratio argument

C. ggplot() is missing data argument

D. coord_flip() should be replaced with coord_cartesian()

Solution

1. Step 1: Check arguments accepted by coord_flip()

   coord_flip() flips axes but does not accept ratio argument.

2. Step 2: Identify correct way to set aspect ratio when flipping

   Aspect ratio should be set using coord_fixed(), not inside coord_flip().

3. Final Answer:

   coord_flip() does not accept ratio argument -> Option B

4. Quick Check:

   coord_flip() has no ratio argument [OK]

Hint: coord_flip() does not accept ratio argument [OK]

Common Mistakes:

• Trying to set ratio inside coord_flip()
• Confusing coord_flip() with coord_cartesian()
• Assuming geom_point() needs extra arguments

5. You want to create a pie chart in ggplot2 using coord_polar(). Which coordinate system transformation should you apply before coord_polar() to make the pie chart correctly represent proportions?

hard

A. Use geom_point() with coord_polar()

B. Use geom_bar() with stat = 'identity'

C. Use geom_bar() with default settings and map fill to factor variable

D. Use geom_line() with coord_polar()

Solution

1. Step 1: Understand pie chart creation in ggplot2

   Pie charts are created by using bar charts with polar coordinates.

2. Step 2: Identify correct geom and mapping

   Using geom_bar() with default settings and mapping fill to a factor variable creates bars representing counts, which coord_polar() converts to pie slices.

3. Final Answer:

   Use geom_bar() with default settings and map fill to factor variable -> Option C

4. Quick Check:

   Pie chart = geom_bar() + coord_polar() with fill mapped [OK]

Hint: Pie charts use geom_bar() + coord_polar() with fill mapped [OK]

Common Mistakes:

• Using geom_point() or geom_line() which don't create bars
• Setting stat='identity' unnecessarily for counts
• Not mapping fill to a factor variable