The Comprehensive Guide to Semicircles: Technical Specifications, Data Visualization, and Comparative Analysis

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A semicircle is a fundamental geometric shape that is half of a complete circle, with a radius that is half the diameter of the full circle. Understanding the technical specifications, data visualization techniques, and comparative analysis methods for semicircles is crucial for various scientific and engineering applications.

Technical Specifications of Semicircles

Radius and Diameter

The radius of a semicircle is half the diameter of the full circle. Mathematically, if the diameter of the full circle is d, then the radius of the semicircle is r = d/2.

Circumference

The circumference of a semicircle can be calculated using the formula:
$C = \pi r$

where r is the radius of the semicircle.

Area

The area of a semicircle can be calculated using the formula:
$A = \frac{1}{2} \pi r^2$

where r is the radius of the semicircle.

Sector Angle

The sector angle of a semicircle is always 180 degrees or $\pi$ radians.

Trigonometric Functions

In a right-angled triangle formed by the diameter and radius of a semicircle, the following trigonometric relationships hold:
– $\sin(\theta) = \frac{y}{r}$
– $\cos(\theta) = \frac{x}{r}$
– $\tan(\theta) = \frac{y}{x}$

where x and y are the coordinates of a point on the semicircle, and r is the radius.

Semicircles in Data Visualization

semicircle

Semicircles are commonly used in data visualization to create pie charts or donut charts that display proportional data.

Semicircle Pie Charts

To create a semicircle pie chart, you can follow these steps:
1. Determine the total value of the data set.
2. Calculate the proportional value for each category.
3. Assign a unique color to each category.
4. Arrange the categories in a semicircular layout, with the proportional slices representing each category.
5. (Optional) Include the total value in the center of the semicircle.

Semicircle Donut Charts

To create a semicircle donut chart, you can follow these steps:
1. Determine the total value of the data set.
2. Calculate the proportional value for each category.
3. Assign a unique color to each category.
4. Create a donut chart with a specified inner radius.
5. Arrange the categories in a semicircular layout, with the proportional slices representing each category.
6. (Optional) Include the total value in the center of the semicircle.

Tableau Semicircle Visualization

In Tableau, you can create a semicircle pie chart or donut chart by using a dual-axis chart. Here’s how:
1. Create a circle chart on one axis.
2. Create a blank chart on the other axis.
3. Adjust the mark properties to create a semicircle.
4. (Optional) Add a legend and adjust the colors and labels.

Altair Semicircle Donut Chart

In Altair, you can create a semicircle donut chart using the following code:

import altair as alt
import pandas as pd

# Sample data
data = pd.DataFrame({
    'category': ['A', 'B', 'C'],
    'value': [30, 20, 10]
})

# Create the semicircle donut chart
chart = alt.Chart(data).mark_arc(innerRadius=50, outerRadius=100).encode(
    theta=alt.Theta('value:Q', stack=True),
    color=alt.Color('category:N')
)

# Display the chart
chart

Note that in Altair, it is not currently possible to combine coloring per category with a half pie/donut chart.

Comparing Semicircular Data Sets

When comparing two semicircular data sets, it is important to consider the appropriate scaling and analysis techniques.

Scaling and Transformation

  1. Scale the data appropriately to the range [0, 2π].
  2. Analyze the scaled data using appropriate statistical methods.
  3. Back-transform the results to the original scale.

Watson-Wheeler Test

For fully circular data, the Watson-Wheeler test can be used to compare two semicircular data sets. However, transforming the data to [0, 2π] and back-transforming the results may not be appropriate for semicircular data.

Appropriate Scaling and Analysis

The choice of scaling and analysis techniques depends on the specific characteristics of the semicircular data sets. It is important to consider the underlying assumptions and limitations of the chosen methods to ensure accurate and meaningful comparisons.

Additional Resources and References

  1. Semicircle Pie Chart – Tableau Community: https://community.tableau.com/s/question/0D54T00000C66mrSAB/semi-circle-pie-chart
  2. Semicircle in a Quadrant – YouTube: https://www.youtube.com/watch?v=-DN3zjHceNw
  3. Semicircle Donut Chart with Altair – Stack Overflow: https://stackoverflow.com/questions/72083560/semi-circle-donut-chart-with-altair
  4. Half-Circle Pie Chart (Semicircle) – Tableau Community: https://community.tableau.com/s/question/0D54T00000C5xCZSAZ/halfcircle-pie-chart-semicircle
  5. Comparing Semicircular Data – Cross Validated: https://stats.stackexchange.com/questions/419838/compare-semi-circular-data