## Thiessen Polygon Tool

You can access sample input files for download here: DropBox

## What is Thiessen Polygons?

Thiessen Polygons, or Voronoi Polygons, are geometric constructs dividing a plane based on proximity to given points. Named after Alfred Thiessen, each polygon encapsulates points closer to its origin than any other. Formed by connecting midpoints between adjacent points, these polygons establish equidistant boundaries to neighboring points. A fundamental tool in spatial analysis and geography, Thiessen Polygons efficiently delineates regions of influence around specific points. Their application extends across various fields where understanding spatial relationships and proximity zones is crucial, including meteorology for precipitation analysis, geomorphology for drainage basin studies, and facility location planning. The technique, grounded in Voronoi tessellation or Delaunay triangulation, offers a versatile means to interpret spatial data and visualize the influence zones of given point sets.

## What is the Thiessen Polygon tool?

The Thiessen Polygon tool is a user-friendly application designed for drawing Thiessen Polygons, also known as Voronoi Polygons, on maps or images. Developed to facilitate ease of use, the tool allows users to input station data, including latitude, longitude, and optional variables like rainfall. It provides options to customize text positions and visual elements. The process involves drawing polygons, saving images, exploring map tiles, viewing/exporting area data, and calculating weighted averages. The tool simplifies the implementation of the Thiessen Polygon Method, making it accessible for spatial analysis, geography, and related applications.

The license of this tool is applicable for one year of using and you can renew it by pay 20% of the price for the new year.

All images are contained within the PDF files, which can be accessed through the provided links above.

### Thiessen Polygon Formulas

The Thiessen polygons method, also known as Voronoi polygons or Dirichlet polygons, is a geometric technique used to partition a plane into regions based on proximity to a given set of points. Named after Alfred Thiessen, this method is commonly used in spatial analysis, geography, and other fields where the concept of proximity zones is relevant.

Here are the key steps involved in constructing Thiessen polygons:

#### Input a Set of Points or Sites:

Begin with a set of discrete points scattered across the plane. These points represent measurement locations or observation sites.

#### Calculate Voronoi Diagram:

- The Delaunay triangulation is a method to connect these points in such a way that no point is inside the circumcircle of any triangle formed by the points. Connect the points to create Delaunay triangles, forming a triangulation of the given set.
- Determine the circumcenter of each Delaunay triangle, as it will be a key point in constructing the Voronoi diagram. Illustrate circumcircles for each triangle.
- Connect the circumcenters to form polygons and edges. These edges, along with the original points, create the Voronoi diagram. Each Voronoi polygon corresponds to an area that is closer to its associated point (station) than to any other point in the set.
- Determine the area of individual polygons by applying the Shoelace formula, a mathematical technique for computing the area of a polygon given the coordinates of its vertices. This method involves systematically summing the products of coordinates in a specific pattern.

#### Shoelace Formula

The general case for finding areas of polygons

The general formula for the area of an n-sided polygon is given above. For a triangle this gives:

For a quadrilateral this gives:

For a pentagon this gives:

You might notice a nice shoelace like pattern (hence the name) where x coordinates criss cross with the next y coordinate along. To finish off let’s see if it works for an irregular pentagon.

#### Signed area of a polygon on the Earth's surface

Firstly, we convert the coordinates from degree to radians as below:

Then we calculate area in square kilometers as below:

Atan2 is an angle, θ, measured in radians, such that tan(θ) = y / x, where (x, y) is a point in the Cartesian plane.

The R parameter represents the radius of the Earth and is utilized to scale the computed area to square kilometers. The Earth's average radius is approximately 6,371,009 meters.

### Thiessen Polygon Tutorial

The Thiessen polygons method, alternatively referred to as Voronoi polygons or Dirichlet polygons, is a geometric approach employed to divide a plane into distinct regions determined by their proximity to a specified set of points. Coined after Alfred Thiessen, this technique finds widespread use in spatial analysis, geography, and various fields where defining proximity zones is essential.

#### Buy a license and Installing

The installation procedure for this tool is straightforward and trouble-free. Once you've obtained a license, you'll gain access to a tool called "ID Finder." Share your unique ID with us, and in response, you'll receive the installer for the registered version. Running the installer will seamlessly install the tool without needing an additional activation key. Once installed, you can effortlessly access the device by clicking on the desktop shortcut or running "Thiessen Polygon Software" in your computer's program list.

#### Sample Data

If you require sample data, you can download it from our website. Additionally, the tool includes a convenient button for loading this sample data, which encompasses station names, latitude, longitude, latitude of text, longitude of text, and yearly rainfall data values. Also, the list of polygon's points with latitude and longitude.

#### Input Data

There are two datasets to input: the list of stations and border information. The station list can be entered manually or imported from a file, including mandatory fields such as 'Name of Stations', 'Latitude', and 'Longitude.' If you wish to specify the location of text on the chart, include values for 'Latitude of Text' and 'Longitude of Text.' Additionally, if you want to calculate the variable's average, provide the 'Value' item; otherwise, omit it.

When dealing with border information, you have various options. You can input a polygon with latitude and longitude through a file. Alternatively, you may define a rectangular border by entering its corners or automatically selecting it based on the min/max latitude/longitude of stations, plus or minus a small offset specified as 'dx.'

#### Thiessen Polygon Image

On the left-hand side, there are three tabs, and the first tab is used to draw polygons as images. If you have previously entered the latitude/longitude of text, the 'Use Position' checkbox will become active, allowing you to check it. You also have the option to display the name and area on the image. When using Degree2, the tool treats the data as a plane, resulting in the area being measured in square degrees or the square of the unit corresponding to your latitude (Y) and longitude (X).

However, if you select KM2, the tool considers the data as representing Earth, and the area is calculated in square kilometers. After clicking the 'Draw Lines' button, you can easily save the image to a file. Note that the default image dimensions are 806px width and 581px height, but you can adjust the size and DPI (Dots per Inch) during the saving process.

#### Thiessen Polygon Map

The second tab is similar to the first tab but with an additional feature for maps. You can now change the map tile by selecting from a dropdown list in the combobox.

#### Area Calculation

On the third tab, you can view and export the area for each station in square kilometers and square degrees. This tab will populate after you draw the polygon lines in the first or second tab.

If you have entered values in the list of stations on left-side, you can calculate the weighted average of the station values in this tab. This weighted average can be in square kilometers or degrees.