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https://github.com/cansik/longlivethesquare

An algorithm to find the minimum bounding box.
https://github.com/cansik/longlivethesquare

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An algorithm to find the minimum bounding box.

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## Long Live The Square (LLTS)

An algorithm to find the arbitrarily oriented minimum bounding box in `R²`.



### What is a minimum bounding box?

A [minimum bounding box][1] is a rectangle that encloses all points in a given set of points and has the smallest area of all enclosing rectangles (*Figure 1*).



![Minimal Bounding Box](images/bounding_box.png)



*Figure 1: Minimal Bounding Box*



### How does it work?

The **input** of the algorithm is a **set of points** which are described through a 2d position vector. To decrease the amount of points to process it is advisable to calculate the **convex hull** first.



#### Convex Hull



The [convex hull][2] is a set of points which are the most outtest points of the input set (*Figure 2*). It is like if you would put a rubber band around the points and take the ones which come in contact with it.



LLTS uses the [monotone chain algorithm][3] to calculate the convex hull. This algorithm first sorts the points [lexicographically][4] and then runs through the upper and lower most points to create the upper and lower hull. Those two combined is the convex hull.



![Convex Hull](images/convex_hull.png)



*Figure 2: Convex Hull*



The monotone convex hull algorithm runs with `O(n * log(n))` speed.



#### Minimal Bounding Box

The idea of the LLTS algorithm is to use the property that one edge of the minimum bounding box is **parallel** to an edge of the convex hull.



So first of all the algorithm **connects** all points of the convex hull together to a list of segments.



Because it is more trivial to find the axis aligned bounding box then the arbitrarily oriented we rotate all points to be parallel to the **x-axis** with our **segment**. This can be achived with a rotation matrix and a bit of trigonometry:



```cs

// get angle of segment s

var delta = s.A - s.B;

var angle = -Math.Atan (delta.Y / delta.X);



// rotate vector v

var newX = v.X * Math.Cos (angle) - v.Y * Math.Sin (angle);

var newY = v.X * Math.Sin (angle) + v.Y * Math.Cos (angle);



return new Vector2d (newX, newY);

```



Now for each segment it tries to find the topmost and bottommost **y-value** and the leftmost and rightmost **x-value**. With these four values it is possible to create a bounding box in the rotated coordination system.



To find the minimal bounding box the algorithm now just takes the bounding box with the smallest **area**. It is not necessary to rotate the bounding box edge points back to calculate the **area** because it is not affected by rotational changes.



After cycling through every segment it just rotates back the four points of this bounding box and returns them.



#### Edge Cases

* If there are only one or zero points in the input set the algorithm just returns the set as it is.

* If two points are on the exact same place they will be counted as one point.



### Example Application

The example application is written in C# with the ui framework [Eto.Forms][5] to be platform independent. The project solution is a Visual Studio project and can be opend with [Visual Studio][7], [Xamarin Studio][6] or [Monodevelop][8]. To start it just run the project for your specific platform:



* Windows

 * WPF

 * WinForms

* Mac

 * MonoMac

 * XamarinMac

* Linux

 * Gtk2

 * Gtk3



![Example Application](images/application.png)



*Figure 3: Example Application*



#### Functions

The UI (*Figure 3*) does contain following functions:



* To **add new points** just left click on the drawing pane.

* To **zoom** in and out use the mousewheel (like scrolling).

* To reset the pane click the **Reset** button.

* To add 20 random points click the **Random** button.

* To calculate and show the convex hull click the **Convex Hull** button.

* To calculate and show the minimum bounding box click the **Bounding Box** button.



### About

* *Copyright (c) 2016 Florian Bruggisser*

* *[MIT License](LICENSE)*



### Ports



C++ minimal example including XCode example project [MinimalBoundingBox](https://github.com/schmidt9/MinimalBoundingBox)



[1]: https://en.wikipedia.org/wiki/Minimum_bounding_box "Minimum Bounding Box"

[2]: https://en.wikipedia.org/wiki/Convex_hull "Convex Hull"

[3]: https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain "Monotone Chain Convex Hull"

[4]: https://en.wikipedia.org/wiki/Lexicographical_order "Lexicographical Order"

[5]: https://github.com/picoe/Eto "Eto Forms"

[6]: https://xamarin.com/studio "Xamarin Studio"

[7]: https://www.visualstudio.com/ "Visual Studio"

[8]: http://www.monodevelop.com/ "Monodevelop"