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 is a rectangle that encloses all points in a given set of points and has the smallest area of all enclosing rectangles (Figure 1).

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 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 to calculate the convex hull. This algorithm first sorts the points lexicographically and then runs through the upper and lower most points to create the upper and lower hull. Those two combined is the convex hull.

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:

// 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 to be platform independent. The project solution is a Visual Studio project and can be opend with Visual Studio, Xamarin Studio or Monodevelop. To start it just run the project for your specific platform:

• Windows
• WPF
• WinForms
• Mac
• MonoMac
• XamarinMac
• Linux
• Gtk2
• Gtk3

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

Ports

C++ minimal example including XCode example project MinimalBoundingBox