How to calculate figure rotation efficiently?

Question

I have a Figure represented through a matrix of bytes (bitmap-like matrix). Example Figure is shown on the Picture 1.

The goal is to find the best rotation angle of some given Figure. When Figure is rotated by best angle, the rectangle that is parallel to X and Y axes and inscribes the Figure has the smallest area.

Rectangles that inscribes the figure are shown as light-gray on the pictures. In the Picture 2, you can see that ideal rotation of the Figure is about 30 degrees clockwise.

Now, I know algorithm how to find this angle, but it seems to me that is very inefficient. It goes like this:

1. Loop through angles from 0 to 45.
2. For the current angle, for every figure point calculate new, rotated, location
3. Find bounds of rectangle that contains figure (minimum and maximum x,y) and register it if is the best match so far
4. Next angle

This is a kind of brute-force method and works well and reasonably fast for the small figures. However, I need to work with figures that contain up to 10 million points, and my algorithm becomes slow.

What would be good algorithm for this problem?

Answer 1

It looks like you can find the arbitrarily aligned minimum bounding box using the linear time Rotating calipers algorithm.

Once you have the bounding box, you just need to determine the angle of rotation by calculating the slope of one of the sides.

Answer 2

The first step of your approach is flawed - there are an infinite number of real values between 0 and 45, so it makes no sense to "loop through them". However, your algorithm can be repaired:

• find the convex hull of the polygon

• loop through the finite (!) number of angles given by the outer edges of the convex hull

• now apply the steps 2 to 4 using these angles.

This works because it can be shown that the minimum enclosing rectangle must touch one of the outer edges of the convex hull.