# A Data Processing Framework for Polar Performance Diagrams
## 3 The hrosailing Framework
### How to Use the Convex Hull of a Polar Diagram Slice for Optimal Manoeuvres  
The use of the convex hull for optimal manoeuvres is a well-known fact in the sailing community, but has never been formulated as part of a data-handling framework.  
In its simplest form, a polar diagram gives us information about the possible boat speed for given values of TWA and TWS under some further assumptions, without planning tacks or jibes. If we allow tacks and jibes (or changing of sails), the sailing vessel can outperform these speeds in some directions.  
For given TWS, a diagram representing the top performance in a direction allowing manoeuvres can be constructed as follows: first, take the _convex hull_ of the polar plot of the respective slice, that is the smallest convex set containing the polar plot; then, for each direction, choose the point in the convex hull which is farthest away from the origin. The resulting diagram is the one giving the top performance aimed for. It can be read in a similar way to the usual polar plot of a slice (see Figure 8).  
It is also possible to get an exact description of the manoeuvre necessary to obtain the respective top performance. To do this, first note that in our situation any point \(p\) on the boundary of the convex hull is either a point of the initial set or can be expressed as \(p=\lambda v+(1-\lambda)w=w+\lambda(v-w)\), where \(v\) and \(w\) are points of the boundary of the initial set and \(0\leq\lambda\leq 1\). In other words, the point \(p\) is on a segment between two boundary points \(v\) and \(w\) of the initial set and \(\lambda=\frac{|p-w|}{|v-w|}\) determines the exact position on the segment. If \(p\) is on the boundary of the initial polar diagram, it is sufficient to simply steer in the respective direction. Otherwise, an optimal course would be to steer at a ratio of \(\lambda\) of the time in the direction of \(v\) and a ratio of \((1-\lambda)\) of the time in the direction of \(w\).  
Often a ship has different sails which can be used. In these cases, we need a polar diagram for each possible setting of sails. Then, we can use the same argumentation as above applied to the convex hull of the union of the respective polar plots of slices. The corresponding strategy then may involve a change of sails (see Figure 9).  
Figure 8: Diagram showing how to use the convex hull (gray) to obtain an overall better performance by tacks and jibes.  
For this construction to be precise, we have to make some idealised assumptions. First of all, we assume that the polar diagram precisely describes the capabilities of the boat. Since we are using the best available polar diagram, this assumption is reasonable. Secondly, we assume that the wind conditions are constant in time and position. This assumption is only reasonable if the weather is suitably stable. Lastly, we assume that a direction change can be done without loss of velocity. This assumption can be motivated in particular for longer periods of travel (and with as few direction changes as possible), since then these effects become neglectable.  
To summarise, these kinds of calculations work well if the wind conditions are relatively stable and we keep directions for a reasonable amount of time. In other situations, it may be favourable to use weather routing instead.