VMG & ZMG

What's the problem?

VMG stands for Velocity Made Good.
ZMG stands for aZimuth Made Good.
ZMG is also known as CMG (Course Made Good).

The VMG is the projection of the Boat Speed (BSP) against the direction of the True Wind.
The ZMG is the projection of the Boat Speed (BSP) against the direction of the next mark, which can be independent of the wind direction.

VMG ZMG

The VMG should (ideally) be the same on both tacks. Unless your mark is right upwind or dead downwind, the ZMG is not the same on both tacks, which makes a huge difference.

Progression upwind or downwind is not to be understood as progressing toward a given point. The way to manage and understand those progressions are completely different. We are going to try to explain this difference...

The best VMG is obtained when BSP * cos(TWA) reaches its maximum (or minimum for downwind). The layline in this case is on both tacks given by the same TWA, as long as the wind does not change.

Determining the best ZMG on one leg follows the same rule. The ZMG is the best when the Boat Speed multiplied by the cosinus ot the angle between the route and the mark reaches its maximum value.

Where things begin to be very different is when it's about to determine where to change the course. Unlike in the VMG case, where we can expect the performance to be the same when we appropriately tack or jibe, as the wind is not involved in the ZMG calculation, the speed of the boat will more than probably change on the other leg, and this change in speed will depend on the place where this change happened.
The best ZMG gives the route on which the progression in the direction of the mark is the fastest, and not towards the mark.

A & B
If A & B have left S at the same time, then the course of B has a better ZMG than the course of A, even if it is longer in distance. But what about comparing the two courses again when B will have to change his course to go to the mark? Maybe at that time, A will have a better ZMG than B!
The big question for B is definitelly When to go to the mark ?
What composite route has the best ratio between speed and distance?
What composite route is the fastest?
And this calculation is pretty complex...

...What's the solution?

When the target is between the laylines, upwind or downwind, there is no problem. The solution is the same as for the VMG. The distance to cover on one tack will be longer than on the other, and that's about it.

Otherwise, computing such a route requires a good knowledge of the polars of the boat, and some good processing capabilities.

Considering that:

The polars are generally made out of several curves
Those curves can have several kinds of equations (polynoms, splines, b-splines, Béziers, etc)
For a given kind of curve, the degree or its equivalent may vary
The number of curves for one polar is not defined (1, 2, more...)

... it will be very difficult to setup a generic model for the performance prediction.
If the polars of the boat are available, then the most efficient solution is probably to iterate on several options and find the best one.
The frame not to exit could be defined by the Best ZMG at the origin, and by the layline at the mark.
Iterations to find the best route might concern two areas:

The initial course, to be chosen within a specific range
The point where to change the course and go to the mark

Let's take an example:
The direct course to the mark is called directCourse.
The course having the best ZMG is called bestZMGcourse .
The distance to the mark is called distanceToMark.
In meta-language, the algorithm we will use will look like this:

      
  for (alpha=directCourse; alpha<bestZMGcourse; alpha += ((bestZMGcourse - directCourse) / 10))
  {
    for (dist=0; dist<distanceToMark; dist += (distToMark / 10))
    {
      position = whatWeReachByFollowing(alpha, dist)
      timeToMark = timeFromOriginTo(position) + timeToMarkFrom(position)
    }
  }

We then have to find the course and distance for which timeToMark is the smallest.
In the sample above, we iterate 10 times on the course, and 10 times on the distance. This parameter (10) can be increased, depending on the capabilities of the machine used for the computation.
The functions timeFromOriginTo and timeToMarkFrom are defined after the polars, for a given TWS.

The result of such a computation may vary a lot, depending on the shape of the polars.

Volvo 60' J/105

Volvo 60' J/105

As shown above, finding the right route might be critical on a Volvo 60 footer...

...more to come...

Any comment ?