Atarimaster wrote:Hello,
for some time now, I have been wondering: Is there a handy, easy-to-understand definition of the term »minimum error map projection« resp. »low-error« (as used by Canters)?
Low-error indicates an effort to reduce error compared to other, similar projections, but without trying to claim that the results are the best. Since there are no rigorous criteria for this, the term is used loosely and sometimes irresponsibly.
Minimum-error, formally, means that the developer has
proved, mathematically, that there is no way to get better results given the design constraints. Many of these projections have been devised, including Clarke’s perspective azimuthal, Tissot’s optimization of Albers, and Eisenlohr’s epicycloidal.
However, informally, minimum-error may only indicate
algorithmic convergence. That is, the algorithm used to adjust the projection toward minimum error may simply have reached something close to a minimum, resulting in an empirical solution. The solution may not be
the minimum for any of three distinct reasons:
- The algorithm actually only found a local minimum instead of the global minimum;
- The algorithm does come very close to the true minimum, but due to computational constraints such as round-off error or the need to terminate the calculation in some reasonable time, the true minimum is not reached; or,
- The formulation is only an approximation such that, even in the absence of computational constraints, the true minimum would not be reached.
I am not aware of any published projections that are characterized by (1), though, due to the complexity of guaranteeing a global minimum in numeric processes, presumably several published projections would be found to suffer this deficiency if they underwent lengthy analysis.
Any projection that uses numeric minimization techniques is characterized by (2), including the “minimum-error” projections by Canters, Snyder, Laskowski, and many others.
As for (3), Snyder’s “GS50” projection is a fine example. In this case, Snyder did not construct a continuous level curve of scale around the region of interest, which would have qualified the projection as a true, formal, minimum-error projection. Instead, he used complex polynomials to approximate such a boundary. Hence this projection is subject to (2) as well, and very likely also (1). On the other hand, even if he had constructed a continuous boundary, the improvements would be marginal.
Technically, an empirical minimum-error projection is merely low-error, and yet, since rigorous effort went into characterizing constraints and computing the parameters needed to optimize the projection against those constraints, the results don’t normally differ meaningfully from true minimum error.
You sometimes see the term
optimal used as a way of distinguishing empirical minimum-error from analytic minimum-error, where the latter is called optimal. On the other hand, sometimes optimal just means the result of an empirical convergence (such as Lawskowki’s tri-optimal). I prefer to reserve “optimal” to mean the projection satisfies formal minimum-error constraints.
Summing all that up, I would explain the concept as,
You can measure many different properties on the sphere. When you project to the plane, you change the geometry of the surface, which necessarily compromises some measurements. You can choose which measures are more important to you, and you can optimize for those when you project. To do this, you must be clear about the measures you intend to optimize for, and you must be clear about how you characterize deviations in the map from the true measures on the sphere. Then you construct a projection method that minimizes those deviations, by some defensible definition of “minimize”. If your method necessarily arrives at a unique solution that cannot be improved upon for the same measures using the same definitions, then the result is a minimum-error projection for the measurements you chose.
— daan
