## How are projections blended?

### How are projections blended?

I've been using GeoCart to blend projections, and I understand it does that based on the specified p value where 0 < p < 1. I know the manual says the p value is used in calculating the new blended projection by weighting the two parent projection outputs. I'm assuming this is more involved than starting with a point on the globe (phi and lambda), calculating the projected x and y for either parent projection, and taking a weighted mean of the the two parent x and y values for the blended x and y. Does that p value get applied differently in different projections, depending on the functions a projection uses to calculate its x and y values? Or am I over-thinking it, and it is simply a post-projection weighted averaging of the plotted points of either parent projection?

### Re: How are projections blended?

Welcome to the boards, P.Raposo.

As far as the generating formulæ for the resulting blended projection go, they are just the weighted average of the two projections as you describe IF the two projections differ. If the two projections are the same but have different parameters or other elements, then Geocart applies a weighted average to each of the differing components. It is pretty intelligent about this; if, for example, the differing parameter is a discrete value ranging from 0…9, then if and only if your frame count is 10, Geocart will properly distribute and weight the intermediate frames. There is one other special case bit of handling: If the two projections are the same, but their projection centers differ, then normally the 0–1 parameterization describes a great circle path. If, however, the difference in center is only a meridional rotation, then the parameterization will follow the unchanging parallel rather than the great circle connecting the two endpoints.

The far more difficult matter in blending is the description of the outline in the intermediate projections. It’s easy enough for two rather conventional projections, of course. But Geocart allows for any arbitrary (but described) outline in a projection, and so blending two arbitrary projections implies an arbitrary (but undescribed) boundary. The huge difficulty is in ferreting out that boundary, particularly since there is no guarantee that there will not be topological absurdities. Of course overlapping is not allowed in principle, but in practice Geocart does not try to detect such things. It just fails if it runs into something that “doesn’t work”. Sometimes overlaps are not egregious enough to cause a failure that Geocart can’t work around, and so you can get some very odd things to happen.

The next update will remove some impediments in blending, especially when the domain of the projection has been constricted by the boundaries you set, for example. It will also increase the robustness of the blending in general. I apologize if you are running into a lot of “The projections are incompatible” errors. That will never go away entirely, but at least you’ll soon see a big improvement.

As far as the generating formulæ for the resulting blended projection go, they are just the weighted average of the two projections as you describe IF the two projections differ. If the two projections are the same but have different parameters or other elements, then Geocart applies a weighted average to each of the differing components. It is pretty intelligent about this; if, for example, the differing parameter is a discrete value ranging from 0…9, then if and only if your frame count is 10, Geocart will properly distribute and weight the intermediate frames. There is one other special case bit of handling: If the two projections are the same, but their projection centers differ, then normally the 0–1 parameterization describes a great circle path. If, however, the difference in center is only a meridional rotation, then the parameterization will follow the unchanging parallel rather than the great circle connecting the two endpoints.

The far more difficult matter in blending is the description of the outline in the intermediate projections. It’s easy enough for two rather conventional projections, of course. But Geocart allows for any arbitrary (but described) outline in a projection, and so blending two arbitrary projections implies an arbitrary (but undescribed) boundary. The huge difficulty is in ferreting out that boundary, particularly since there is no guarantee that there will not be topological absurdities. Of course overlapping is not allowed in principle, but in practice Geocart does not try to detect such things. It just fails if it runs into something that “doesn’t work”. Sometimes overlaps are not egregious enough to cause a failure that Geocart can’t work around, and so you can get some very odd things to happen.

The next update will remove some impediments in blending, especially when the domain of the projection has been constricted by the boundaries you set, for example. It will also increase the robustness of the blending in general. I apologize if you are running into a lot of “The projections are incompatible” errors. That will never go away entirely, but at least you’ll soon see a big improvement.