Do Van Der Grinten I and Van Der Grinten II uh... not have the middle Indicatrix?

[daan]

These two projections are extremely troublesome around the origin. 25 years ago, when I implemented them, I spent a lot of time trying to work out a series development for longitude close to the prime meridian and latitude close to 0 within that. I only managed a partial solution.

I can sympathize. In my very crude program, written in BASIC, suitable only for drawing illustrative maps of the whole world, what I've done in such cases is, for a degree or so around the problematic parallel or meridian, to approximate the circles of radius unreasonably large by parabolas. No further terms in the power series.
And, as yet, I've made no attempt to try to calculate the Tissot indicatrix, let alone isoclines, whatsoever.

I like it.

— daan

[Piotr]

Does the official method to compute the reverse for Van Der Grinten exist? If so, how much does it cost to get the license to the official Van Der Grinten formulas?

[daan]

Does the official method to compute the reverse for Van Der Grinten exist? If so, how much does it cost to get the license to the official Van Der Grinten formulas?

Van der Grinten’s patent filing was for a drafting method. There are no “official” formalæ. I use Snyder’s development, which is forward-only.

— daan

[quadibloc]

Van der Grinten’s patent filing was for a drafting method. There are no “official” formalæ.

I realize that you're responding directly to the specific phrasing used. But still, that sounds strange to me. After all, a "drafting method" directly implies the (forwards only) formula to which it is equivalent. Indeed, I just took the same method I used for determining the desired point of intersection between two intersecting circles used for the Nicolosi Globular, and used it again with the Van der Grinten, just changing the locations of the circles to correspond to the projection.
So in my mind-set, the only question is whether Snyder's formula, had I chosen to go to him as a source, would be correct (doubtless, they would be)... the very concept of a formula being "official" or not would not occur. The drafting method is given, it specifies the projection, and the formula is equivalent. That means they're the same thing.

[daan]

So in my mind-set, the only question is whether Snyder's formula, had I chosen to go to him as a source, would be correct (doubtless, they would be)... the very concept of a formula being "official" or not would not occur. The drafting method is given, it specifies the projection, and the formula is equivalent. That means they're the same thing.

And yet, you use a (deliberate) approximation for parts of it.

Snyder’s or your formulation may well be perfectly congruent to van der Grinten’s drafting method, but as far as I know, no one has ever proved Snyder’s rendition. It makes sense to me to talk about “official” (or at least original) formulæ almost any time because second-hand comes with risk of error. Also, even the originator doesn’t always end up with what they intended due to errors in derivation or by using the necessary evil of an approximation. Later someone might come along and correct the error. Now you have two distinct projections usually sharing the same name. Which is official? Depends on context.

There are more reasons. In the large-scale projections world, you have to be specific about which series development or approximation you use. Any official system is specific about that, often even down to how many digits to round to at what steps. Even though this thread is about a small-scale projection, where simple plotting isn’t (usually) going to be visibly affected, your choice of approximation would affect distortion analysis.

In my descriptions of some of my published projections, besides the “pure” mathematical description, I provide series developments for problematic regions so that the person implementing them doesn’t have to worry that their calculations don’t unintentionally (and even undetected) go haywire. Not many people are well equipped to deal with such problems. Are those series developments part of the “official” formulæ? Perhaps yes: they are how the originator described the thing.

There is the pure math, and then there is the approximation we actually use, since computers never give exact results. That lack of exactitude increases for each individual computation unless specifically controlled for. If the originator supplies implementation instructions, I consider that to part of the official definition and formulation.

— daan

[quadibloc]

In the large-scale projections world, you have to be specific about which series development or approximation you use. Any official system is specific about that, often even down to how many digits to round to at what steps.

Oh, indeed. And, of course, in the world of large-scale projections, one has to start considering using an ellipsoid instead of a sphere. And this reminds me of something I read somewhere: that the official Australian geoid is a specific polynomial approximation, rather than the original ideal mathematical formula from which it was derived.

[Piotr]

Does the official method to compute the reverse for Van Der Grinten exist? If so, how much does it cost to get the license to the official Van Der Grinten formulas?

Van der Grinten’s patent filing was for a drafting method. There are no “official” formalæ. I use Snyder’s development, which is forward-only.

— daan

But doesn't Geocart use both forwards and backwards formulas, as well as formulas for Tissot Indicatrix and min/max scale formulas, as well as all the formulas required for all of the possible blends?

[daan]

But doesn't Geocart use both forwards and backwards formulas, as well as formulas for Tissot Indicatrix and min/max scale formulas, as well as all the formulas required for all of the possible blends?

I think you know that the answer to that is “no”.

Geocart does implement some inverse formulæ, generally for the most commonly used projections. In a few cases, the Tissot indicatrix and other analytics also implemented explicitly (such as for the ellipsoidal transverse Mercator). Everything else is approximated using numerical techniques.

— daan

[Piotr]

But doesn't Geocart use both forwards and backwards formulas, as well as formulas for Tissot Indicatrix and min/max scale formulas, as well as all the formulas required for all of the possible blends?

I think you know that the answer to that is “no”.

Geocart does implement some inverse formulæ, generally for the most commonly used projections. In a few cases, the Tissot indicatrix and other analytics also implemented explicitly (such as for the ellipsoidal transverse Mercator). Everything else is approximated using numerical techniques.

— daan

Is that why many blends fail?

[daan]

Geocart does implement some inverse formulæ, generally for the most commonly used projections. In a few cases, the Tissot indicatrix and other analytics also implemented explicitly (such as for the ellipsoidal transverse Mercator). Everything else is approximated using numerical techniques.

Is that why many blends fail?

Most blends that fail do so either because (a) the topology has overlaps, or (b) Geocart wasn’t able to synthesize a coherent boundary. The boundary synthesis is complicated and the code is buggy.

If the topology does not overlap itself and if Geocart can synthesize a usable boundary, then the blended projection will normally succeed.

— daan