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

[Piotr]

What?

[Piotr]

Can you state a source where Van Der Grinten I and Van Der Grinten II are defined as having no middle Indicatrix?

[daan]

Can you state a source where Van Der Grinten I and Van Der Grinten II are defined as having no middle Indicatrix?

Nobody made such a claim.

— daan

[Piotr]

Can you state a source where Van Der Grinten I and Van Der Grinten II are defined as having no middle Indicatrix?

Nobody made such a claim.

— daan

So, why does Geocart depict Van Der Grinten I and Van Der Grinten II as having no middle Indicatrix?

[Atarimaster]

Piotr, it’s simply a bug. The same bug was to be found in a different projection a while ago (can’t remember anymore which projection it was).
A workaround is to set the center longitude to 0°0'0.0036"E, but I’m sure the bug will be fixed in the next Geocart version.

Edit: Apparently, the workaround only works for van der Grinten I, van der Grinten II seems to need a higher value, e.g. 0°5'E.

[Piotr]

Here's a zoom into the Van Der Grinten II, up to 0.2° north, east, south, and west, rendered in Draft. Each white pixel is undefined.

[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’m a little more experienced at this now. They’re fixed, but it was still a lot of frustrating work. The projections defined by graphical procedures using straight-edge and compass all suffer regions of problematic numerics (Fournier, Nicolosi globular &c).

— daan

[quadibloc]

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.

[Piotr]

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.

Wouldn't it cause a problematic discontinuity in the map?

[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.

Wouldn't it cause a problematic discontinuity in the map?

It causes a discontinuity unless more steps are taken to smooth it out. The discontinuity isn’t normally “problematic” unless the series development is not sufficiently accurate.

— daan