On the design of globe gores

[daan]

This is the slide deck for a presentation I gave at NACIS 2001. While the screen shots are outdated, and of course Geocart 3 is considerably more capable, the principles themselves have not changed.

Download is 4.5 MB.

Happy mapping!
— daan

[gearmar]

I would be very interested in reading through the slide deck. The hyperlink is broken. Can you please re-attach the link?

Thanks, Mark

[daan]

Well hrmph. I am not sure how that got lost. I have updated the link in my original post. Enjoy!

— daan

[gearmar]

Thanks very much. I see it now.

[Piotr]

globe gores under 1 parameter is American Polyconic, while as the parameter approaches Infinity, it becomes Sinusoidal. In between are compromise projections.

[Piotr]

If the parameter is below 1, globe gores is self-intersecting (but still bijective, as it doesn't fail to render). So if the paper is so stiff that its warpage is below 1, then you have to print the map in multiple pieces and put it together to make the self-intersecting globe gores.

[daan]

If the parameter is below 1

BUG.

Fixed.

globe gores is self-intersecting (but still bijective, as it doesn't fail to render)

Geocart does not perform some mathematical check for bijective property. It simply tries to render. It may not find a continuous boundary, in which case it will not render at all. It might find a continuous, closed boundary, but the boundary might intersect itself, which ought to be disallowed. However, Geocard does not try to detect self-intersection. In some of those cases, it will draw; in others it will not. If the boundary is continuous and not self-intersecting, it will draw, but the interior could be quite messed up, depending on all kinds of things.

— daan

[Piotr]

If the parameter is below 1

BUG.

Fixed.

But what if the paper is so stiff that its warpage is below 1, then does that mean users would have to downgrade Geocart or edit the source code themselves (if it ever becomes open-source) to do this?

globe gores is self-intersecting (but still bijective, as it doesn't fail to render)

Geocart does not perform some mathematical check for bijective property. It simply tries to render. It may not find a continuous boundary, in which case it will not render at all. It might find a continuous, closed boundary, but the boundary might intersect itself, which ought to be disallowed. However, Geocard does not try to detect self-intersection. In some of those cases, it will draw; in others it will not. If the boundary is continuous and not self-intersecting, it will draw, but the interior could be quite messed up, depending on all kinds of things.

— daan

What's the problem with self-intersecting boundaries? If you were to extrapolate the sinusoidal projection away from the plane chart, you get a self-intersecting boundary as well, but the actual area doesn't overlap. Is Geocard a mobile version of Geocart?

[Piotr]

Make globe gores support complex numbers, positive i warpage is that the paper warpage is more conformal, while negative i warpage is that the paper warpage is more equal-area.

[daan]

What's the problem with self-intersecting boundaries?

It implies that the range of the map overlaps itself, which is no longer bijective. (The boundary is allowed not to be bijective and is allowed to touch itself. It must not cross itself.)

If you were to extrapolate the sinusoidal projection away from the plane chart, you get a self-intersecting boundary as well, but the actual area doesn't overlap.

I don’t follow “away from the plane chart”.

— daan