On the design of globe gores

[Piotr]

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

When you blend A with B, if the parameter is below 0.0 that's extrapolation of A away from B, and if the parameter is above 1.0 that's extrapolation of B away from A. So, when sinusoidal is extrapolated away from the plane chart, it is no longer the pole that are compressed to a point, but instead a lower latitude (both north and south). Instead, the pole is an inverted line.

This can also be simulated in Geocart:

Make a sinusoidal map.
Make a plane chart map, set it at a conformal scale of 0.5 and an angle of 180°.
Blend the projections.
Set the blend at a conformal scale of 4.

[daan]

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.

When you blend A with B, if the parameter is below 0.0 that's extrapolation of A away from B, and if the parameter is above 1.0 that's extrapolation of B away from A. So, when sinusoidal is extrapolated away from the plane chart, it is no longer the pole that are compressed to a point, but instead a lower latitude (both north and south). Instead, the pole is an inverted line.

This can also be simulated in Geocart:

Make a sinusoidal map.
Make a plane chart map, set it at a conformal scale of 0.5 and an angle of 180°.
Blend the projections.
Set the blend at a conformal scale of 4.

Yes. A self-intersecting boundary is fine; it’s a self-intersecting region that causes problems. Self-intersecting boundaries are not uncommon. A typical set of globe gores, for example, intersects itself at the equator for each meridian not at the outer edge. The meridian, because it is duplicated, could be considered continuous and impinging itself at one point, or could be considered to be crossing itself at one point. Your construction is a little more exotic.

— daan

[Piotr]

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.

When you blend A with B, if the parameter is below 0.0 that's extrapolation of A away from B, and if the parameter is above 1.0 that's extrapolation of B away from A. So, when sinusoidal is extrapolated away from the plane chart, it is no longer the pole that are compressed to a point, but instead a lower latitude (both north and south). Instead, the pole is an inverted line.

This can also be simulated in Geocart:

Make a sinusoidal map.
Make a plane chart map, set it at a conformal scale of 0.5 and an angle of 180°.
Blend the projections.
Set the blend at a conformal scale of 4.

Yes. A self-intersecting boundary is fine; it’s a self-intersecting region that causes problems. Self-intersecting boundaries are not uncommon. A typical set of globe gores, for example, intersects itself at the equator for each meridian not at the outer edge. The meridian, because it is duplicated, could be considered continuous and impinging itself at one point, or could be considered to be crossing itself at one point. Your construction is a little more exotic.

— daan

The extrapolated map, after the 60th latitude also has the region mirrored: as the boundary outline self-intersected, it flipped direction, so the interior becomes mirrored.

Why did you then say, "It might find a continuous, closed boundary, but the boundary might intersect itself, which ought to be disallowed.", when this map shows that a boundary self-intersection is allowed?

[daan]

Why did you then say, "It might find a continuous, closed boundary, but the boundary might intersect itself, which ought to be disallowed.", when this map shows that a boundary self-intersection is allowed?

Because I wasn’t talking about intersection that could be interpreted instead as merely impinging.

— daan

[Piotr]

Why did you then say, "It might find a continuous, closed boundary, but the boundary might intersect itself, which ought to be disallowed.", when this map shows that a boundary self-intersection is allowed?

Because I wasn’t talking about intersection that could be interpreted instead as merely impinging.

— daan

So how would it numerically be computed which is which?

[daan]

So how would it numerically be computed which is which?

I don’t try. I render the boundary, and if that results in a closed boundary, and if I can find an inverse toward the top of the boundary, then I “trust” the projection and proceed with full rendering. That doesn’t mean the projection is bijective. If it’s not, you can get bizarre artifacts.

For example, I made a programming error. This resulted:

Spirograph.jpg (79.26 KiB) Viewed 1838 times

I rather like it.

— daan

[Piotr]

So how would it numerically be computed which is which?

I don’t try. I render the boundary, and if that results in a closed boundary, and if I can find an inverse toward the top of the boundary, then I “trust” the projection and proceed with full rendering. That doesn’t mean the projection is bijective. If it’s not, you can get bizarre artifacts.

For example, I made a programming error. This resulted:
Spirograph.jpg

I rather like it.

— daan

How many polygons is that?

[daan]

How many polygons is that?

We won’t know unless you count them. I haven’t.

— daan