Spherical Geometry, Beach Style

When you’ve got a Pottery Barn basket girlfriend in your SoCal beach house, and you’re lying together under a colossal throw blanket watching the sun go down on the maritime horizon, what landmass are you looking toward? The answer might surprise you.

Counterintuitive Latitudes

One’s first instinct might be to take a map of the world and trace the same line of latitude (a special instance of what’s technically called a “rhumb line”) until you hit the nearest landmass across whatever ocean you’re looking at. In the case of the Western Hemisphere, including the United States, the Washington Post once published this helpful map depicting which parts of the coast correspond to the latitudes of which countries across the sea:

In the case of California’s South Coast, this method yields the result that your position corresponds to Japan. Follow a rhumb line due west from anywhere in California, in fact, and the next landmass you’ll hit is Japan.

The Washington coast looks out toward Russia; to be directly due east of China you have to be in Mexico’s Baja California!

Another interesting aspect of this map is that it reveals just how far south the United States lies compared to Europe. As far north as the Outer Banks along the East Coast you’re at latitudes that correspond to North Africa, not anywhere in Europe. Portugal and Spain correspond to latitudes far enough north to catch parts of Nova Scotia. Almost as far north as the Torngat Range of Labrador you’re only looking at Britain and Ireland, rather than any truly nordic part of Europe; for that you have to go up to Greenland’s latitude.

This, of course, is a product of climatology; the Gulf Stream warms Europe like no other continent of its latitude, and west coasts such as Europe are always under more maritime influence than their east-coast counterparts, for the simple reason that prevailing winds (from the west in temperate zones) come off the ocean there, whereas on east coasts prevailing winds come from more extreme continental interiors. Hotter and more humid summers are one effect, but an even more pronounced effect is colder winters; eastern North America has the coldest winters of any region of its latitude, excepting East Asia, which not coincidentally is the only region in the temperate world downwind of an even vaster interior than North America’s.

Anyway, looking at the map above again, it gets interesting in the southern hemisphere. Notice that large parts of Chile correspond to Australia and New Zealand; land is so scarce in large parts of the Pacific that distances upward of 5000 miles must be traversed to find any along those latitudes.

It gets even worse for landlubbers in the Americas’ southernmost reaches; notice the southernmost part of Argentina’s east coast corresponds to Chile, and the southernmost part of Chile’s west coast corresponds to Argentina. This is an artifact of that portion of South America being literally the only land in the world at those latitudes. Indeed, there’s a large stretch of latitudes along the Drake Passage that don’t cross any land at all; it’s all ocean.

What’s really over the Horizon?

Notice that heretofore we’ve used rhumb lines. This is helpful in figuring out which pieces of land are the same distance from the equator as you, but this method does not in fact tell you what you’re looking at across the beach. When you look out along the beach, you’re looking along a straight line, and straight lines on a globe are not rhumb lines, but rather great circles: relative to the longitude and latitude grid, they curve. A common and trivial example of this phenomenon is the great-circle routes followed by airliners.

Another factor is that beaches are angled; when Malibu Barbie looks out across the water from home, she’s not even looking due west, but rather south, owing to the beachfront stretching from west to east, not north to south. So, given these two factors — spherical geometry, and angled beaches, what is she looking at? Well, if she’s looking straight out from Santa Monica Pier, she’s looking at New Zealand; from her house, she’d be looking at Antarctica. On the flip side, if she goes out to San Miguel Island’s beaches she may well be looking at Japan, though along a great-circle route that almost touches the Aleutian Islands rather than a rhumb line across the central Pacific.

Indeed, about the only continent you can’t “see” from some beach or another in California is Europe. Helpfully, Andy Woodruff made these maps that chart the great-circle routes that various coastlines look out at, so you can see what exactly it is you’re looking towards from a particular stretch of coast:

In particular, this map makes it clear the concentrations of vantage points; i.e. how common is it to be looking out towards a given continent on a given stretch of coast. Denser and more vivid lines correspond with more vantage points.

There are a few surprises: most of the United States’ East Coast actually looks out toward Africa, not Europe, owing primarily to how the coast is angled. Only a few beaches, mostly spits of land that jut out from the coast at odd angles, look out toward Europe, and even then mostly to Portugal rather than Northern Europe. South America, on the other hand, has an abundance of beaches looking out toward northern Europe, owing, again, to how the coast is angled.

Indeed, so sharp is the effect of the angling that it’s much more commonplace to be looking out toward Australia from North America’s East Coast than it is Europe. This owes to the straight lines that you’re looking out along bypassing the southern reaches of Africa and then proceeding onward across the Indian Ocean to Australia; there’s very little land along these lines, so they hit nothing but ocean until nearly antipodal distances are reached.

Curiously, this has the effect of making Australia a very commonly viewed continent from both coasts of the United States; many west coast beaches look out at Australia too!

Europe and North America have relatively few vantage points worldwide, perhaps owing to how they’re nearer to the centers of the “land hemisphere”, the half of the world that has the greatest concentrations of land, which tends to block these lines from proceeding across truly great distances.

South America, on the other hand, has abundant viewing points from all continents. Ditto for Australia and to a lesser extent Africa. Asia is a bit of an exception, with abundant viewing points despite being located in the land hemisphere, but this perhaps owes to its large size.

The real champion, though, might be Antarctica; located near the center of the water hemisphere, with clear paths to every continent. Quite a few vantage points toward Antarctica are located deep in the Arctic, i.e. at antipodal distances!

Interesting stuff, I think.

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