The Antipode of a Location

The popular childhood question: “if I dug a tunnel straight through the Earth, where would I come out?” The answer is the antipode of your starting location — the geometrically opposite point on the surface, exactly halfway around the globe. The arithmetic is simple; the geographic answers are surprising.

This article covers the formula, the famous antipodal pairs, why most antipodes turn out to be ocean, and the technical relevance of antipodes to coordinate-distance algorithms.

The formula

Given a point at latitude φ and longitude λ, its antipode is:

φ_antipode = −φ
λ_antipode = λ + 180° (mod 360°, normalised to [−180°, 180°])

Equivalently, the longitude is shifted by 180° in either direction, with the wrap to keep it in the conventional range.

A worked example. The Empire State Building at (40.7484°N, 73.9857°W):

φ_antipode = −40.7484° = 40.7484°S
λ_antipode = −73.9857° + 180° = 106.0143°E

So the Empire State Building's antipode is at approximately (40.7484°S, 106.0143°E) — in the Indian Ocean, roughly 800 km west of Perth, Australia.

Famous antipodal pairs

A selection of well-known cities and their antipodes:

| City | Coordinates | Antipode (approximate) | | ----------------------------- | --------------------------- | --------------------------------------------- | | London, UK | (51.5°N, 0.1°W) | Antipodes Islands, NZ (51.5°S, 179.9°E) | | New York, USA | (40.7°N, 74.0°W) | Indian Ocean off WA (40.7°S, 106.0°E) | | Madrid, Spain | (40.4°N, 3.7°W) | South Pacific Ocean (40.4°S, 176.3°E) | | Wellington, NZ | (41.3°S, 174.8°E) | Atlantic Ocean off Spain (41.3°N, 5.2°W) | | Buenos Aires, Argentina | (34.6°S, 58.4°W) | Eastern China (34.6°N, 121.6°E) | | Shanghai, China | (31.2°N, 121.5°E) | Atlantic Ocean off Argentina (31.2°S, 58.5°W) | | Tokyo, Japan | (35.7°N, 139.8°E) | South Atlantic Ocean (35.7°S, 40.2°W) | | Cape Town, South Africa | (33.9°S, 18.4°E) | Pacific Ocean near Hawaii (33.9°N, 161.6°W) | | Singapore | (1.3°N, 103.8°E) | South Pacific Ocean (1.3°S, 76.2°W) | | Hawaii, USA | (21.3°N, 157.8°W) | Botswana (21.3°S, 22.2°E) |

The pattern is striking: the vast majority of city antipodes fall in ocean. The reasons are historical and geometric.

Why most antipodes are ocean

About 71% of Earth's surface is ocean. Naïvely, you'd expect that 71% of antipodes are ocean. The reality is more extreme — about 96% of all land antipodes are ocean, with only ~4% of Earth's surface being “land at the antipode of land.”

The explanation: Earth's continents are concentrated in the northern hemisphere (about 67% of all land area). The antipodes of northern-hemisphere continents fall in the southern hemisphere — and the southern hemisphere is overwhelmingly ocean (the Pacific, Southern, and Indian Oceans dominate). North America's antipode is the Indian Ocean off the southwest coast of Australia; Europe's antipode is the South Pacific off New Zealand; Asia's antipode is largely ocean off the coast of South America.

The exceptions where land does fall opposite land:

• South America ↔ Asia: the substantial overlap of Argentina / Chile with China. Buenos Aires is the antipode of an area near Shanghai; most of Argentina is opposite eastern China.
• Borneo ↔ South America: parts of Borneo align with the northern Andes.
• New Zealand ↔ Spain: the small overlap between the North Island and central Spain. The Antipodes Islands, ironically, are not opposite Spain — they're opposite London.
• Hawaii ↔ Botswana: a small overlap.

A printed map showing land-land antipodal pairs is striking because of how little the overlap is.

The Antipodes Islands

A literal example of the “land opposite land” phenomenon: the Antipodes Islands, a small uninhabited volcanic group about 860 km southeast of New Zealand's South Island. They were named in 1800 by Captain Henry Waterhouse based on their being almost exactly antipodal to Greenwich, England — the historical home of the Royal Observatory.

Coordinates: 49.7°S, 178.8°E. The actual antipode of Greenwich (51.5°N, 0.0°E) is at (51.5°S, 180.0°), which is the south of the Antipodes Islands by about 200 km in ocean. The naming captures the geometric concept even if it isn't precisely geographically exact.

The islands have a Royal Navy meteorological station and are otherwise sub-Antarctic wilderness. They remain a small symbol of the antipodal concept three centuries after the British named everything “Antipodean” in popular geography.

Why antipodes matter for distance algorithms

Vincenty's formula is the standard ellipsoidal-geodesic distance computation. It uses an iterative scheme that converges to millimetre accuracy for most pairs of points — but fails to converge near antipodal points. Specifically, when the input pair is within about 50 km of being exactly antipodal, the iteration oscillates rather than settling.

The geometric reason: at antipodes, the geodesic isn't unique. Every great circle through the centre of Earth passes through both endpoints; any of them is “the” shortest path. The iteration has no preferred direction to converge toward, and the longitude update bounces back and forth.

Robust implementations of Vincenty (the Coordinately src/lib/coords/vincenty.ts module included) detect non-convergence after a maximum number of iterations (~100–200) and fall back to the spherical haversine formula, which always produces a result. The fallback result has ~0.5% accuracy rather than millimetre — acceptable because antipodal-pair distances are rarely needed at sub-percent precision anyway.

Modern algorithms like Charles Karney's 2013 method handle antipodes natively (using a reformulated equation system that doesn't depend on the offending iterative variable). The GeographicLib reference implementation of Karney's algorithm is the modern successor; PROJ 6+ and PostGIS use it.

The half-Earth distance

For exactly antipodal points, the geodesic distance is half the Earth's circumference. On a WGS 84 ellipsoid, that's approximately 20,037 km along a meridian (because the meridional circumference is 2π · b where b is the polar radius, ≈ 6,357 km). For an equatorial circumference, the half distance is 20,038 km (using a = 6,378 km).

The two numbers are within 1 km of each other; for everyday purposes, “~20,040 km” is the antipodal distance. Mathematically, the exact distance depends on the specific path (along a meridian vs along the equator vs along some intermediate great circle), but the variation across paths is below 1 km for a 20,000 km distance.

Antipodal maps

A classical cartographic curiosity: an antipodal map is an overlay of Earth and its mirror image, designed to show where land sits opposite land. Two transparent globes (or paper maps) are mounted so one is rotated 180° relative to the other in both latitude and longitude. Where land appears on both layers at the same projected position, that's a land-on-land antipodal point.

The result is visually striking because of how little the continents overlap. The biggest overlaps are:

• Argentina with China and Mongolia (~1.5 million km²)
• Indonesian island Borneo with northern South America
• A small region of Botswana with Hawaii
• Bits of Spain with New Zealand

The remaining ~96% of land has ocean at its antipode. Modern online antipodal-map tools (e.g., antipodesmap.com) let you visualise this for any input point.

Antipodal time-zone curiosities

Because the antipode shifts longitude by 180°, antipodal points have time zones offset by 12 hours. London (UTC+0 winter, UTC+1 summer) has its antipode in UTC+12 (the time zone including most of New Zealand). New York (UTC−5) has its antipode in UTC+7 (Indian Ocean timezone).

The midnight-noon symmetry: when it's noon at a location, it's midnight at the antipode. The pattern is exact up to small adjustments for daylight saving and the politically-drawn time-zone boundaries.

Common misconceptions

“If I dug straight down, I'd come out in China.” Depends entirely on where you start. From most of the continental US, you'd come out in the Indian Ocean (antipode of CONUS is mostly ocean off Australia). From Spain, yes — central New Zealand. From Argentina, China is roughly correct. The folk wisdom is locally specific.

“The antipode of the equator is the equator.” True — every equatorial point has its antipode on the equator, just on the opposite side of the globe. The equator is the only great circle that maps to itself under the antipodal transformation.

“The antipode of the North Pole is the South Pole.” Yes — the only point-antipode pair where both endpoints are distinguished, named locations. Every other point's antipode is somewhere most people couldn't name without looking it up.

“Antipode is just (−lat, −lon).” No — it's (−lat, lon ± 180°). The latitude flips sign; the longitude shifts by 180° (wrapping around the antimeridian). Negating both gives a different (unrelated) point that's often mistaken for the antipode.

“The antipodal distance is always 20,000 km.” Approximately. The exact distance depends on the ellipsoid model and the specific great-circle path chosen between the antipodes (all such paths are nominally the same length on a sphere; small differences exist on the ellipsoid). On WGS 84, the distance varies from about 20,003 km to 20,038 km depending on the path. The variation is small but real.

“Antipodes are exact on the WGS 84 ellipsoid.” On a perfect sphere, the antipode formula (−φ, λ ± 180°) is exact. On the WGS 84 ellipsoid, the formula gives the antipode to within sub-millimetre precision because the antipodal transformation is symmetric on any geocentric ellipsoid. The “Earth is an ellipsoid” complication doesn't affect antipode computation in practice.

“The Pacific Ocean is dominant because Earth's landmass is biased.” Exactly so — the Pacific covers about 30% of Earth's surface but contains only a handful of significant landmasses (Australia, New Zealand, Indonesia, Japan, the Philippines, Pacific island nations). When you flip the Earth's land hemisphere through the geocentre, you land in the Pacific. This is also why the Pacific dominates discussions of trans-equatorial flight routing, ocean shipping, and global climate — Earth's geography is structurally asymmetric, and antipodes are one of the simplest demonstrations of that asymmetry.

“You can dig a hole through Earth and come out at the antipode.” Geometrically yes — practically very, very no. Earth's core is molten iron at ~6,000 K with extreme pressure. The deepest borehole ever drilled (the Kola Superdeep, Russia) reached only ~12 km — about 0.2% of the way to the centre. The popular childhood thought experiment is the introduction to the antipode concept; the actual physics of boring through Earth has no engineering path.