Meridian arc

Definition and properties

On a perfect sphere a meridian arc is a circular arc whose length is proportional to the central angle between the two latitudes. The real Earth is better approximated by an oblate ellipsoid of revolution: curvature and thus meridian length vary with latitude. For this reason the linear length of one degree of latitude is not constant; it averages about 111 kilometres and changes by a few hundred metres between equator and poles. Determinations of meridian arcs therefore reveal how much the planet departs from a sphere and constrain the parameters of reference ellipsoids.

Mathematical formulation

On an ellipsoid with semi-major axis a and eccentricity e the length of the meridian arc between latitudes φ1 and φ2 is given by the integral of the meridian radius of curvature M(φ):

arc length = ∫φ1φ2 M(φ) dφ, where M(φ) = a(1 − e²) / (1 − e² sin²φ)^(3/2).

Practical calculations use closed-form series expansions in eccentricity or flattening, or numerical integration. Series are convenient for map projections and coordinate conversions because they yield sums of trigonometric terms in latitude; numerical methods are used when high precision is required over long distances.

History and measurement

Meridian arcs have played a central historical role in determining the size and figure of the Earth. Around 240 BC Eratosthenes compared solar zenith observations at Syene and Alexandria to produce an early estimate of Earth's circumference. In the 18th century scholarly expeditions measured long meridian arcs to test theoretical predictions: one team led by Pierre Louis Maupertuis measured an arc in Lapland, and another (including Charles Marie de La Condamine) worked in what is now Ecuador. In the late 18th century the French republic undertook a detailed survey between Dunkirk and Barcelona by Jean-Baptiste Delambre and Pierre Méchain; their results contributed to the original definition of the metre.

Isaac Newton predicted on theoretical grounds that a rotating fluid Earth would be flattened at the poles (an oblate spheroid) and gave an early estimate of the flattening. Subsequent triangulation and astronomical measurements confirmed the oblate character and refined its parameters. In modern times satellite geodesy (GNSS, satellite altimetry and gravity missions) has provided global and highly precise constraints on meridian lengths and the reference ellipsoid.

Applications

Meridian arc lengths are fundamental for converting latitude differences into linear distances, for establishing and scaling geodetic networks, and for defining national and global map datums and reference ellipsoids. Cartographers and surveyors use formulas for meridian arcs when projecting geographic coordinates onto flat maps, when computing distances in cadastral work, and when integrating local surveys into global coordinate systems such as WGS84. The distinction between a mathematical ellipsoid and the geoid (the equipotential surface approximating mean sea level) is clarified by combining meridian arc data with gravity measurements.

Computation and modern practice

Accurate meridian arc computation requires an adopted ellipsoid or a model of the geoid and may involve corrections for local gravity anomalies when relating physical distances to mean sea level. Modern practice blends astronomical, gravimetric and satellite observations. Satellite missions that measure Earth's gravity field and shape have largely supplanted long ground arcs for global parameter determination, but ground-based meridian arcs and triangulation networks remain important for regional control and historical consistency of map datums.

Further reading and resources

• Introduction to geodesy
• Longitude and meridians
• Geometry of arcs
• Arc segment (mathematics)
• Curve and curvature concepts
• Imaginary lines on the globe
• Models of the globe and map projection basics
• Reference ellipsoid definitions
• Geoid versus ellipsoid
• Determination of the figure of the Earth
• Satellite geodesy overview
• Alexandrian science and observers
• Eratosthenes and ancient measurements
• Earth circumference estimates
• Solstices and astronomical references
• Local solar time and observations
• Zenith, altitude and solar observations
• Ancient Egyptian observation sites
• Newton's Principia and terrestrial shape
• Oblate spheroid, flattening and ellipsoid theory