Meridian arc

A meridian arc is an important idea used in geodesy and navigation. It describes the curve between two points on Earth that share the same longitude. This means if you imagine lines running from the North Pole to the South Pole, a meridian arc would be a small part of one of those lines close to the Earth's surface.

People have studied meridian arcs for many years. They use special tools to measure these curves directly in the field, and they also use math and geometry to calculate their lengths. Both ways help us understand the shape and size of our planet better. This knowledge is useful for maps, GPS, and many other tools we rely on every day.

The study of meridian arcs has helped scientists learn more about Earth. By measuring these curves, they can figure out how far places are apart and how the ground is shaped. This information is important for many jobs, from making maps to planning journeys across the world.

Measurement

Measuring meridian arcs helps us understand the shape of the Earth. By measuring these curves, scientists can figure out the best shape to represent our planet, called a reference ellipsoid. These measurements, taken at different places around the world, help create a model that fits the whole Earth.

Long ago, people used simple tools to measure these arcs. Today, we use advanced tools like satellites to get even more accurate results. These measurements are important for many things, like mapping and navigation.

19th century

After the French Revolution in 1789, there was a big effort to create new measurement standards. Scientists wanted to measure the curve of Earth called the meridian passing through Paris to help define the length of a metre. This work was led by the French Academy of Sciences.

Teams of scientists travelled to measure this distance from north to south. They faced many challenges, including difficult travel and political troubles after the Revolution. They measured parts of the meridian in France, from Dunkirk to Barcelona, and used these measurements to help create the metric system. These careful measurements helped scientists understand more about the shape of our Earth.

The nautical mile

Long ago, a nautical mile was thought to be the distance of one small part of Earth's curve called a minute of arc. Today, it is exactly defined as 1,852 metres. For most everyday uses, people still measure distances using the latitude scale on maps, where one minute of latitude is treated as one nautical mile.

Calculation

See also: Latitude § Meridian arc

On a sphere, the length of a meridian arc is simply the distance along a circle. On an Earth-shaped ellipsoid, for short arcs, the length can be estimated using the Earth's radius and circular distance formulas.

For longer arcs, the length is found by subtracting the distances from the equator to two points at different latitudes.

This idea is important for making maps, especially for a type of map called the transverse Mercator projection.

The exact formulas involve special math functions and series, but they help scientists and mapmakers calculate distances very accurately on the Earth's surface.

Quarter meridian

The distance from the equator to the pole is called the quarter meridian. It was used in the past to help define the metre and the nautical mile.

The quarter meridian can be calculated using special math formulas. For Earth, the quarter meridian is about 10,001,965.729 metres long.

The inverse meridian problem for the ellipsoid

Sometimes we need to solve the opposite problem: if we know the distance between two points, how can we find their latitude? One way to do this is by using a step-by-step method called Newton's method. This method keeps improving its guess until it gets very close to the right answer.

There is also another way using special math patterns called series. These patterns help us calculate the latitude when we know the distance between points. A famous mathematician named Adrien-Marie Legendre showed that measuring distances on Earth is related to shapes called ellipses. This idea helps solve problems about measuring distances on our planet.