Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 193

That is, the length of a quadrant of a meridian is approximately 10,000 km, and
then the circumference of the Earth is approximately 40,000 km. The length of a
“meter” was originally defined as a ten-millionth of this length. In 1793, this length
became the standard in France. However, it was later found that the first prototype
meter bar was short by a fifth of a millimeter because of miscalculation of the
flattening of the Earth. The polar circumference of the Earth is therefore shown to
be about 8,000 m more than 40 million meters.
When the arc length is short (e.g., X

45 km, calculations are accurate to
0.001 m), the meridian can be considered a circular arc. The radius of the circle
is the meridional radius of curvature M m at the mean latitude B m ﾼ

<

1

of
the arc. The central angle equals the difference in latitude between the two
endpoints, namely

2 B 1 þ

ð B 2 Þ

ʔ B ﾼ B 2 B 1 . Its computational formula is given by:

M m ʔ

B
ˁ

X ﾼ ,

ˁ 00 ﾼ 206264.80624710 00 ,or

o

57.29577951308 o .

with ˁ ﾼ

Formula for Length of a Parallel Arc

A parallel (circle of latitude) is a circle, so the arc length along a parallel is the
length of the circular arc of its subtended angle at the center (difference in
longitude).
In Fig. 5.15 , let P 1 and P 2 be two points on the parallel. Their latitude is B,
difference in longitude is l, the radius of the parallel is r, and P 1 K

ﾼ N is the radius

of curvature in the prime vertical; we have:

r ﾼ N cos B : ð 5 : 44 Þ

Hence, the formula for the length of a parallel arc can be written as:

r l N cos B l S ﾼ

ˁ ﾼ ˁ :

ð 5 : 45 Þ

Variations in the Unit Meridian and Parallel Arc Lengths with Latitude

The formulae for the arc elements of the meridian and the parallel are given by:

dX ﾼ MdB

:

dS ﾼ rdL

The meridian radius of curvature M increases gradually with increasing latitude,
whereas the radius of the parallel r decreases sharply with increasing latitude.
Hence, the meridian arc length of the unit latitude difference increases slowly