It's the same one degree difference on the map, so why are the distances different?
The longitude lines meet at the poles, and the earth is not a perfect sphere: the map scale cannot be directly expressed as kilometers.
Sky coordinates and observations
Traveling one degree of longitude on the equator is much further than traveling the same degree of longitude at high latitudes. To connect two cities, we have to consider a path along the surface, rather than a straight line through the earth.
Guess first
If we go 1° along the longitude at the equator and 75° north latitude, are the kilometers the same?
Show answer and explanation
no the same — The latitude circles have different radii.
Then connect experience and science
The longitude scale will become more crowded as we go further
Moving from the equator to the poles, the same circle of latitude shrinks, and the distance represented by one degree of longitude decreases.
1° longitude = πRp/180
The length of latitude is not equal everywhere
The oblateness of the Earth causes the curvature of the meridians to vary slightly with each degree of latitude.
1° latitude = πRm/180
Make the same circle and go further to the equator
All locations share the angular velocity of the Earth's rotation, but the farther away from the axis of rotation, the greater the linear velocity.
v = ωRp
Measure the distance between two points along the ground
First use the average spherical radius to roughly calculate, and then use the Andoyer ellipsoid to approximate the correction. The near-opposite situation is degraded and precise results cannot be faked.
s≈D[1+fH₁sin²Fcos²G−fH₂cos²Fsin²G]
Rp is the radius of the latitudinal circle, and Rm is the radius of meridional curvature, both in kilometers. The arc length per degree is multiplied by π/180; the linear velocity of rotation also requires the angular velocity ω. The distance between two points on the earth's surface is approximated by an ellipsoid, and the two lengths per degree cannot be directly added.
give it a try
First look at the length of each degree of 42° latitude, and then compare the surface distance from Paris to Washington in the original book.
No need to memorize symbols
- Rp/Rm
- Latitude circle radius/meridian radius of curvature, km
- ω
- The rotational angular speed in the book is 7.292114992×10⁻⁵rad/s
- s
- Reference ellipsoid surface distance approximation
Algorithm, accuracy and scope of application
Supplement Chapter 11: Length per degree of latitude and longitude, linear velocity of rotation, spherical distance and Andoyer ellipsoid correction. The longitude is due west in the book, and is due east on this site. The two-point difference will be inverted at the same time without changing the square term. Coincidence points return 0; near-antipodal degeneration returns unavailable instead of dividing by zero.
my understanding check
Does one degree of longitude at high latitudes get shorter, which means the Earth's angular velocity slows down?
No, the latitude circle is smaller — The same angular velocity can correspond to different linear velocities.
Is the distance between two points in this lesson a straight line through the earth?
No, it’s the approximate distance on the surface — The chord length cannot be used as a surface path.
What happens when the near-antipodal point formula degenerates?
Clarify reporting limits — More complete geodetic methods suitable for this situation are needed.
Angles are not kilometers. Latitude, the shape of the earth, and path definition all affect actual distance.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 11 chapter