16 days ago

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Projected Coordinate System Datums are among the top reasons people think we live on a sphere.

_____
One more time, any calculations based on latitude and longitude will yield the graticule it was projected from.

To calculate those speeds, guess what they used?

ALL distances based on the graticule.
the formula for calculating the speed of a point moving along a circle's circumference, which is based on the circle's radius and the rotational speed:
𝑣=𝜔𝑟
where 𝜔 (omega) is the angular velocity, and 𝑟 is the radius of the circle at the latitude in question.

Angular Velocity (𝜔): Earth rotates 360 degrees per day, which translates to an angular velocity of 360∘24 hours=15∘ per hour.

Radius at Tropic of Cancer (𝑟):
The radius of the path of a point on the Tropic of Cancer can be calculated from the Earth's radius at the equator multiplied by the cosine of the latitude (𝑅cos⁡(latitude)

Assuming Earth's average radius 𝑅 is about 6,371 kilometers, the radius at the Tropic of Cancer is 6371×cos⁡(23.5∘).
With R≈6,371 km and latitude ≈23.5∘
the radius 𝑟 at the Tropic of Cancer becomes 6371×cos⁡(23.5∘) kilometers.

Convert this radius into miles if required
(1 km = 0.621371 miles).

Use the radius to calculate the speed:
𝑣= 15∘ ℎ𝑜𝑢𝑟 × (𝑟 in miles or kilometers)

At the equator, the distance per degree of longitude is about 69 miles (or about 111 kilometers).
This is derived by dividing the Earth's equatorial circumference by 360 degrees.
The formula to calculate the actual distance per degree of longitude at a given latitude is:
𝐷=cos⁡(latitude)×69 miles

ALL BASED ON THE GRATICULE.
ALL distances based on the graticule.

Now, consider the Sun.

Makes perfect sense, the sun is going the same speed in the south as it does in the north, but with a different tangential velocity . You know speed and tangential velocity are different, right?
V=wr VS V = d/t
Of course, since the Gleason map represents the exact same data as the globe, which is the same as every other map that uses the required longitude and latitude, the tropics are figured to be about 22,859 miles long. Because the math is based, of course, on the coordinate system....Its easy
Circumference of tropics l=2πR⋅cos(ϕ)
where
R [mean radius of the Earth, approximately 6378.137 kilometers]
ϕ is the latitude of the Tropic of Cancer in radians.

Or more accurately for your geoid.....
l=2πR⋅cos(ϕ)⋅(1−0.00669438sin2(ϕ))−0.5
(I mean, we are better globers than you as well)
Because YOU have to account for Earth's oblateness (flattening at the poles) using the eccentricity factor 0.006694380.00669438, and it subtracts 0.50.5 to adjust for possible overestimation.

Of course, you can always go backwards, and derive the latitude at your current tropic by
l = 2π cos(φ) ⋅ 6378137 ⋅ (1 - 0.00669438 sin^2(φ))^(-0.5)
Given l = 36788 km, we can plug this value into the formula and solve for φ.
36788 = 2π cos(φ) ⋅ 6378137 ⋅ (1 - 0.00669438 sin^2(φ))^(-0.5)
We have to denote φ = cos(φ) and φ = sin(φ), so our equation becomes:
36788 = 2φ ⋅ φ ⋅ 6378137 ⋅ (1 - 0.00669438 φ^2)^(-0.5

Right?

The Answers (FE Only)
---------------
1) What do all maps project?
All maps project a coordinate system onto a flat surface. The most commonly used coordinate system is the geographic coordinate system, which models the Earth as a sphere or ellipsoid. This system uses latitude and longitude to specify locations on the Earth’s surface. Different map projections then transform these spherical coordinates into a two-dimensional representation, each with its own set of compromises regarding area, shape, distance, and direction.

2) How is latitude determined?
Latitude is determined by measuring the angle between the horizon and a known celestial object,
such as the Sun at its highest point (noon) or Polaris (the North Star) at night. This angle corresponds directly to the latitude of the observer. For example, if Polaris is 30 degrees above the horizon, the observer is at approximately 30 degrees North latitude.

3) How is longitude measured - what observations were crucial in establishing the farthest southern longitudes, as requested by the Board of Longitude in 1714?
Since The geographic coordinate system relies on the positions of celestial bodies to determine latitude and longitude, Taking angles to the stars at various longitudes was a fundamental aspect of properly mapping and refining the geographic coordinate system. They supposedly studied the variations of the night sky, as different stars become visible, and the lunar values - the angular distance between the Moon and a specific star - then compared them with distances listed in the nautical almanac.

4) What are the resultant differences in geodesic surveying, commonly referred to as 'deflection of the vertical', used for in regard to celestial navigation?
The deflections of the vertical are critical for accurate celestial navigation and are accounted for in various ways, including through the use of ephemerides and navigational almanacs.
They incorporate the deflections of the vertical to adjust the observed altitude. The deflections of the vertical are critical for precise celestial navigation. While they are not directly added to the almanac or ephemeris, correction tables or algorithms are used to adjust observed celestial altitudes. These corrections ensure accurate navigation by accounting for local gravitational anomalies, enhancing the reliability of positional data derived from celestial observations. Modern navigational systems and digital tools integrate these corrections
automatically, streamlining the process for navigators.

5) What are any of the 3 stated purposes of astrogeodetic surveying based on? [Based on the document "The Use and Abuse of Vertical Deflections,"]
Transformation of Astronomical Coordinates to Geodetic Coordinates: Converting coordinates determined by observing celestial bodies to those based on an ellipsoidal Earth model.
Conversion of Astronomic Azimuth to Geodetic Azimuth: Adjusting azimuths (directional angles) derived from astronomical observations to correspond with geodetic references.
Reduction of Vertical and Horizontal Angles to the Spheroid: Correcting measured angles to account for the curvature of the Earth, ensuring they align with the spheroidal model used in geodetic systems.

[Source for the paper]( https://www.semanticscholar.org/paper/THE-USE-AND-ABUSE-OF-VERTICAL-DEFLECTIONS-Featherstone/8e1900b750cd2687e2688972b9b4dc5c5773ac92)
Featherstone, W. E. (1999). The use and abuse of vertical deflections. In Proceedings of the Sixth South East Asian Surveyors’ Congress, Fremantle, Western Australia.

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