Extreme Points of a Great Circle - (Part 3, Great Circles)

The Northernmost and Southernmost Points of a Great Circle

If you travel from Amsterdam (P1 in Figure 1) to San Francisco (P2) or the other way around, then you first go towards the north for a while, and then towards the south for a while. All great circles except for the equator have a northernmost point (PN) and a southernmost point (PS). You can calculate them as follows.

Fig. 1: Northernmost and Southernmost Point on a Great Circle

Certain Direction from a Point - (Part 2, Great Circles)

Suppose you want to know where you go if you start from a particular town in a particular direction and keep going straight (along a great circle). You can calculate the coordinates of points along the route as follows:

Fig. 2. Mercator and Hondius

Definition of a Great Circle - (Part 1, Great Circles)

[Since great circle calculations are so important on all spheres (like Earth, planets, polar coordinates, etc.) I have to add here such a text. This follows mostly /1/]

Fig. 7. Arthur H. Robinson 1979

["Arthur H. Robinson (January 5, 1915 – October 10, 2004) was an American geographer and cartographer, who was professor in the Geography Department at the University of Wisconsin–Madison from 1947 until he retired in 1980. He was a prolific writer and influential philosopher on cartography.

One of Robinson's most notable accomplishments is the Robinson projection. In 1961, Rand McNally asked Robinson to choose a projection for use as a world map that, among other criteria, was uninterrupted,[9] had limited distortion, and was pleasing to the eye of general viewers.[10] Robinson could not find a projection that satisfied the criteria, so Rand McNally commissioned him to design one.

Robinson proceeded through an iterative process to create a pseudo-cylindrical projection that intends to strike a compromise between distortions in areas and in distances, in order to attain a more natural visualization. The projection has been widely used since its introduction. In 1988, National Geographic adopted it for their world maps but replaced it in 1998 with the Winkel tripel projection."]

1950's Computer Language "George"

[The Laning and Zierler system (sometimes called "George" by its users) was one of the first operating algebraic systems, that is, a system capable of accepting mathematical formulae in algebraic notation and executing equivalent machine code.

The system accepted formulas in a more or less algebraic notation. It respected the standard rules for operator precedence, allowed nested parentheses, and used superscripts to indicate exponents.

It was among the first programming systems to allow symbolic variable names and allocate storage automatically. The system also automated the following tasks: floating point computation, linkage to subroutines for the basic functions of analysis (sine, etc.) and printing, and arrays and indexing. It could also solve automatically ordinary differential equations using Gills' variation of the 4th order Runge-Kutta Method, that was an inbuilt language feature.

Dr. J. Halcombe Laning

It was implemented in 1952-53 and published in 1954 for the MIT WHIRLWIND computer by J. Halcombe Laning and Neal Zierler. It was made during a time with similar UNIVAC A-2, IBM Speedcoding and a number of other systems that were proposed but never implemented.

The following text is a reprint of a MIT's summer session report 1954.]