[color=#999999][color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/uT3czZnE]Earth and Sun[/url].[/color][br][br][/color][color=#ff0000][center][b]Warning[/b][br]The content of this page may offend the sensibilities of [i]flat Earth beliefs[/i] [url=https://en.wikipedia.org/wiki/Flat_Earth#Modern_flat_Earth_beliefs][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url].[/center][/color]Imagine you place two different points, A and B, on a plane. What is the route to go from A to B without changing direction? And what is the route of minimum length to go from A to B? You know that both questions have the same answer: the straight segment AB. But this happens because these routes are on a flat surface. If A and B are on a spherical surface, the answer is no longer the same.[br][br]RHUMB LINE. This construction will help you interpret much better the most well-known of the terrestrial globes, a result of the [i]Mercator projection[/i] [url=https://www.geogebra.org/m/yjxp7xm3][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url]. The Mercator map owes its popularity to the fact that any constant direction (for example, northeast) is represented as a straight segment on the map, facilitating navigation with a compass. However, on the globe, we can see that, in reality, this fixed course follows a helix, called a [color=#cc0000][i]rhumb line[/i][/color], which connects the two poles (or a circle if the points have the same latitude or longitude).[br][br]ORTHODROMIC. The shortest path between two points on the Earth's surface does not correspond to the rhumb line but to the arc of the great circle, that is, centered at the Earth's center, connecting them. This arc is called the [i][color=#e69138]orthodromic [/color][/i](specific term for the [i]geodesic curve[/i] [color=#ff0000][url=https://en.wikipedia.org/wiki/Geodesic][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url][/color] on the spherical surface). The arcs of the meridians or of the equator are particular cases of orthodromics. A parallel arc, on the other hand, is not an orthodromic. Note that the circumference of the great circle, being a closed line, connects the two points by two different routes: the orthodromic (the shortest "straight" route) and the remaining arc (the longest "straight" route).[br][br]CIRCULAR. When the term "circular route" is used, especially in hiking, it actually refers to a "closed route" (not necessarily circular), meaning it ends where it began. This characteristic is often appreciated by hikers as they do not have to backtrack or return by other means to the starting point. In this construction, in addition to the rhumb line and the orthodromic, you can see the only circumference on the Earth's surface that has the given points as diametrical extremes. It's a [color=#1e84cc]circular [/color]route (this time, literally). In the construction, you can observe the transformation that this circle undergoes when projected onto the Mercator map.[br][br]On the terrestrial globe in the construction, there are two points, one orange and one green. The points of the same color on the Mercator map are their projections. You can choose the position of both points. If you want precision when using the sliders, use the [b]arrow keys[/b] on your keyboard. There are six interconnected modes to choose the position:[br][list][*]Enter their [b]coordinates [/b]with the lower (E-W longitude) and right (N-S latitude) slider. Keep in mind that some latitudes may fall outside the limits of the Mercator map. To get the coordinates of a specific location, search for it on Google Maps [url=https://www.google.com/maps/@30.3374442,4.9036185,4z?entry=ttu][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url] and right-click on it.[/*][*]Move the upper slider, which scrolls through a list of [b]cities [/b]ordered by country. The point will occupy the position of the chosen city.[br][/*][*]Move the point directly on the earth [b]globe[/b]. Use the longitude and latitude sliders to fine-tune the precision. To rotate the entire globe, drag it with the right-click (if you release the mouse button before stopping the drag, the globe will enter automatic animation).[/*][*]Move the point directly on the Mercator [b]map[/b]. Use the longitude and latitude sliders to fine-tune the precision.[/*][*]By directly moving either of the points, either on the globe or on the map, the corresponding text will display the nearest population from the list of cities. Click on that [b]text [/b]to choose that city.[br][/*][*]Activate one of the three [b]checkboxes [/b]with predefined routes. In this case, the information for the country and precise location will appear.[br][/*][/list]Activate (or deactivate) the [color=#cc0000]Rhumb Line[/color], [color=#e69138]Orthodromic [/color]and [color=#1e84cc]Circular [/color]checkboxes to display (or hide) the routes, in red, yellow, and blue, respectively. Use the [color=#9900ff]violet [/color]buttons and the [color=#9900ff]violet [/color]slider (speed) to control the animation. The construction also shows the kilometers of each route and the constant value of the angle at which the rhumb line intersects the parallels (both on the Mercator map and on the earth globe).[br][br]The three predefined routes are:[br][list][*][b]The longest straight overland route[/b] [url=https://www.geogebra.org/m/yjxp7xm3#material/acghuujj][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url] connects the Portuguese coast with the Chinese coast.[/*][*][b]The longest straight sea route[/b] [url=https://www.geogebra.org/m/yjxp7xm3#material/djfy6rqf][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url] does not follow the orthodromic but the complementary arc, that is, the longest arc of the great circle between the two points. It connects the coast of Pakistan with the Russian coast opposite Alaska. You can watch a video about a very similar route here [url=http://geogebra.es/From%20India%20to%20USA%20in%20a%20straight%20line.mp4][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url].[/*][*][b][/b][b]The straight route Washington - Mecca[/b] follows the [i]qibla[/i][i] [url=https://en.wikipedia.org/wiki/Qibla][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url] [/i](i.e., it points toward the Kaaba in Mecca) from Washington. We have predefined this route due to the curious anecdote [i][url=https://geogebra.es/La%20mezquita%20de%20Washington.mp4][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url][/i] related to the construction of the mosque headquarters of the Islamic Center of Washington, which illustrates well the common confusion between the rhumb line ("straight" on the map) and the orthodromic ("straight" on the globe). Currently, Google has an application [i][url=https://qiblafinder.withgoogle.com][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url][/i] to find the qibla from anywhere in the world.[br][/*][/list]Note that all these routes are "ideal"; they do not consider geographical features, human constructions, or natural phenomena such as winds or currents. In practice, orthodromic routes can only be accurately followed by artificial satellites in circular orbit, although a modern ship, in good weather, can closely adhere to an orthodromic maritime route.[br][br]The construction lends itself to various uses. For example, why not try to discover what the longest "straight" overland route is from where you are?[br][br][color=#999999]For better execution, it's recommended to [url=https://www.geogebra.org/material/download/format/file/id/r6zymu4x]download de GGB file[/url].[/color]

Mercator map displayed corresponds to a partial view, from latitude -78.6° to 83.7°, of the image created by Daniel R. Strebe (2011). If the GGB file is downloaded, you can access the full image, from -85° to 85° [i][url=https://commons.wikimedia.org/wiki/File:Mercator_projection_Square.JPG][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url][/i]. [br][br]Of the infinite rhumb lines that (in general) pass through two given points, the shown rhumb line is the only one that does not exceed half a turn (180°) on the sphere, that is, among them, the shortest one. As you can see, this choice entails a sharp change of the chosen course just at the border of the 180° difference between the longitudes of the chosen points.[br][br]The list of cities covers 842 populations from 200 countries and includes all their capitals. Don't like this list and want to create your own? Of course, it's very simple. To do this, just [url=https://www.geogebra.org/material/download/format/file/id/r6zymu4x]download the GGB file[/url] and replace this list (of lists):[br][center]cities= {{"Country", "City", 0, 0}, {"Afghanistan", "Kabul", 69.18, 34.52}, {"Albania", "Tirana", 19.83, 41.33}, ...}[/center]respecting the first list {"Country", "City", 0, 0} that serves as the header and taking into account that the numbers correspond to the longitude and latitude (without the ° symbol for degrees) of the city, in that order. Don't forget to enclose the texts of countries and cities in quotes.[br][br]Once you have replaced that list, you must adjust the sliders of the city list ([color=#ff7700]nA[/color] and [color=#38761d]nB[/color]) to the length of the new list. Initially, this length is 843, as there are 842 cities in the list plus 1 header. That's all.[br][br][br][br][color=#999999][color=#999999]Author of the construction of GeoGebra: [color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url][/color][/color]. I want to thank [url=https://www.geogebra.org/u/jamora]José Antonio Mora[/url] and [url=https://www.geogebra.org/u/javier+cayetano]Javier Cayetano[/url] for their proposals related to this topic, as well as, naturally, the work of [url=https://www.geogebra.org/u/ccambre]Chris Cambré[/url].[/color]