How far is Viadana from Ceuta? ( study of the routes of Air Navigation using Spherical geometry <ul><li>Problem </li></ul><ul><li>How can we describe the air course between two cities very distant? </li></ul><ul><li>To visualize the problem we can use a globe. </li></ul><ul><li>The segment is the shortest line between 2 points and, if we have to cross a square, it is better to move following a straight line rather than any other way. </li></ul><ul><li>To go from Viadana to Ceuta we cannot follow a straight line because: </li></ul><ul><li>the roundness of the Earth and </li></ul><ul><li>on the land surface there aren’t straight lines </li></ul><ul><li>For this problem we need new knowledge about the geometric model that is more suitable to solve this question. The surface of a sphere is a good model for this problem. </li></ul>
This line passes a few kilometers away from the center of the Earth except at the poles and the equator where it passes through Earth's center. Lines joining points of the same latitude trace circles on the surface of the Earth called parallels , as they are parallel to the equator and to each other. The north pole is 90° N; the south pole is 90° S. The 0° parallel of latitude is designated the equator , the fundamental plane of all geographic coordinate systems. The equator divides the globe into Northern and Southern Hemispheres. The Longitude (abbreviation: Long., λ , or lambda) of a point on the Earth's surface is the angle east or west from a reference meridian to another meridian that passes through that point. All meridians are halves of great ellipses (often improperly called great circles ), which converge at the north and south poles. To locate the position of Ceuta and Viadana we have to use a geographic coordinate system that enables every location on the Earth to be specified by a set of numbers. A common choice of coordinates is latitude , longitude The geographic latitude (abbreviation: Lat., φ , or phi) of a point on the Earth's surface is the angle between the equatorial plane and a line that passes through that point and is normal to the surface of a reference ellipsoid which approximates the shape of the Earth .
Geographic_coordinates_sphere.svg A line passing near the Royal Observatory, Greenwich (near London in the UK ) has been chosen as the international zero-longitude reference line λ = 0° , the Prime Meridian . Places to the east are in the eastern hemisphere, and places to the west are in the western hemisphere. The antipodal meridian of Greenwich is both 180°W and 180°E. In 1884, the United States hosted the International Meridian Conference and twenty-five nations attended. Twenty-two of them agreed to adopt the location of Greenwich as the zero-reference line. The Dominican Republic voted against the adoption of that motion, while France and Brazil abstained. To date, there exist organizations around the world which continue to use historical prime meridians which existed before the acceptance of Greenwich became common-place. The combination of these two components specifies the position of any location on the planet.
COORDINATES OF OUR TOWNS ITALY - Istituto d’Istruzione Superiore “S.G. Bosco” – Viadana - IT (Coordinator): 44°55′36″N 10°31′12″E SPAIN - Instituto de Ensenanza Secundaria “Luis de Camoens” - Ceuta - ES (Partner): 35°53′17″N 5°18′43″W FRANCE - Lycee Francois Bazin - CHARLEVILLE-MEZIERES - FR (Partner): 49°46′00″N 4°43′00″E IRLANDA DEL NORD - South West College - Omagh - GB (Partner): 54°35′50″N 5°56′20″W ICELAND - Kopavogur Institute of Education - Kopavogur - IS (Partner): 64°7′N 21°46′W TURKEY - Ozel Sarıyer Doga Lisesi - ISTANBUL - TR (Partner): 41°01′N 28°58′E VIADANA CEUTA BELFAST CHARLEVILLE KOPAGOVUR ISTANBUL longitudine α 10,5200° -5,3120° 5,9389° 4,7167° -21,7667° 28,9667° latitudine β 44,9267° 35,8880° 54,5972° 49,7667° 64,1167° 41,0167°
We have one point P on the surface of the Earth (Figure 3); it’s completely located by two angles (α, β), α is the longitude and β is the latitude: −180° < α ≤ 180° −90° ≤ β ≤ 90°. Excepting the 2 poles (with latitudes + 90° e −90° and longitude whatever ) this coordinate system set a correspondence among the points on the Earth surface and the angles (α ,β ) . The best way to solve this problem is to consider the spherical system which have its origin on the centre of the Earth and the two angles we described (longitude and latitude) as coordinates of the point P ( α, β). We have that the distance OP has these coordinates: OP = [cosα ⋅ cosβ , sinα ⋅ cosβ , sinβ ] and d ( P , Q ) = arccos cos(α1 −α2 ) ⋅ cosβ1 ⋅ cosβ2 + sinβ1 ⋅sinβ2 .
If there is + before the angle of longitude, we call it East, and North for the latitude. Some examples: Longitude Latitude Rome +12° 27’ +41° 55’ Milan +09° 11’ +45° 29’ New York −70° 15’ +40° 45’ Buenos Aires −70° 40’ −33° 30’ Sydney +151° 10’ −33° 55’ We choose, as measurement unit of distances, the Earths’ equatorial radius (about 6378 km = 1 RT, which is correspondent of the angle of 1 radiant). So that d ( P , Q ) = arccos cos(α1 −α2 ) ⋅ cosβ1 ⋅ cosβ2 + sinβ1 ⋅sinβ2 . Viadana 1672 Ceuta 1125 2254 Belfast 694 1744 544 Charleville 2925 3331 1872 2235 Kopagovur 1560 3021 2272 2118 4116 Istanbul
Siano P (α1,β1) e Q (α2,β2) due punti della superficie sferica. La loro distanza è la lunghezza dell'arco di circonferenza massima di estremi P, Q e coincide con la misura (in radianti) dell'angolo PC ˆ Q . Con il prodotto scalare si procede al calcolo della distanza PQ . Poiché: Calcoliamo prima di tutto il prodotto scalare dei due vettori: La misura, in radianti, di d ( P , Q ) rappresenta, in raggi terrestri, la distanza tra i due punti. Si giunge quindi alla risposta al problema iniziale: d (Roma, New York) = 1,041 RT. Tale distanza di 1,041 radianti corrisponde a circa 6640 km (Figura 6). Math is B.E.A.U.
And if we have to visit all the Partners’ Countries how is the order for the shortest distance???