Finding Distance Using the Earth's Grid System
Date: 08/05/97 at 22:28:15 From: Nathan Grand Subject: Finding distance using the earth's grid system Dr. Math, My question is: How do I find the distance between two points on the earth using degrees of latitude and longitude, and how do I find the shortest way to get there? I have a feeling that my question may lie in the way an airplane taking off from Washington, U.S.A and travelling to Japan banks off the North Pole instead of going straight to Japan. Please help me in my quest to find the answer.
Date: 08/06/97 at 08:21:13 From: Doctor Jerry Subject: Re: Finding distance using the earth's grid system Hi Nathan, Assuming the earth is a sphere and you want the great circle distance between two points: Let (x,y,z) be a point on a sphere of radius a. The spherical coordinates of (x,y,z) are (a,phi,theta), where phi is like latitude, but it is measured from the positive z-axis. The angle phi varies between 0 and pi. The angle theta is like longitude, but is measured from the positive x-axis (towards the positive y-axis). The angle theta varies between 0 and 2*pi. If you are given the spherical coordinates (a,phi,theta) of a point, the (x,y,z) coordinates are given by x = a*cos(theta)*sin(phi) y = a*sin(theta)*sin(phi} z = a*cos(phi). So, if you have a point on the earth's surface, you know the number a, which is the radius of the earth, you know or can easily calculate phi from the latitude, and you know or can easily calculate theta from the longitude. So, suppose you are given the points (a,phi_1,theta_1) and (a,phi_2,theta_2). The first thing to do is to calculate the rectangular coordinates r_1=(x_1,y_1,z_1) and r_2=(x_2,y_2,z_2) of these points, as above. After that, letting alpha be the angle between the lines joining the center (0,0,0) of the sphere to r_1 and r_2, use the dot product of the vectors r_1 and r_2 to calculate alpha. You will find that alpha= arccos(r_1.r_2/a^2) = arccos(cos(phi_1)*cos(phi_2) +cos(theta_1-theta_2)*sin(phi_1)*sin(phi_2)). The great circle distance between r_1 and r_2 is d = a*arccos(alpha). -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 08/06/97 at 12:17:01 From: Doctor Anthony Subject: Re: Finding distance using the earth's grid system This can be done quite easily by using the scalar product of two vectors to find the angle between those vectors. If the vectors are OA and OB where A and B are the two points on the surface of the earth and O is the centre of the earth, the scalar product gives OA*OB*cos(AOB) = R^2*cos(AOB) where R = radius of the earth. Having found angle AOB the distance between the points is R*(AOB) with AOB in radians. To find the scalar product we need the coordinates of the two points. Set up a three-dimensional coordinate system with the x-axis in the longitudinal plane of OA and the xy plane containing the equator, the z-axis along the earth's axis. With this system, the coordinates of A will be Rcos(latA), 0, Rsin(latA) and the coordinates of B will be Rcos(latB)cos(lonB-lonA),Rcos(latB)sin(lonB-lonA),Rsin(latB) The scalar product is given by xA*xB + yA*yB + zA*zB = R^2cos(latA)cos(latB)cos(lonB-lonA)+ R^2sin(latA)sin(latB) Dividing out R^2 will give cos(AOB) cos(AOB) = cos(latA)cos(latB)cos(lonB-lonA)+sin(latA)sin(latB) This gives AOB, and the great circle distance between A and B will be R*(AOB) with AOB in radians. The great circle path is the shortest route, but that means you cannot steer a single magnetic course to follow this path. If the two points are in the Northern Hemisphere, you would find that the path tends north of the straight line path drawn on a conventional map. There are various map projections which allow you to plot the great circle as a straight line, and you will find that this line cuts lines of longitude at varying angles along its length. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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