Maximum Difference, Longitude and LatitudeDate: 04/10/2001 at 00:32:40 From: Mini Thomas Subject: Maximum longitude and latitude difference between two points separated by a given distance I want to find the maximum longitude difference and maximum latitude difference between two points A and B on Earth that are separated by a distance of 1000km. I tried the formula theta = cos (LatA)* cos(LatB)* cos(LonA-LonB)+ Sin(LatA)* Sin(LatB), where theta in radians =1000KM/EarthDistance. But I am stuck finding max(LatA - LatB) and max (LonA-LonB) which can be applied to within this distance. Can you please help me? Date: 04/10/2001 at 14:31:14 From: Doctor Rick Subject: Re: Maximum longitude and latitude difference between two points separated by a given distance Hi, Mini. Do you want the maximum latitude and longitude differences for *any* pair of points separated by 1000 km? If A is within 1000 km of the north or south pole, then B could have any longitude whatsoever; the maximum longitude difference is thus 180 degrees. The maximum latitude difference is the same regardless of where point A is, and it isn't hard to calculate. If you draw a circle of radius 1000 km (that is, a radius following the curvature of the earth) around point A, this circle will be tangent to latitude lines at points due north and south of A. These points are therefore the extremes of latitude difference, and the latitude difference there (in radians) is just 1000 km divided by the radius of the earth (R = 6367 km). If you're interested in the maximum longitude difference for a *given* point A, then we have more work to do. As with latitude, we find the points of maximum longitude difference by looking for the points where the circle drawn around point A is tangent to a line of longitude. At these points, the line joining A and B is perpendicular to the line of longitude at B. In other words, A can be reached from B by starting out due east or west and following a great circle route. But since lines of latitude are not great circles, A and B are NOT at the same latitude (unless they are on the equator, which is a great circle). To find the longitudes in question, pass a line through the center of the earth O, perpendicular to the line (circle) of longitude we are looking for. Let this line intersect the surface of the earth at C. Every line (great circle) from C to the line of longitude meets it at right angles, just as lines of longitude meet the equator at right angles. The line through C and A therefore meets the line of longitude at right angles; thus the point where this line meets the line of longitude is the point B that we seek. Since angle AOB is (1000 km/R) radians (as in the latitude calculation), and angle COB = pi/2 radians, and the four points are coplanar, the angle COA must be (pi/2 - 1000/R) radians. Here is where we are so far: given the latitude and longitude of point A, we need to find the point C on the equator that is (pi/2 - 1000/R) radians of arc away from A. (Actually, there will be two such points, one east and one west.) Then we add (or subtract, depending on which point we found) pi/2 radians to the longitude of C, and this is the longitude of point B. Subtract the longitude of A to find the maximum longitude difference at A's latitude. To find point C, we can use vector algebra, discussed in this connection in several places in our archives, including this answer: Distance Between Two Points on the Earth http://mathforum.org/library/drmath/view/51722.html Letting the radius of the earth be 1 unit, we can identify point C (which is on the equator) as (1, 0, 0), on the x-axis of my coordinate system. Then point A will be (cos(lat1)*cos(lon1), cos(lat1)*sin(lon1), sin(lat1)) where lon1 is the *difference* in longitude between C and A. The scalar product of these two vectors is equal to the product of their magnitudes times the cosine of the angle between them. Thus, cos(COA) = A.C = cos(lat1)*cos(lon1) We know lat1, and we know that angle COA is pi/2 - 1000/R. Thus we can solve for lon1, the longitude difference between A and C: lon1 = arccos(cos(pi/2-1000/R)/cos(lat1) = arccos(sin(1000/R)/cos(lat1) The longitude of point B in the coordinate system that I have chosen is pi/2. Therefore the maximum longitude difference is max_dlon = pi/2 - lon1 = arcsin(sin(1000/R)/cos(lat1)) Let's check out this formula. When lat1 = 0 (A is on the equator), we expect the maximum longitude difference to be 1000/R. In fact, since cos(0) = 1, we get the arcsine of the sine of 1000/R, which is 1000/R. When lat1 = pi/2 - 1000/R (so that B could be at the north pole), we expect the maximum longitude difference to be pi/2 (since the 1000-km circle is tangent at the pole to the longitude line 90 degrees from A's longitude). In fact, since cos(pi/2 - 1000/R) = sin(100/R), we have max_dlon = arcsin(1) = pi/2. At higher latitudes, cos(lat1) < sin(1000/R), and the arcsin of a number greater than 1 does not exist. As we saw above, at these latitudes it is actually possible for point B to have any longitude. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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