Area of a Latitude-Longitude Rectangle
Date: 07/31/2003 at 16:36:02 From: Stan Subject: Area of Section on Earth's Surface for "Spherical Rectangle" Given the latitudes and longitudes of four points on the Earth's surface, how do you calculate the surface area enclosed by the four points? The four points are chosen so that they lie on only two lines of latitude and two lines of longitude. For example, suppose the points are (15N,25E), (15N,40E), (60N,25E), and (60N,40E). The enclosed area is defined by the polygon constructed using the four segments of latitude and longitude lines connecting adjacent points. The area calculation is complicated by the fact that the latitude and longitude line segments are actually curves on the Earth's surface and that the distance between points separated by X degrees of longitude varies according to the cosine of the latitude. I assume that one needs to first convert the point locations to spherical coordinates and then compute the area.
Date: 07/31/2003 at 21:11:50 From: Doctor Rick Subject: Re: Area of Section on Earth's Surface for "Spherical Rectangle" Hi, Stan. What you describe is not really a spherical polygon. That term is reserved for figures made of arcs of great circles. Lines of longitude are great circles, but lines of latitude are not (except for the equator). Therefore your figure is not a spherical polygon. Actually, if a rectangle is defined as a quadrilateral with four right angles, there is no such thing as a spherical rectangle. The sum of the angles of a spherical quadrilateral is GREATER than 360 degrees. The following page from the Dr. Math Archives describes how to find the area of a true spherical rectangle in the sense of a spherical quadrilateral with four EQUAL angles (greater than 90 degrees). It is NOT what you want (if you really know what you want and you have described it accurately). Spherical 'Rectangles' http://mathforum.org/library/drmath/view/60748.html Now we can lay aside the question of terminology and consider your figure, whatever we call it. I'll call it a lat-long rectangle. I helped a student with the same problem some time ago. We started with the formula for the area of the earth between a line of latitude and the north pole (the area of a spherical cap, listed in the Dr. Math FAQ on Geometric Formulas). A = 2*pi*R*h where R is the radius of the earth and h is the perpendicular distance from the plane containing the line of latitude to the pole. We can calculate h using trigonometry as h = R*(1-sin(lat)) Thus the area north of a line of latitude is A = 2*pi*R^2(1-sin(lat)) The area between two lines of latitude is the difference between the area north of one latitude and the area north of the other latitude: A = |2*pi*R^2(1-sin(lat2)) - 2*pi*R^2(1-sin(lat1))| = 2*pi*R^2 |sin(lat1) - sin(lat2)| The area of a lat-long rectangle is proportional to the difference in the longitudes. The area I just calculated is the area between longitude lines differing by 360 degrees. Therefore the area we seek is A = 2*pi*R^2 |sin(lat1)-sin(lat2)| |lon1-lon2|/360 = (pi/180)R^2 |sin(lat1)-sin(lat2)| |lon1-lon2| - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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