Spherical 'Rectangles'Date: 05/13/2002 at 06:07:22 From: Saira Subject: Great-Circle Formulae Sir, Suppose I have two points on earth, (Lat1, Lon1) and (Lat2, Lon2). If I consider these to be the the top left and bottom right corners of a rectangle, how can I find the other two corner points? And the lengths of the sides of the rectangle? I am familiar with the Great Circle Formulae and the terms but have been unable to find any solution to the above problem. Kindly help me. Thanks Date: 05/13/2002 at 09:33:29 From: Doctor Rick Subject: Re: Great-Circle Formulae Hi, Saira. I might be able to help with this, but first I need to understand exactly how you wish to define a rectangle on the surface of the earth. A quadrilateral whose sides are great circles cannot have four right angles (which is what the word "rectangle", in its Latin derivation, implies). The sum of the angles will be something greater than 360 degrees; in fact the excess is directly proportional to the area of the quadrilateral. So I suppose I would define a "spherical rectangle" as a figure consisting of four great-circle segments such that the four angles are all equal (and greater than 90 degrees). Intuitively, I can see that opposite sides will then have equal lengths, in agreement with our plane-geometry understanding of rectangles. Perhaps you could tell me the reason you need to do this? It might indicate that some simpler solution will suffice. I don't anticipate this being an easy problem, if we require full generality and accuracy. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 05/13/2002 at 23:08:16 From: Saira Subject: Great-Circle Formulae Sir, You define a "spherical rectangle" as a figure consisting of four great-circle segments such that the four angles are all equal (and greater than 90 degrees). I exactly want that sort of rectangle. Now I have two corner points, top left and bottom right, and I want to find the other two points and the height and width of the rectangle. Surely, height and width will be the Great Circle distances. I need it because I am developing an application in which the user can enter the latitudes and longitudes of the top left and bottom right corners, and I have to display the so-called rectangle on the display. Kindly help me. I shall be very thankful to you. Date: 05/14/2002 at 09:10:36 From: Doctor Rick Subject: Re: Great-Circle Formulae Hi, Saira. There are many rectangles (in the spherical domain as well as in plane geometry) that have the given diagonal. In the plane analog of what I think you seek, we want the particular rectangle whose sides are horizontal and vertical. This task is easy: just use the x and y coordinates of the given points. It's easy to draw an analogous figure on the sphere by using the latitudes and longitudes of the two points; but this will not be a "spherical rectangle" as we have defined it. It will, in fact, have four right angles, and only the "vertical" sides will be great circles (unless one of the points is on the equator). I assume this is *not* what you seek; you want a true spherical rectangle. I'm not sure yet *why* you need to go to this trouble, but I'll work on it anyway. Here is an outline of an algorithm that will give you what I think you seek. First find the midpoint of the diagonal. Then draw the line of longitude ("vertical line") through this center point. Next we can find the great circle that passes through the "upper left" vertex and is perpendicular to the vertical line, and likewise the great circle that passes through the "lower right" vertex and is perpendicular to the vertical line. Now we have the "horizontal" sides of the rectangle. In the same manner we can draw the "horizontal" line (*not* a line of latitude!) through the center point. This will be the great circle through the center point and perpendicular to the vertical line. Then draw the great circle that passes through the "upper left" vertex and is perpendicular to the "horizontal" line, and likewise for the "lower right" vertex. We've got all the great circles defined; all that remains is to find the intersections of pairs of great circles to obtain the other two vertices of the rectangle. Note that it is not sufficient to get the vertices; you will need the great circles too in order to draw the "rectangle", so it's good that the algorithm I have outlined generates the great circles along the way. You can use vector algebra to find the great circle through a given point and perpendicular to a given great circle. First find the normal to the given great circle; if you know two points A and B on the great circle, then the vector N = AxB is normal to it. Then take the cross-product of this vector with vector C corresponding to the given point; this vector M = (AxB)xC is normal to the great circle we seek. Then to find the intersection of this great circle with the given great circle, take the cross- product of N and M. Thus the intersection point is along the vector (AxB)x((AxB)xC) See this item in the Dr. Math Archives for a related problem: Distance from a Point to a Great Circle http://mathforum.org/dr.math/problems/bellamy.5.24.00.html The only other tool I think you'll need is an algorithm to find the midpoint of the great circle between given points A and B. For this I will point you to a useful resource for navigation formulas: Aviation Formulary V1.35: Intermediate points on a great circle http://williams.best.vwh.net/avform.htm#Intermediate - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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