



 
We have two responses for you Dave, There are three obvious approaches. (BTW, I'm not quite sure why you're using so many significant digits when half of them are obviously made up  I doubt that the latlong's of Toronto and Montreal differ by integers to 5 sigfigs. It doesn't even make things much easier. But let it pass.)
Good Hunting!
I don't know why you had difficulty finding a useful web page. I googled "spherical distance formula" and found http://www.movabletype.co.uk/scripts/latlong.html Just enter your coordinates and click "see it on a map" and you are shown that your the points you've chosen are somewhat west of Mississauga and a bit west of the Montreal airport, with the greatcircle distance between them equal to 525.5 km. (which agrees favorably with the official distance between the cities). The web page gives you the formula used. To understand the formula, of course, you would have to be able to understand how spherical coordinates work. If instead of using spherical coordinates you prefer to use an inner product, I discuss that method at the end. Everybody seems to use the "average earth radius" of 6371 km. (= 3958.8 mi); for greater accuracy you would have to learn more about the earth radius (from Wikipedia or from somebody who knows something about geography). It is not clear to me what "greater accuracy" means in this context  roads curve around and go up and down; airplanes deviate by dozens of miles from the shortest path. For the "tunnel distance" it is perhaps easiest to switch from spherical to cartesian coordinates, then use the standard formula for the distance between two points. For this you need a calculator with trig functions (or use some program like Excel). If the latitude and longitude of point A are lata and longa, while those of point B are latb and longb, then the cartesian coordinates on the unit sphere are x = cos(lata)*cos(longa) u = cos(latb)*cos(longb) dist = SQRT[(xu)^{2} + (yv)^{2} + (zw)^{2}] (= the chord length between the two points on a unit sphere). To find the tunnel distance, multiply dist by the earth radius (6371). Note that for points that are as close together as Montreal and Toronto, the tunnel distance is only slightly less than the greatcircle distance (measured along the earth). The numbers I found are 525.36 and 525.50. Finally, once you have the cartesian coordinates of the points, the easy way to find the greatcircle distance between them by using their inner product: Chris  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 