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    Re: Cosine formula to solve a right spherical triangle?
    From: Frank Reed
    Date: 2020 Jan 22, 19:36 -0800

    Fred, you wrote:
    "why is the point where your latitude line meets their longitude line not a right angle?"

    By far the best way to see this is by playing around with a common globe. Find a cheap one that you're comfortable damaging with thumbtacks. You can get nice 4-6 inch globes as gift items at Target stores (I've seen them there recently) and no doubt other places for about $10. Nice to have an ordinary globe that doesn't take up much space anyway.

    Imagine you're Charles Lindbergh. Stick a pin in Long Island east of New York City. Stick another pin at Paris, France. Now grab a rubbed band and stretch it from one pin to the next. Pluck the rubber band out a little from the globe and let it snap back. That's the shortest path across the Atlantic, a portion of a great circle. Notice how it aligns with the lines of latitude and longitude. Leaving Long Island, that path is inclined well away from due east. And arriving in Paris, it's inclined a long way from due west. There's a spot where that great circle path does hit a line of longitude at a right angle out in the middle of the Atlantic. It happens to be the point of highest latitude between start and endpoint. Makes sense! In great circle trivia exams, this point is known as the "vertex". If you extend a portion of a great circle, following it from its maximum latitude all the way back to the equator, you'll find that the initial course as measured from east or west is identical to the latitude of the "vertex" in degrees. That makes sense, too, right?

    Instead of snapping a rubber band, you can consider the tunnel through the Earth from one point, A, to another, B. That's the true three-dimensional shortest path between A and B. Once you have the tunnel imagined, reach down and pull its mid-point up to the surface of the Earth. Pull vertically, directly away from the Earth's center. Where it reaches the surface, call that point C. You now have two tunnels, from A to C, and from C to B. Repeat this process a few thousand times. You've pulled your whole tunnel up to the surface. It's directly above the original tunnel at every point, so that's the shortest point along the Earth's surface from A to B. The original tunnel, all the points of the great circle arc, the start and endpoints, A and B, and the center of the Earth, all lie in one geometric plane. And that's the most important property of a great circle: a great circle is co-planar with the center of the Earth.

    Frank Reed

       
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