NavList:

Frank wrote

Note that this is the dip short for distances up to a mile or two. Beyond that the unrefracted drop-off angle is exactly equal to the distance divided by two. That's how much the curved surface of the Earth is falling away from a flat plane tangent to the surface at the observer's location. This relationship applies at any distance, including two miles or ten miles away from the observer and also a thousand miles or halfway around the world, too. That's surprising at first, but when you draw it out it makes good sense.

Imagine you want to build a whimsical signpost in your garden with signs showing the direction to nearby points and a few distant locations, too. One sign says "barn" and points east, another says "town" and points north. The next sign says "Manhattan" and points WSW. The last two signs say "Bermuda" and "Hawaii". The first points SSE and the Hawaii sign points a bit north of west. But Bermuda and Hawaii are below the horizon so the signs should point somewhat downward. How much? Well, from my location, Bermuda is about 630 nautical miles away and Hawaii (the big island) is about 4350 nautical miles away. That's 10.5° distance to Bermuda and 72.5° to Hawaii. So if I face west (actually about 10° north of west for the straight line to Hawaii from here), I should look 36.25° below horizontal to see Hawaii. That's where my Hawaii sign should point. And my Bermuda sign points SSE 5.25° below horizontal. In fact, even my sign pointing at Manhattan should be angled 2.0° downward.

Note that this is the dip short for distances up to a mile or two. Beyond that the unrefracted drop-off angle is exactly equal to the distance divided by two. That's how much the curved surface of the Earth is falling away from a flat plane tangent to the surface at the observer's location. This relationship applies at any distance, including two miles or ten miles away from the observer and also a thousand miles or halfway around the world, too. That's surprising at first, but when you draw it out it makes good sense.

Imagine you want to build a whimsical signpost in your garden with signs showing the direction to nearby points and a few distant locations, too. One sign says "barn" and points east, another says "town" and points north. The next sign says "Manhattan" and points WSW. The last two signs say "Bermuda" and "Hawaii". The first points SSE and the Hawaii sign points a bit north of west. But Bermuda and Hawaii are below the horizon so the signs should point somewhat downward. How much? Well, from my location, Bermuda is about 630 nautical miles away and Hawaii (the big island) is about 4350 nautical miles away. That's 10.5° distance to Bermuda and 72.5° to Hawaii. So if I face west (actually about 10° north of west for the straight line to Hawaii from here), I should look 36.25° below horizontal to see Hawaii. That's where my Hawaii sign should point. And my Bermuda sign points SSE 5.25° below horizontal. In fact, even my sign pointing at Manhattan should be angled 2.0° downward.

Thanks for this information. You may have solved a problem that has been puzzling me. Recently, on another, forum I posted a reply to a silly thread about changing the gauge of the NZ railway system from Cape gauge to standard gauge. The writer wrote in a deadly serious manner but did not explain why such a change was necessary. In my reply I stated that the long term plan was to buid a straight  tunnel from Wellington to Auckland. The railway would be on the Breitspurbahn gauge.

If we assume that the earth is a perfect sphere and that the tunnel runs just below the surface  a Mercator's sailing can be used to determie the bearing and length of the rumb line aka loxodrome. A calculator or traverse tables can be used. The rumb line has a vertical curve but I want a tunnel that is straight in all dimensions. In other words I want the chord between Auckland and Wellington. Your method allows me to  calculte the ditance and dip.

Of interest is the maximun depth of the  tunnel. This can be found by the railway engineering technique known as stringlining. I have a photo  of this method in use but following a disk crash some months ago I did not fully restore the index so cannot easily find the photo.

I haven't done the calculations yet.