# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: horizon visibility puzzle**

**From:**Paul Hirose

**Date:**2001 Jun 21, 4:58 PM

I got the same result as Hal and Aubrey. The low hill blocks the view of the horizon. The formula I used was distance to the horizon, from Bowditch but converted to metric: d = 3.92 * sqrt(h), where d is km and h is meters. Here is how I reasoned out the problem. From the 777 m hill the horizon is 109 km away. From the 401 m hill it's 79 km away, or (adding the 35 km between hills) 114 km from the rear hill. This is further than you can see from the rear hill, so the forward one is blocking your line of sight to the horizon. I was delighted at the two responses I got. Exactly what I was hoping for. All three of us used the same tool: distance to the horizon. We all thought in terms of the horizon distance from each hill. However, each person utilized different imagery. Hal pegged a point on the horizon from the high hill, then went out there and looked back at the low hill. Aubrey started with the hills beside each other, then separated them until the horizons coincided. I kept the hills 35 km apart, but adjusted the horizon distances to a common frame of reference. It's fascinating and comforting that individuality emerges even when solving a cut and dried mathematical problem with the same formula. On the sci.engr.surveying newsgroup, where I first saw this, the one worked solution I saw was in terms of depression angles of the lines of sight to the horizon and the forward hill. That's a surveyor's mind-set at work. Speaking of surveying, one of my textbooks gives a formula which closely approximates the amount a horizontal line of sight rises above a level surface. In "universal" form (same units for height and distance), it's h = 6.75e-8 * d * d I.e., in 10 km your line of sight rises 6.75 meters. This takes the curvature of Earth and refraction into account. You can use this to solve the "two hills problem". Imagine yourself on the ocean, in line with the two hills. With your eye at water level you can barely see the top of the 777 m hill. (Ignore the other one for the moment.) Your distance must be such that h = 777. Solving the formula, d = 107 km. (Just 2% different from Bowditch formula.) The 401 m hill is 35 km closer, so its distance is 72 km. Your horizontal line of sight rises only 353 m at this distance, so it hits the hillside. Intuition tells me this formula is closely related to the ones for dip and dip short. I've fiddled around trying to rearrange it to look like the dip formula, but so far have not been able to get the plug into the socket. -- paulhirose@earthlink.net (Paul Hirose)