Math Question

[mastreb]

Hi Jen,

The simple answer is that you cannot know the percentage of obscurity of an object at a distance without knowing both that object's height and how far it is from you. This is because the percentage increases by the tangent as the distance increases, not by a fixed amount.

Here's what you can do:

The basic horizon formula is:

SquareRoot(height above surface / 0.5736) = distance to horizon. This is in feet and miles.

So:

Hobscured=(Dobject-Sqrt(Hboaterjen/0.5736))^2*0.5736

where Dobject is the distance from you to the object in miles, and Hboaterjen is your height off the surface in feet. The answer Hobscured is how many feet of the object height are obscured below the horizon.

In order to calculate a percentage, you must know how tall the object is.

Matt

[Hamin' X]

All joking aside, you really have no idea how much of what you are seeing is below the horizon, unless you know what it is that you are looking at. For example: If your eyes are 6 ft from the surface of the water, you can see ~3 miles to the horizon. Now, if you are looking at an island that is three miles away, you will see the entire thing, top to beach. However, if you just see some land in the distance, are you looking at the top 1000' of a 5000' mountain on a continent, or a rock sticking out of the water a few feet? How will you know?

~Rich

[seahouse]

An equation should be able to be made that I can do that's easy

Boaterjen--All of the above formulae will work, the units have been changed to protect the innocent.

With all the usual assumptions (there are lots, also made in the formulae previously mentioned) the following will get you a reasonably close estimate. Note that under some conditions temperature inversions alter the speed of light (yes, it's true) and bend your line of sight to follow the curvature of the earth, so you could theoretically see for much greater distances.

If your units are in feet, and nautical miles (after all, you are on the water, so they should be anyway) the formula becomes much simpler, namely -- the distance to the horizon approximately equals the square root of your height in feet. That’s rounding the optical line of sight multiplier of 1.06 in Kevin’s radar link above to be 1.(Ooooo. A six percent error )

So if you take your eye height as being 6 feet above the water, then the distance to the horizon point is the square root of 6, or 2.5 nautical miles.

Therefore the simplest formula I could come up with for you to use is…

To find the obscured height of an object (in feet), given the distance to the object (in nautical miles), simply subtract 2.5 from that distance, and square the result.

Boom. That’s it, that’s all. Note that the drop-off over longer distances increases exponentially, and also that the number you get is a straight line of sight, and in the real world, as mentioned above, your line of sight will curve, so you will be able to see at least that much of the object.

Hope this was helpful – Brian.

[seahouse]

Hey Rich -

And don't forget to factor in the undulation of the geoid in calculating the true height of msl above the WGS84 elipsoid. Sorry, I've been trying to figure out why my GPS has so much alttitude error.

As I pointed this out in this old post, sea level varies by as much as 500', from gravitational and other differences.

Then adjust for the curvature of the earth, and the 500’ difference in sea level that occurs depending on where you are in the world.

So they're workin' on that!

-Brian.

[Catigale]

I'll post the page from the Pilot guide which is similar to the stuff above if I can find an old one...

[Hamin' X]

Still won't tell her how much she can't see that is below the horizon.

~Rich

[Russ]

Don't forget, with a henway, you can see twice as far over the horizon.