Distance to Horizon

vCalc Reviewed
Equation / Last modified by Administrator on 2018/07/10 07:05
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ID
vCalc.Distance to Horizon
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7265d448-d8d5-11e4-a3bb-bc764e2038f2

The Distance to the Horizon calculator computes the straight line distance to the horizon from a specified height (h) using the mean equatorial radius of the Earth.

INSTRUCTION: Choose units and enter the following:

  • (h) This is the height of the observation.

Distance to Horizon: The calculator returns the distance to the horizon in meters.  However, this can be automatically converted to other distanced units via the pull-down menu.

The Math / Science

The formula for the distance to the horizon is:Horizon.png

     `d = sqrt(h*(2*Re + h))` 

where:

Derivation

The graphic shows a right triangle, which is formed when looking to the horizon at any elevation (h).  The sides of the right triangle are:

  • d - the distance to the horizon
  • h + Re - the distance from the observation point to the center of the Earth
  • Re - the radius of the Earth at the horizon point.

Using the Pythagorean theorem, we know:

   d2 + Re2 =   (h + Re)2 

Now we can use simple algebra to isolate the distance to the horizon (d).Ellipsoid_OblateSpheroid_Sphere.png

  1. Expand the right hand side:    d2 + Re2 =  h2 + 2•h•Re + Re2 
  2. Subtract Re2 from both sides:  d2 = h2 + 2•h•Re 
  3. Factor h out of the right side:     d2 = h•(h+2•Re)
  4. Take the square root of both sides to get:

                                             `d =  sqrt(h*(h+2*Re))`

Unfortunately the Earth is not a perfect sphere.  It has mountains and valley, but it also had a bulge around the Equator.  This bulge makes Earth more of an Oblate Spheroid that is the shape one gets when rotating an ellipse about an axis.  In the Earth's case, the ellipse is rotated around the polar axis (see diagram).  

For this reason, this distance to the horizon equation is only an approximation.  Better models would take into account the changes in terrain and Earth's oblateness.

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