distance to horizon
STUARTe
horizon is from his plane.
neglect refraction. assume the ground below is not mountainous.
neglect terms like 1/R and 1/R*R where R is the earth's radius. find
a simple expression that approximates the distance to the horizon.
David W. Cantrell
> a pilot knows his altitude in miles.
Let h be his height above the surface of a spherical planet of radius R.
> he wants to know how far the
> horizon is from his plane.
> neglect refraction. assume the ground below is not mountainous.
> neglect terms like 1/R and 1/R*R where R is the earth's radius. find
> a simple expression that approximates the distance to the horizon.
His distance to the horizon is exactly Sqrt(h (h + 2R)). Were you wanting a
expression even simpler than that?
David
Gene Wagenbreth
in miles. Actual distance is about 23% greater. So from 10,000 feet horizon is
100 miles away (crudely) or 123 miles (less crudely).
G
STUARTe
using R = 4000 miles you get an expression which is a constant
multiplied by a simple function of r.
David W. Cantrell
Perhaps your r is my h.
In any event, you seem to be thinking of an approximation similar to that
already posted by Gene. But it requires taking a square root, and if you're
going to have to take a square root, then IMO you might as well get the
distance to the horizon exactly!
David
Sjouke Burry
and r=earth radius(in meters 6378000):
Distance=sqrt((a+r)*(a+r) -(r*r))
or:
acos=reversed cosine,
angle=acos(r/(r+a))
distance=(r+a)*sine(angle)
For a eyeheight of 1.80 m: 4791 m
For 10 km: 357 km
Dont shoot me is your figures are different.
STUARTe
>
> - Show quoted text -
yes. i said r when i should have said h.
gene's answer is correct but sqrt(h(h+2R)) can be further simplified
by setting R=4000 and h<<2R
sqrt(h(8000)) or approximately 90*sqrt(h)
Laurence Finston
In traditional linear perspective, the horizon line represents an
infinite distance from the focus, so the problem you set can't be
solved. A horizon line isn't necessary for linear perspective, but is
used in typical constructions where the ground plane represents the
earth (considered to be planar) and the line from the focus to the
center of vision (vanishing point 90 deg.) is parallel to the ground
plane. The horizon line is the line through all the vanishing points
for sets of parallel lines that lie in the ground plane or in planes
parallel to the ground plane. These sets of parallel lines appear to
converge, but the place where they appear to converge is at an
infinite distance from the focus. The height of the horizon line
above the ground plane in a perspective construction is the height of
the focus above the ground plane. However, a ground plane is also not
necessary for a perspective construction. A perspective construction
can be made for a focus floating in space without reference to a
ground plane. Nor is a horizon required to find vanishing points. If
a pilot is flying high enough so that the curvature of the earth is
perceptible, then I don't think the term "horizon" would be applicable
anymore. For one thing, the pilot would probably be looking down. If
not, a direction of view, i.e., focus to center of vision, parallel to
the plane tangent to the earth's surface at the intersection point of
the line from the pilot's eye (assuming a one-eyed pilot) to the
center of the earth (assuming a perfectly spherical earth) would
result in a very distorted projection indeed. In traditional
perspective, the plane of projection is always perpendicular to the
line focus -- center of vision. Changing the direction of vision
implies changing the plane of projection, and vice versa.