Math / Geometry Problem

[raptor1956]

In the western desert of Utah, on the Bonneville Salt Flats, is an art installation known as "The Tree of Life".  It's an odd looking thing in that particular setting but intriguing nonetheless.  So I Google it and it's reported as 27m tall, but I got to thinking ... how would one confirm that?  The piece is formed of a concrete trunk with branches onto which several large hollow concrete spheres are hung.  So, as I was thinking about this I asked myself how would a surveyor measure the height to the top -- the top is one of the spheres. 

I suppose if you had a total station you could sight on the center of the top sphere (the point closest to you) to get the elevation and distance using the laser range finder that's built in and then you could move to either side of the center and measure two more points with three points defining a circle.  To improve accuracy you could do this numerous times both side-to-side and up-and-down.  Then, once you have an accurate measure of the sphere radius it's a simple task to then calculate how much higher to the top.

But, what if you didn't have a total station and instead you had a theodolite, how would you determine the height using that?  You might sight on the center of the top sphere as before and doing so from a couple different locations should permit you to calculate the height to the point on the sphere you sighted on but you'd not know the radius of the sphere so calculating the height to the top would be a challenge.  You might move the theodolite in azimuth to the limb or edge of the sphere to measure the angle of the tangent and do that at both sides but that would not be the full width of the sphere unless you were very far away where other errors figure in.

Question:  How would you measure then calculate the radius of the sphere to complete the height calculation? 


Brian

[NivagSwerdna]

Make a wooden frame with a 45 degree angle.  Walk away from the base until the top is parallel to the slighting to the 45 degree angle... measure to base.

A

y/x = tan theta

thing  (and tan 45 degrees = 1)

?

PS
Looks fun.  We were in Utah and other places for the 2017 Eclipse... Absolutely awesome.

[T3sl4co1l]

If you're using a range finger, then sphere diameter is easy:

Sight in the leftmost edge of the sphere and note the angle.  Sight in the rightmost edge and note the angle.  Measure the closest distance to the sphere, which will be the face perpendicular to you.

Similar triangles, tangent lines and a bit of trig yield the sphere diameter, and from this you can calculate the correction for the actual height (highest altitude) without being able to see the highest point.

Assuming of course the sphere in question is spherical, and is the highest point, which would be difficult to verify without a tall ladder or airplane.

Or vertical sight angles work just fine too, or any other axis, as long as the sighted points are diametrically opposite.

Tim

[raptor1956]

Make a wooden frame with a 45 degree angle.  Walk away from the base until the top is parallel to the slighting to the 45 degree angle... measure to base.

A

y/x = tan theta

thing  (and tan 45 degrees = 1)

?

PS
Looks fun.  We were in Utah and other places for the 2017 Eclipse... Absolutely awesome.

For the eclipse I drove up to Idaho to a place I figured would be in the middle of no where but dead center on the path of totality.  Driving up there on that day the traffic was way heavier than I expected and I found my secret little spot was already filled with nearly a hundred others that must have thought as I did. 

Here's my video from that day...




Brian

[raptor1956]

If you look at the eclipse video I linked you can see the magnetic poles at 10:30 and 4:30...


Brian

[raptor1956]

Make a wooden frame with a 45 degree angle.  Walk away from the base until the top is parallel to the slighting to the 45 degree angle... measure to base.

A

y/x = tan theta

thing  (and tan 45 degrees = 1)

?

PS
Looks fun.  We were in Utah and other places for the 2017 Eclipse... Absolutely awesome.

Playing around with your idea on my CAD program and 45 degrees would be a problem, but 5 degrees would allow me to be further away and given that the sphere is over 20m up and a theodolite can't see straight up you'd need to be further away.  But, the math still works and with a decent theodolite the resultant accuracy should be within a couple cm for radius determination and perhaps 4cm for true height.


Brian

[raptor1956]

Well I solved the problem for a general solution no matter what the angle to the limb is.  If I can measure the distance 'd' to the closest point using a handheld laser range finder and you can pick one up for less than $30USD these days then if you measure the angle from left side to right side of the sphere the equation to calculate the spheres radius is:

r =  -dtan° / (tan° - sin°tan° - cos°)

where ° is one half the measured angle from limb to limb.


Brian

[raptor1956]

Looked at the problem again and I see an equivalent but simpler solution to the problem.

r = -dsin°/(sin° - 1)

Sometimes the simpler solution is right in front of you and you miss it...


Brian