For maybe a couple of hundred years prior to the mid 1980s surveyors used the method of Stadia (Wiki) to determine distances. Stadia accuracy was not as good as using a tape measure (or chain (Wiki)) but for many uses, like making a topographic map, was the standard distance measurement method Starting in the mid 1980s Electronic Distance Measuring (EDM) replaced stadia and tapes because it was both much faster and much more accurate. EDM has been replaced be the Total Station. A Total Station (Wiki) combines EDM and a
Theodolite (Wiki) that has digital reading of the scales and can output digital data with the horizontal and vertical angles as well as the distance to the target. A Partial Total Station does not have the digital outputs. The accuracy that can be obtained from a single sight is about 1/300 in distance and about 0.06 feet in elevation per hundred feet of length. For a traverse the closure will be about 0.8 feet * SQRT(miles).
The telescope used for Stadia has three horizontal hairs. One that is the normal leveling hair and is centered. Above and below the leveling hair are the two Stadia hairs separated by a total angle of TAN-1(1/100) = 0.572939 degrees = 0.01 radians. This way when the rod is 100 feet away there will be a separation of 1 foot on the rod. In the metric system if the rod (Wiki) is 100 meters away the rod separation will be 1 meter.
Older external focus instruments, like the Leitz 115A transit, use the formula D = 100 * AB + 1'
where D is the number of feet from the instrument center to the rod with AB being the rod separation in feet and for a level shot.
Modern internal focus instruments, like the K&E 76 0000 Alidade, use the formula D = 100 * AB.
This is explained in the K&E publication Care and adjustment of K&E Surveying Instruments.
In this case the Horizontal distance (Ft.) is 100 * (A-B) +1
Where 100 is the Stadia constant and 1' is the instrument constant C (may be zero on modern instruments), the separation A minus B is in feet
The correction to correct the top and bottom readings (A-B) for the Vertical angle (a) is:The K&E Alidade has the trig functions built in for horizontal and vertical distances.
Corrected separation = COS(a) *(A-B) [ = (A' - B')]
Then (S) the Slope Distance = 100 * COS(a) * (A - B) +1
(H) the Horizontal Distance = COS(a) * [ 100 * COS(a) * (A - B) + 1] 500.27' = COS(13.7) * [ 100 * COS(13.7) * (5.3') + 1]
(V) Vertical Distance = SIN(a) * [ 100 * COS(a) * (A - B) + 1] 121.19' = SIN(13.7) * [100 * COS(13.7) * (5.3') + 1 ]
Range Finders - Stadimeter
Stadia Principles - although I have not checked I think the equations and mine are the same
Stadia System - uses the expression "fixed hair tacheometer" which can be applied to any instrument that has the stadia cross hairs. Includes procedure to determine stadia constants for an instrument where they are not known.
Plane Surveying - Stadia - a 14 page pdf chapter on the use of stadia for measuring distance and it's applications
Foresight -CDS Tutorial 2 - getting stadia shots into the program and getting the program do make a topo plot
Stadiametric rangefinding (Wiki) uses the method of stadia in a slightly different manner.
Stadimeter (Wiki) -
406829 Range Finder, Bradley Allen Fiske, Jul 9, 1889, cooperative Classification G01C1/00 - dual electrical potentiometers
483999 Range and Position Finder, Bradley Allen Fiske, Oct 11, 1892, G01B2210/283 - dual electrical potentiometers
523721 Method and Apparatus for Range Finding, Bradley Allen Fiske, Jul 31, 1894, G01C3/22 - all optical
864237 Range Finder, Bradley Allen Fiske, Aug 27, 1907, G01C3/10 - all optical
1943 Stadimeter -
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