by Stelios Zacharias » Nov 22, 2006 6:08 am
Let's say I want to survey a deep shaft and we are not happy with the accuracy of azimuth readings when the inclination is in the range of 70-90 degrees (as it will be with the shaft). We cannot measure plumb-lines as the shaft is rigged with re-belays set so as to avoid the one being directly below the other.
In an effort to get better azimuth measurements between two stations, I figured that if we take an azimuth reading from the from station (A) and from the to station (B) to a third point (C) on the other side of the shaft (which is about 10m wide), given that we know the distance and inclination between A and B (call it X meters), this could be used to calculate the real azimuth between the two stations in the case that the high inclination gives an inaccurate reading.
This is a doddle to draw (compass and protractor stuff), but it should also be possible to work it out mathematically - I could have done it while still at high-school, but the use it or lose it maxim has worked against me.
The solution will come if we set up a 2D Cartesian problem as follows:
-two lines intersect at C - we know their gradients (given by converting the azimuths to Cartesian)
-we want to find the gradient of the line (to be reconverted to polar coordinates later) which links the two lines thus drawn at the point where they are only X meters apart. There are going to be two such lines and it should (from context) be easy to figure out which of the two it is.
The cop-out of course is to tell me to go buy a disto whatsit and measure AC and BC, and knowing all distances reduce it to a pure trig question, but I don't have a disto and am not likely to get one.
Questions:
-Has anyone else come across this problem and solved it, ie is there an equation available into which the numbers (two azimuths and one distance) can be inserted to get the third azimuth
-Has anyone solved it another way?
Am I fretting too much about something which will yield too small a benefit in terms of accuracy?
Many thanks for your time,
Stelios