12S56 Circle data        Collected 11/03/2009

 

 

Data

AT            TO                  Angles (deg)              Distance (m)

A              B                    0.0001                        34.371

                 C                    67.5996                      34.359

 

B              C                    0.0001                        38.233

                 A                    56.1876                      34.371

 

C               A                    0.0001                        38.362

                 B                    56.2187                      38.233

                 a                    315.2607                    8.027

                 b                    338.1861                    23.402

                 c                     20.5806                      40.271

                 d                    43.8662                      40.753

                 e                    70.5921                      33.013

                 f                     93.4452                      20.656

                 O                    33.8481                      20.702

 

Solution:

Solution adjustment.  The first step in the analysis is to make the angles consistent (i.e., sum to 180 degrees).  These adjustments are usually made by distributing the "misclose" (the difference from 180 deg), into each angle inversely proportional to the line lengths.  In our case the lengths are all about the same length so we subtract 0.002 deg to each angle. (This corresponds to mis-pointing by ~1.0mm over the 33-39 meter distances).  The distance measurements all agree in the forward and back directions except for one 1 mm difference.  The first measurement was adopted.

(a) Using the geometry from the figure above at site 00, we can write two equations for the radius:

The division of these two equations results in the R being canceled and using the expansion of we can write

By expansion, this equation reduces to:

Using the estimate of a1, we can then solve for the radius R.

For each corner point the results are:

(b) To find the radius to each of the intermediate points, we use the data from site C.  The cosine rule is used to solve for r and the sine rule to solve for y.  To solve these equations we use:

 

(c) The position of the sprinkler at the center (CEN) and computed by geometry.  If the spigot had been exactly at the center, the distance to it would have been 20.677 m (compared to the measured value of 20.702 m).  The difference in position places the spigot 0.028 m from the center at y =-27 deg.

 

The total results are shown in the figure below.  (ÒSouthÓ is the direction from the center of the circle to point A, ÒEastÓ at right angles to this direction.

 

The residuals to the mean radius and a function of the angle at the center are in the figure below:

 

 

This project was solved using Matlab code Proj_3_09.m.  The output of the code (in addition to the figures above is:

 

 

12S56 Project Number 3

Sum of angles in triangle is 180.0057 deg, adding -0.0019 to each angle

-------12S56 2009------------------------

Results for each angle/distance pair

Alpha    1  33.8125  Radius 1 20.677

Beta     1  33.7836  Radius 2 20.677

Gamma    1  22.4044  Radius 2 20.677

Mean radius 20.677

-----------------------------------

 Point      Radius    Drad  Angle

    B    20.677     0.000  67.5639

    a    20.749     0.072  270.0690

    b    20.637    -0.040  223.2940

    c    20.689     0.011  138.9328

    d    20.691     0.014  92.3788

    e    20.645    -0.032  38.5476

    f    20.608    -0.069  352.5300

Sprinkler Position  0.028 (m) at -27.39 deg