1.4. Example 1.22: Calculation of Tacheometric Constants

The following readings were taken with a tacheometer on to a vertical staff, calculate the tacheometric constants.

Horizontal Distance (m) Stadia Readings (m) 45.00 0.885 1.110 1.335 60.00 1.860 2.160 2.460

This section introduces the concept of tacheometric readings, which are used to determine horizontal distances based on angles and distances recorded with a tacheometer. The readings taken at two different horizontal distances (45 m and 60 m) with associated stadia readings show how the instrument can measure differences in staff positions, allowing a surveyor to compute necessary constants for further calculations.

Think of a tacheometer like a high-tech measuring stick that combines a ruler and a protractor. Just like you might measure how tall someone is by using a wall as a reference and a tape measure from a certain distance, the tacheometer uses these measurements to estimate distances to objects or points without directly measuring them.

Solution: D = KS + C ................................(1) 1 1 D = KS + C ................................(2) 2 2

The tacheometric equation relates the horizontal distance (D) to the stadia readings (S) through two constants (K and C). The constants help to calibrate the readings taken by the tacheometer. In simple terms, K determines the scale of the measurement, while C accounts for any systematic biases in measurement. These equations show two separate tacheometric readings that lead to the same horizontal distance calculation.

Imagine you're baking cookies. The recipe requires specific ratios of ingredients (like flour and sugar) to get the right taste (the final outcome, which would be the distance in our case). K acts like the amount of flour you add, modifying how sweet or savory the cookie will be. C shows how much sugar was already in the batter, helping adjust the overall flavor without needing to change what you started with.

So, 45 = K (1.335 – 0.885) + C 45 = K (0.45) + C ....................(3) 60 = K (2.460 – 1.860) + C 60 = K (0.6) + C ………….….(4)

In this step, we set up two equations (3 and 4) based on the readings provided. Equation 3 comes from the first distance measurement of 45 m, and Equation 4 from the second measurement at 60 m. Each equation uses the differences in stadia readings to form a linear relationship between K and C. This creates a system of equations that can be solved simultaneously to find K and C.

Think of it like solving a mystery using clues. Each clue (the equations formed) gives you important pieces of information (the values of K and C). By gathering all st clues to figure out who dunnit (solving for K and C), we get a clearer picture of the situation.

Equating C from equations 3 and 4, we get- 45 – K (0.45) = 60 - K (0.6) 0.15K = 15 K =100 Now put the value of K in either equation 3 or 4, we get C =0

Here we manipulate the equations to isolate C by equating it from both equations (3 & 4). This leads to the discovery of the value of K as 100. Substituting K back into either of the original equations allows us to solve for C, which turns out to be 0. Understanding these constants is crucial for accurate distance calculations in surveying work.

Imagine if you were given coordinates to places on a map but needed to understand the specific distance between them. By solving these equations, you can determine the exact measurement and make precise calculations for your travel plan, just like using K and C in surveying.

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