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Calculate the value of K and C, if the measurements are taken between two points of 50m and 130m distance apart and the stadia readings will be 0.024m, 0.824m respectively.
100, 0
47.6, 100
0, 100
100, 47.6
In tacheometric surveying, the horizontal distance D between the instrument station and the staff station is given by the distance equation D=Ks+C, where K is the multiplying constant (f/i) and C is the additive constant (f+d). Using the two given conditions with staff intercept s and distance D, simultaneous equations are solved to determine K=100 and C=47.6 m.
In tacheometric surveying, the horizontal distance D between the instrument station and the staff station is given by the distance equation D=Ks+C, where K is the multiplying constant (f/i) and C is the additive constant (f+d). Using the two given conditions with staff intercept s and distance D, simultaneous equations are solved to determine K=100 and C=47.6 m.
D=Ks+C — General tacheometric distance equation
K=if — Multiplying constant
C=f+d — Additive constant
K=s2−s1D2−D1 — Formula to derive K from two observation points
Tacheometry relies on the principle of isosceles triangles, where the ratio of the distance from the vertex to the base and the length of the base remains constant. The stadia hair readings give the staff intercept s, which when multiplied by the multiplying constant K and added to the additive constant C yields the horizontal distance D.
The multiplying constant K is typically equal to 100 in an ideal tacheometer equipped with an anallactic lens.
The additive constant C varies according to telescope design, but reduces to 0 when an anallactic lens is fitted.
Subtracting equation (i) from equation (ii) yields 130−50=K(0.824−0.024)⇒80=K(0.800)⇒K=100.
Substituting K=100 into equation (i): 50=100(0.024)+C⇒50=2.4+C⇒C=47.6 m.
Allows direct determination of horizontal distance without using physical measuring tapes.
Particularly suitable for steep, rough, or inaccessible terrains.
Accuracy depends strongly on precise staff intercept measurements.
Observational errors in staff reading lead directly to distance inaccuracies.
Contour mapping and topographic surveying.
Preliminary route location surveys for roads, railways, and canals.
Option A (100, 0) represents an anallactic telescope where the additive constant C=0.
Option C (47.6, 100) reverses the values of K and C, which is incorrect.
Option D (0, 100) is invalid as K cannot be zero for meaningful distance measurement.
B is correct — Solving 50=0.024K+C and 130=0.824K+C simultaneously yields K=100 and C=47.6 m.
Remember that if an anallactic lens is provided in the tacheometer telescope, K=100 and C=0 directly, eliminating the additive term.