If h is the difference in height between end points of a chain of length ℓ, the required slope correction is:
🔊 Understanding Slope Correction
When measuring distance on sloping ground, the measurement is taken along the slope. However, for mapping and design purposes, we always need the true horizontal distance. The measured slope distance (\(\ell\)) will always be longer than the true horizontal distance (L).
Therefore, a slope correction must be applied, and it is always negative (subtractive) because we are correcting a longer measured length to a shorter true length.
🔬 Derivation of the Formula
The relationship between the measured length (\(\ell\)), the horizontal length (L), and the difference in height (h) forms a right-angled triangle. According to the Pythagorean theorem:
$$ \ell^2 = L^2 + h^2 $$
Rearranging for the true horizontal distance L:
$$ L = \sqrt{\ell^2 - h^2} = \ell \left(1 - \frac{h^2}{\ell^2}\right)^{1/2} $$
Using binomial expansion for \((1-x)^n \approx 1 - nx\) (since h is much smaller than \(\ell\)), we get:
$$ L \approx \ell \left(1 - \frac{1}{2}\frac{h^2}{\ell^2}\right) = \ell - \frac{h^2}{2\ell} $$
The slope correction (\(C_{slope}\)) is the difference between the measured length and the true length:
$$ C_{slope} = \ell - L \approx \ell - \left(\ell - \frac{h^2}{2\ell}\right) $$
$$ C_{slope} \approx \frac{h^2}{2\ell} $$
Since the correction must be subtracted, its value is \( - \frac{h^2}{2\ell} \). The question asks for the magnitude of the correction.
📊 Summary of Key Surveying Corrections
| Correction Type | Formula | Sign | Description |
|---|---|---|---|
| Slope | \( \frac{h^2}{2\ell} \) | Always Negative | Corrects for measuring on a slope to find the true horizontal distance. |
| Temperature | \( \alpha(T_m - T_0)\ell \) | Positive or Negative | Accounts for tape expansion/contraction due to temperature changes. |
| Pull (Tension) | \( \frac{(P - P_0)\ell}{AE} \) | Positive or Negative | Corrects for the tape stretching more or less than standard tension. |
| Sag | \( \frac{W^2\ell}{24P^2} \) or \( \frac{w^2\ell^3}{24P^2} \) | Always Negative | Corrects for the tape sagging under its own weight between supports. (W=total weight, w=weight per unit length) |
| Standardization | \( \frac{C \cdot L'}{L} \) | Positive or Negative | Corrects if the tape's actual length is different from its designated length. (C=correction per tape length) |