1.7. Example 1.25: Gradient Calculation between Points

To determine the gradient between two points P and Q, a tacheometer was set up at R station, and the following observations were taken keeping the staff vertical at P and Q. If the horizontal angle PRQ is 36020՛ and RL of HI is 100 m, determine the average gradient between P and Q. Staff station Vertical angle Stadia readings (m) P +4040՛ 1.210, 1.510, 1.810 Q -0040՛ 1.000, 1.310, 1.620

In this example, we are calculating the gradient between two points P and Q using a tacheometer set up at station R. The procedure involves taking vertical angles and stadia readings for both points. The vertical angle at point P is +40°40', indicating an upward angle, while at point Q, it is -00°40', indicating a downward angle. The readings of the staff positioned at both points give us the necessary measurements that we will use later in our calculations.

Imagine you are standing on a hill (point R) and looking down at two different places (points P and Q). If you want to explain to a friend how steep the hill is between these two spots, you'd measure how high or low the land is from where you're standing to each spot. This is similar to our setup, where we're measuring angles and distances to understand the slope.

In the first observation (From R to P) S = 1.810 – 1.210 = 0.6 m θ = + 4040՛ Horizontal distance D = KS cos2θ + C sinθ = 100 x 0.6 x cos24040՛ + 0 = 59.60 m

Here, we first calculate the staff difference (S) which is the difference between the highest and lowest staff readings at Point P. This gives us a vertical distance of 0.6 m. Next, we use the vertical angle (θ) and the tacheometric constant (K) to calculate the horizontal distance (D) from R to P. The formula involves using the cosine of the angle to find the effective horizontal projection of the slope. Here, we find that the distance D is 59.60 m.

Think of measuring how far you are from your friend standing at a distance on a playground. If you're both on uneven ground, you wouldn't just use the straight distance—you'd have to consider the angle of the slope you're both standing on. This is what we do with our calculations by considering angles and heights.

Vertical Distance V = KS sin2θ/2 + C sinθ = 100 x 0.6 x Sin (2 x 4040՛) / 2 + 0 = 4.865 m

In this calculation, we find the vertical distance (V) to point P using a formula that considers the sine of twice the vertical angle (θ). This step is vital because it allows us to understand how much elevation there is over the horizontal distance we just calculated. In our case, we derive that the vertical distance is 4.865 m.

If you're climbing a tree, you'd use a rope to measure the height from the ground to your position in the tree. You aren't just climbing straight up; you need to know how much height you've gained compared to how far out you've gone from the trunk. This calculation reflects that vertical gain based on angles.

In the second observation (From R to Q) S = 1.620 – 1.000 = 0.62 m θ = - 0040’ Horizontal distance D = KS cos2θ + C sinθ = 100 x 0.62 x Cos20040՛ + 0 = 61.99 m

In this observation, we perform similar calculations, this time for point Q. The staff difference is 0.62 m, representing the height difference between the readings at Q. Then, we calculate the horizontal distance again using the cosine of the angle. We find that the distance D from R to Q is 61.99 m.

Imagine now you’re sliding down a slide that’s tilted—sometimes the angle can be downward, making you go faster to the bottom. This observation is reflecting how we measure distance differently based on whether we're looking up or down, like when we measure distances for points P and Q.

Vertical distance V = KS sin2θ/2 + C sinθ = 100 x 0.62 x sin (2 x -0040’) / 2 + 0 = 0.721 m

For point Q, we calculate the vertical distance just like we did for point P, now applying the downward angle for Q. This quantifies how much lower point Q is from the height we're referencing at R. Here, we calculate a vertical distance of 0.721 m.

It’s like measuring how far down you go when you’re climbing down a staircase. Just as you want to know how low you've gone, we measure the vertical drop here to see how elevated point Q is in relation to our reference point at R.

Distance D = PR = 59.60 m Distance D = QR = 61.99 m Angle PRQ = 36020՛ PQ^2 = PR^2 + QR^2 – 2 x PR x QR x cos36020’ PQ = 37.978 m Difference of elevation between P and Q RL of P = RL of HI + V – h = 100 + 4.865 – 1.510 = 103.355 m RL of Q = RL of HI – V – h = 100 – 0.721 – 1.310 = 97.969 m Difference = 103.355 – 97.969 = 5.386 m Average gradient between P and Q = Difference in RL between P & Q / Distance PQ = 5.386 / 37.978 = 1 / 7.051

Finally, we calculate the actual horizontal distance between points P and Q using the distance formula. After determining the relative heights (R.L) of both points, we find the elevation difference. The average gradient between P and Q is then calculated by dividing the elevation difference by the distance between the points, indicating the steepness of the hill or slope.

This is akin to understanding how steep a hiking trail is. If you know how much you've climbed and how far you've gone horizontally, you can tell if the trail is easy-walking or steep. That's exactly what this calculation achieves—an understanding of the gradient between two points.

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