1.2.2. Establishing a point by at least two independent measurements

Horizontal control points in surveying are located by linear and/or angular measurements. If two control points are established by surveying measurements, a new point (third point) can be established with the help of these two known control points by taking two linear or two angular measurements, or by one linear and one angular measurement.

In surveying, control points are essential reference markers on the ground. These points are determined through precise measurements either in a straight line (linear measurements) or at angles (angular measurements). Once two control points are established, they serve as anchors from which a third point can be accurately placed. This is done using the properties of triangles, as the relationships between the points can be calculated mathematically. If you know the distances or angles between the first two points, you can find the position of the third point, allowing the surveyor to expand their map without needing to personally visit each new location.

Consider laying down a triangle of three points on a flat surface, like a table. If you put two points down and know the distance between them, you can easily find where the third point will go using a ruler. This is similar to using two landmarks to determine a new location on a map; once you have two reference points, locating others becomes simpler and much more precise.

In other words, indirectly the location of the new point is established using the geometry or trigonometry of the triangle formed by these three points.

The positioning of the third point is derived from geometric principles, particularly those related to triangles. A triangle's angles and sides have defined relationships expressed through trigonometry. When the lengths of two sides of a triangle and the included angle are known (or vice-versa), various mathematical formulas can be used to calculate the unknown sides or angles. This relationship allows surveyors to accurately locate the third point based on the two established control points, making use of trigonometric functions like sine, cosine, and tangent.

Imagine you're trying to find a hidden treasure in a park. You know the distance between two well-known trees (control points), and you also know the direction to move from one tree at a specific angle to find the treasure. Using these details—distances and angles—you can determine exactly where to dig, similar to how surveyors use triangles to pinpoint locations.

Point R can be established using the distances PR, and R’R. In Fig. 1.2(b), R is established using the distances PR and QR. In Fig. 1.2(c), R is established by the angle RPQ and distances PR or by establishing angle RPQ and distance QR. In Fig. 1.2(d), R is established using the angles PQR and QPR.

Surveyors have multiple ways to determine the location of the new point 'R' based on different combinations of distances and angles from the two known control points 'P' and 'Q'. Depending on the available data, they can choose to rely solely on distance measurements (like how far point R is from P and Q) or utilize angles (the angles formed at points P and Q) combined with distances to pinpoint the exact location of R. Each method allows for flexibility, making it easier to suit the specific conditions of the surveying task.

Think about using various methods to find your way in a city. If you only know how far you are from two landmarks, you might navigate with distances alone. However, if you also know the direction you need to travel from one landmark at a specific angle, it gives you a clearer picture and helps you find your destination more easily. This is akin to how surveyors may either depend on distance or angle to locate new points.

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