Calculate the coordinates of a known point P in an orthogonal coordinate system to a reference orthogonal coordinate system preserving shapes (angles) during transformation, in 2-D space.

Determine the elevation and grade value of an unknown point K which lies along a profile vertical curve (convex in this example).

*Commonly in profile vertical curves, a parabola is applied as an approximation to circular curve, considering that for large radius values and target precision, these elements practically coincide.

Determine the elevation of an unknown point X which lies between two known points A and B along an alignment tangency. The profile of tangency AB is also a tangent.

Determine the coordinates of an unknown point M (impossible to occupy) visible from two (2) previously surveyed reference stations A, B of a control baseline, only by observing angles α and β subtended by lines of sight from stations A and B to the intersected point M. Intersection technique is commonly implemented when the unknown point to be observed is inaccessible during a survey.

Calculate the maximum offset value ΔΖmax of an extrusion or intrusion detail from a reference plane (mainly flat object) so that the maximum error on the image plane due to radial shift does not exceed a certain limit Δρmax. (optical axis perpendicular to object plane)

Determine water content w and unit weight γsat of a sandy soil sample when it becomes saturated, at the same void ratio.

Calculate the ratio of the density of coal ash to the density of the water (Specific gravity of coal ash Gs)

**Determine water content within a soil sample as a percentage, by drying the soil in the oven at 105 ^{o}C for 24 hours.**

Calculate the surface area of a simple (non-self-intersecting) *n sides* polygonal shape with known Cartesian coordinates in the plane for all of its vertices. The main concept of the method is to divide the main polygon in *n* trapezoids and to cross-multiply corresponding coordinates to find the area enclosing the polygon (green trapezoids), and subtract from it the surrounding trapezoids (red) to find the area of the polygon within. It is also called the shoelace formula because of the constant cross-multiplying for the coordinates making up the polygon, like tying shoelaces. Mainly applicable in topography.

**Determine the coordinates of an unknown station P from three (3) previously coordinated reference points A, B and C visible from station P, only by measuring the angles subtended by lines of sight from station P to the three coordinated points. Resection technique is commonly implemented when the existing reference points are not accessible during a survey.**

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Calculation Examples