Right angled triangle
Geodetic surveying stands on measuring angles, distances and works with them as inputs for further calculation. For understanding how to make a report from field data, surveying engineer needs to be familiar with math and geometry behind them.
(3.1) a2 + b2 = c2
(3.2) sin α = a / c = opp / hyp
(3.3) cos α = b / c = adj / hyp
(3.4) tan α = a / b = opp / adj
Graph below reminds that sinus starts in [0, 0], that means sin(0) = 0. If you are not confident, whether sin α is computed from opposite or adjacent, draw a triangle with very small α and the question becomes non-existent.
Oblique triangle and other objects
Diagram below depicts that one can compute the area of any triangle by formula
(3.4) A = bh / 2.
Similar logic can be applied to trapezoid:
(3.5) A = h (a + c) / 2.
Finally circle, cylinder and cone.
(3.6) C = 2πr is circumference of circle,
(3.7) A = πr2 is area inside circle,
(3.8) V = Ah = πr2h is volume of the cylinder and
(3.9) V = Ah / 3 = πr2h / 3 is volume of the cone.
Polar and cartesian coordinate system
- Cartesian coordinates are described by pair [x, y]
- Polar coordinates are depicted by pair [r, θ]
It is a common practice of a student or engineer to work with cartesian coordinates. Cartesian coordinates are described by pair [x, y]. On the other hand, measurements in the field and calculations conducted by surveying engineer depends on polar (radial) coordinates as well. Polar coordinates are depicted by pair [r, θ]. It is essential that an engineer can convert both coordinates between each other. Picture below helps to understand conversion from radial to cartesian coordinates:
Angle and slope in coordinate system
Picture above reminds how line is described in cartesian coordinate system and what slope m does mean:
- There is a relation between slope m and angle θ: tan θ = m.
- Two lines that are parallel have identical value of slope m.
- Parameter b, which is part of line equation, can be read in the chart.
Circle in cartesian coordinate system
is described by
(3.6) x2 + y2 = r2 or
(3.7) (x - a)2 + (y - b)2 = r2 respectively.
Other useful trigonometric formulas
In the book, there are tens more formulas listed to compute areas, volumes or other properties. Those are unneeded to remember or use since a few formulas shown above are almost always sufficient.
Some more less helpful trigonometric equations are listed below:
(3.8) tan α = sin α / cos α
(3.9) sin2α + cos2α = 1
(3.10) sin 2α = 2 sin α cos α
(3.11) cos 2α = cos2α - sin2α
(3.12) a / sin α = b / sin β = c / sin γ (law of sines inside oblique triangle)
(3.13) c2 = a2 + b2 - 2ab cos γ (law of cosines inside oblique triangle)
If you try to solve following examples, you will find out there is indeed no need for tens of magic equations from the book:
- A triangle has sides of 2, 3 and 4 metres. Compute area of the triangle.
- A triangle has sides of 2, 3 metres and angle of 30 ° between them. Compute area of the triangle.