Geometry fundamentals

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.

Almost all computing tasks from the book can be transformed into tasks on right angled triangle.
If one is familiar with right angled triangle, there is no need to remember or search for a lot of formulas on oblique triangle.    Right angled triangle

(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.  Area of triangle

Similar logic can be applied to trapezoid:

(3.5)   A = h (a + c) / 2.  Area of trapezoid

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.  Circle, cylinder, 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:  Polar to cartesian conversion

Angle and slope in coordinate system  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.  Circle in cartesian coordinate system

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)

Example

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.

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