# 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.**

(3.1) a^{2} + b^{2} = c^{2}

(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 = πr^{2} is area inside circle,

(3.8) V = Ah = πr^{2}h is volume of the cylinder and

(3.9) V = Ah / 3 = πr^{2}h / 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) x^{2} + y^{2} = r^{2} or

(3.7) (x - a)^{2} + (y - b)^{2} = r^{2} 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) sin^{2}α + cos^{2}α = 1

(3.10) sin 2α = 2 sin α cos α

(3.11) cos 2α = cos^{2}α - sin^{2}α

(3.12) a / sin α = b / sin β = c / sin γ *(law of sines inside oblique triangle)*

(3.13) c^{2} = a^{2} + b^{2} - 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.