## 1. Definition

A horizontal angle is measured perpendicular to gravity. Three points are involved:

• Backsight (BS)

• Foresight; FS

• At

The angle is between two vertical planes; one contains the BS and At points, the other contains the FS and At. The At point is generally occupied by a surveying instrument, Figure G-1. Figure G-1 Horizontal Angle

If we look down on the horizontal plane with the three points, Figure G-2, we see there are two horizontal angles between the vertical planes: HAng1 and HAng2. Figure G-2 Two Horizontal Angles

This causes an ambiguous situation unless we can specifically identify which angle we mean. Recall that the four parts of an angle are:

• Start

• Direction

• Magnitude

• End

HAng1 and HAng2 have the same start (BS) and end (FS) but different direction and magnitude. All four parts must be clearly defined. Merely including magnitude without a direction doesn’t resolve the ambiguity. For example, a 135° angle from BS to FS could still be interpreted two ways, Figure G-3. Figure G-3 Magnitude without Direction

Similarly, expressing a direction, to the right (clockwise) or left (counterclockwise), without a magnitude is also unclear, Figure G-4. Figure G-4 Direction without Magnitude

There are a few different ways to specify horizontal angles which help minimize confusing interpretations.

## 2. Types of Horizontal Angles

### a. Interior/Exterior

On a closed non-crossing polygon, Figure G-5, horizontal angles can be either interior (red) or exterior (blue). Figure G-5 Interior and Exterior Angles

The geometric condition for a non-crossing polygon is Equation (G-1). Equation (G-1)

Interior and exterior don’t make sense if the polygon is open, Figure G-6(a), or if the polygon crosses itself, Figure G-6(b). a. Open Polygon b. Crossing Polygon Figure G-6 Interior/Exterior?

In the latter case, the polygon turns itself inside-out and Equation (E-1) does not apply. Open and crossing polygons need some other way to express an angle.

### b. Right/Left

Right or left is the rotational direction from the BS point to the FS point. Consider standing on the At point, looking at the BS point: an angle to the right means you physically turn your body to the right to see the FS point. An angle right is clockwise, left counter-clockwise, Figure G-7. Figure G-7 Clockwise/Counter-clockwise

Figure G-8(a) shows a closed non-crossing polygon with interior angles to the right; Figure G-8(b) to the left. a. Angles Right b. Angles Left Figure G-8 Interior Angles

The dashed lines in Figure G-8 show the angle measurement sequence. In Figure G-8(a), interior angles to the right measurement progresses counter-clockwise around the polygon.

We can also have exterior angles to the right, Figure G-9(a), and to the left, Figure G-9(b). a. Angles Right b. Angles Left Figure G-9 Exterior Angles

Right or left help with angle definitions on the open and crossing polygons. In Figure G-10, progressing in the direction of the dashed line, the open polygon has angles to the right. Figure G-10 Angles Right on Open Polygon

Figure G-11 shows a crossing polygon using angles left progressing in the direction of the dashed line. Figure G-11 Angles Right on Crossing Polygon

Note how the angles change from “interior” to “exterior” although there is no consistent polygon interior.

### c. Deflection Angles

A deflection angle is how much the next line deflects from an extension of the previous line. It consists of two parts: (1) magnitude and (2) direction. For example, in Figure G-12:

Line BC deflects from line AB 45° to the right, written as 45°R.

Line CD deflects from line BC 30°L. Figure G-12 Angles Right on Open Polygon

A deflection angle ranges from 0° (no deflection) to 180° (going back along the preceding line).

The mathematical angle condition for a closed non-crossing polygon, Figure G-13, is: Equation (G-2)

Right deflection angles are positive (+), left are negative (-). Figure G-13 Deflection Angles Around Polygon

Equation (G-2) is also the mathematical condition for a closed polygon which crosses itself an even number of times. For an odd number of crossings, Figure G-14, the mathematical condition is: Equation (G-3) Figure G-14 Odd Crossings Polygon

How does reversing travel direction around a polygon change the deflection angles? Figure G-15 shows that the deflection angle value is the same, but its direction is reversed.  a. Clockwise Travel b. Counterclockwise Travel Figure G-15 Reversing Travel Direction

## 3. Example Problem

Given this polygon with combination of angle types: Determine

#### a. Deflection angle at A from D to B Defl angle A = 180°00’ - 113°20’ = 66°40’ = 66°40’ R

#### b. Angle left at B from A to C Int angle B = 180°00’ - 92°42’ = 87°18’

#### c. Deflection angle at D from A to C Need interior angle at D to compute the deflection angle. To compute interior angle at D, subtract the sum of the other interior angles from (n-2)x180°.

Need interior angle at C. Int angle C = 360°00’ - 259°35’ = 100°25’ Int angle D = (4-2)x180°00’ - (113°20’ + 87°18’ + 100°25’) = 360°00’ - 301°03’ = 58°57’ Defl angle D = 180°00’ - 58°57 = 121°03’ = 120°03’ L 