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).

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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


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.


Qt SVG Document

Figure G-7

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