## Special Telemeters and Custom Nomographs

*Tutorials and setup instructions for specialist or custom-made Telemeters.*

## Wndsn Angular Size Pendant

A tool for celestial navigation and rangefinding, the Wndsn Angular Size Pendant^{[1]} enables you to do more than merely guessing distances. Find an object of known size at the distance you need to measure and determine its angular size.

Inspired by the medieval Kamal, a celestial navigation tool that greatly facilitated latitude sailing, the Wndsn XPD Angular Size Pendant fixates the distance to the measuring scale by way of creating a necklace that is holding the scales at a set distance, thus enabling accurate measurements.

The scales on the pendant are calculated in a way that enables the user to measure one-handed, no further setup required other than wearing the pendant around their neck. The pendant subtends angular sizes of up to 13° with a precision of 0.2° or 12 arcminutes.

### How to set it up

To install, loop the string through the provided hole at the top. Measure a length of exactly 19.1 cm (7.5 in) from eye to scale. The cord around your neck fixates that length.

### Accuracy

Accuracy is determined by two elements, provided that the string length is respected:

- The reading of the scale and approximation of the corresponding mark
- The estimation of the height of the measured object

For training and reference purposes, you may want to create a table of the exact height of common objects.

Note that you can measure in any unit (cm, in, ft) or system (metric, imperial, custom), the factors are always the same and return your distance in the same unit you used to approximate the object measured.

### FAQ

**Q: Why is the string length so important, and what's special about the 57.3/3?**

It says "baked-in trigonometry"; also, on the tool itself, there is a hint: `For d = 57.3 cm, arctan(s/d) in deg = s => 1° = 1 cm`

The explanation is the definition of radian^{[2]}, or rad: The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3° (when the arc length is equal to the radius). The circumference subtends an angle of 2π radians.

Now, at a radius (hence the string length) of 57.3 cm/3, 1/3 cm in measured size subtends exactly 1°; `D = S * d / α`

where:

- D = distance from eye to observed object of known size
- S = known size of the object observed
- d = distance from tool to observer's eye
- α = the angular size of the object observed at the distance d

**Q: How do I use the degree scale?**

The degree scale measures the angular size^{[3]} of the object observed, in other words the measured height in degrees, (easily converted to multiplication factors as per the instructions on the instrument) which can be used when the actual height of the object is unknown.

Also note that: `α = (S * 57.3) / D`

Relationships between angular size and multipliers (valid for d = 57.3 cm):

1° = x 57 2° = x57 /2 = x 28.5 3° = x57 /3 = x 19 4° = x57 /4 = x 14.25 5° = x57 /5 = x 11.4 6° = x57 /6 = x 9.5 7° = x57 /7 = x 8.1 8° = x57 /8 = x 7.1 9° = x57 /9 = x 6.3 10° = x57/10 = x 5.7 11° = x57/11 = x 5.2 12° = x57/12 = x 4.75 13° = x57/12 = x 4.4

**Q: Can I measure the width instead of the height of an object?**

You can measure any dimension; width, height, etc., as long as it's on a plane that is perpendicular to you, the trigonometry doesn't care where in space the triangle is located.

**See also:**

**References:**