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Wndsn Expedition Team

Ex Mensura, Scientia

Wndsn Shadow Square Calculator

High-precision shadow square calculator. Companion to Wndsn Telemeters: Low tech, high utility graphical distance & altitude computers from the Wndsn applied science lab.

By way of using the shadow square on a Wndsn Quadrant, we can obtain information about angles and convert between ratio, tan, arctan, cot, arccot, and angle. Drag the degree line on the Quadrant, or enter a reading in the form.

The shadow square of the Wndsn Quadrant. Example: 8/12 on the shadow square (umbra recta) results in a tangent value (on the slope scale) of 1.5 where arctan(1.5) ≅ 56°.

Input

The reading on the shadow square, or an angle instead

Or enter the angle (the reading wins when both are given):

Or enter a tangent or a cotangent (the reading and the angle win):

Reset

Results [JSON] [XML] [CSV]

The angle

  • Arctangent (the angle) = 56.30993°
  • Arccotangent = arctan(cot) = 33.69007°, the complement of the angle, read from the 90° edge

The ratio

  • Tangent = umbra versa / umbra recta = 1.50000
  • Cotangent = umbra recta / umbra versa = 1 / tan = 0.66667

On the Quadrant

  • Outer band, 24 parts: umbra recta 16.00000, umbra versa 24.00000
  • Inner band, 20 parts: umbra recta 13.33333, umbra versa 20.00000

The longer side of the reading is the full band, as on the instrument.

The display precision set is .5. To change that, edit the variable &digits=x in the URL where 0 <= x <= 9.

Explanation

[Load sample values (JSON) (XML) (CSV).]

The horizontal scales are called 'umbra recta', which is Latin for true, or straight shadow and describes the tangent. The vertical scales are named 'umbra versa', from Latin meaning turned, or upright shadow; the cotangent.

Note that cot(x) = 1/tan(x), so cotangent is the reciprocal of a tangent while arctan(x) is the angle whose tangent is x. On this page, the arccotangent is arctan(cot): the complement of the angle (90° minus the angle), as read from the 90° edge of the Quadrant.

Using the Quadrant

Drag the degree line, or tap the Quadrant to move it there. Hold Shift while dragging to snap to whole degrees. With the line's handle focused, the arrow keys move it in small steps. Every move updates the form and the results. And since the address follows along, a bookmark or a shared link brings back exactly this reading. A reading on the shadow square wins over an angle, and an angle wins over a tangent or a cotangent. But every field also shows its result, so the form always agrees with the Quadrant.

URL and API parameters

umbra_recta, umbra_versa
the reading on the shadow square, each greater than 0 and up to 96; both are needed
angle
the angle in degrees, greater than 0 and less than 90; used when the reading is missing
tan, cot
a tangent and a cotangent, each greater than 0; either one is enough. Used when the reading and the angle are missing
calc=yes
calculate; without it the page shows the example reading 8 / 12
digits
display precision, 0 to 9 (default 5)
format
json, xml or csv for the API; leave it out for this page

See also