Wndsn Expedition Team

Wndsn Sight Reduction Calculator

Sight reduction turns a sextant altitude into a line of position. The calculator solves the navigational triangle for the computed altitude Hc and the true azimuth Zn, and with an observed altitude Ho it gives the intercept. A worked example reduces the same sight with Pub. No. 229, Sight Reduction Tables for Marine Navigation, step by step, for the navigator who works with the tables.

Move the body or the pole on the sphere, or enter the latitude and the declination, then the LHA or the GHA with the longitude. Add Ho for the intercept.

Explanation

[Load the examples from the introduction to Pub. No. 229: linear interpolation (JSON) (XML) (CSV), and with the double second difference.]

From altitude to line of position

A sextant measures how high a body stands above the horizon. That altitude places the observer on a circle on the Earth, centered on the point where the body is in the zenith. Since the circle is far too large to draw, the navigator chooses a position near the true one and works out the altitude the body would have there. This is the computed altitude Hc, and with it comes the direction of the body, the true azimuth Zn.

The intercept is the difference Ho − Hc. A minute of arc equals a nautical mile, so the intercept is a distance: toward the body when Ho is larger, away from it when Ho is smaller. Measured along Zn from the chosen position, it marks a point of the line of position, which runs at right angles to Zn. This is the intercept method of Marcq Saint Hilaire.

The navigational triangle

On the celestial sphere the body forms a spherical triangle with the elevated pole and the observer's zenith. Its side from pole to zenith is the co-latitude 90° − φ. From pole to body it is the polar distance 90° − δ, counted with δ negative when the declination has the contrary name. The angle at the pole is the meridian angle t, the LHA counted east or west up to 180°. Given these two sides and the angle between them, spherical trigonometry yields the third side, the zenith distance 90° − Hc, and the angle at the zenith, the azimuth angle Z:

• sin(Hc) = sin(φ) · sin(δ) + cos(φ) · cos(δ) · cos(LHA)
• tan(Z) = cos(δ) · sin(LHA) / (cos(φ) · sin(δ) − sin(φ) · cos(δ) · cos(LHA))

These are the formulas Pub. No. 229 was computed with, and the calculator uses them too, taking the zenith distance from its half-angle form so it stays exact near the zenith. Z is counted from the elevated pole, east or west, and turns into Zn by the rules printed with the tables: in north latitude Zn = Z when the LHA is greater than 180° and Zn = 360° − Z when it is less; in south latitude Zn = 180° − Z and Zn = 180° + Z.

Using reduction tables

Pub. No. 229 tabulates the triangle for whole degrees of its entering arguments. But a real position rarely falls on whole degrees, so the navigator works from an assumed position (AP): its latitude is the whole degree nearest the dead reckoning position, and its longitude, within 30′ of the dead reckoning longitude, makes the LHA a whole degree. Entered with these and the whole degrees of the declination, the tables give the tabular altitude ht and the azimuth angle Z. Beside each altitude stands d, the altitude difference to the next degree of declination.

What remains is the declination increment, the minutes of the declination beyond the whole degree. The Interpolation Table multiplies d by Dec. Inc./60′ in two parts, the tens of d and then its units and decimals. When the tables print d in italics followed by a dot, the altitude curves too much for a straight line between two degrees, and the double second difference (DSD) adds a correction. The DSD is the difference between the d below and the d above in the same column, and the correction is always added to the altitude as printed. Z is interpolated by eye or with the same table.

The worked example follows these steps for the sight entered above, with the values as the tables give them, in tenths of a minute. The intercept is measured from the AP, and the line of position is plotted from there.

The tabular altitude is negative. The tables print it without its sign, on the other side of the C-S Line, with the supplement of Z (180° − Z). The example shows it with its sign, and d and the corrections carry the signs that go with it.

For this sight the tables give an Hc that differs from Hc calculated for the AP by 0.0′.

Near the horizon the tables continue past the C-S Line, the step line that marks where the altitude changes sign. A negative tabular altitude is printed without its sign on the far side of that line, with the supplement of Z. The worked example names the page and the side where it is printed.

Using the sphere

The sphere shows the sight as it stands in the sky. Drag the red body and the LHA and the declination follow; drag the north pole along the meridian and the latitude follows. With the body or the pole selected, the arrow keys move it in steps of 0.1°, or 1° with Shift and 1′ with Alt. Drag anywhere else to turn the sphere, and Reset view brings it back. The Coordinates button swaps the triangle for the coordinates it is made of, drawn in red: the latitude φ along the meridian and the declination δ along the hour circle, the meridian angle t along the equator, and the altitude h and the azimuth Zn in the horizon system.

Every move fills in the form, and the results and the worked example follow. When the GHA and the longitude are given, the GHA follows the body and the longitude stays as entered. And the address follows along, the view of the sphere included, so a bookmark or a shared link brings back the same sight.

Entering angles

Every angle can be typed as decimal degrees or as degrees and minutes, the way the Nautical Almanac gives them: 5 45.5 N, 5°45.5′N, N 5:45.5, and 5.7583 all mean the same declination. Seconds are accepted as a third number. South and west can be written with the letter or with a minus sign. The LHA comes from the GHA and the longitude whenever both are given.

URL and API parameters

latitude, longitude

the position, north and east positive (or N, S, E, W)

declination

the declination of the body, north positive (or N, S)

lha

the local hour angle, 0 to 360

gha

the Greenwich hour angle, 0 to 360; with the longitude it replaces lha

ho

the observed altitude, -5 to 90, for the intercept

calc=yes

calculate the values given; without it the page opens with the first example from the tables

digits

display precision, 0 to 9 (default 1): decimals of a minute of arc, and of a degree for azimuths

layer

what the sphere shows: triangle (the default) or coordinates

view

the direction the sphere is seen from, as azimuth and elevation in whole degrees (95,17); without it the sphere faces the body

format

json, xml, or csv for the API; leave it out for this page

In JSON and XML, input holds the values given, in degrees, and output holds the solved triangle and, with ho, the intercept in nautical miles with its direction and the point of the line of position. Positions and arcs are in degrees with three more decimals than digits, azimuths and angles in degrees with digits decimals. A value that does not exist is null.

Sources

• NGA: Pub. No. 229, Sight Reduction Tables for Marine Navigation (its introduction and its Interpolation Table)
• NGA: Pub. No. 9, The American Practical Navigator (Bowditch)
• U.S. Naval Observatory: The Nautical Almanac (the Greenwich hour angle and declination of the bodies; published with HM Nautical Almanac Office)
• Wikipedia: Intercept method
• Wikipedia: Sight reduction

See also

• Wndsn Sun Calculator and Wndsn Star Calculator
• Wndsn Dip Calculator
• How tos: Wndsn Quadrant Telemeters
• Wndsn Quadrant Telemeters as well as the print edition of the manual in the store