How the chart plot problems are worked

What every problem is built from

Every problem is generated by this site's engine from latitude and longitude, courses, speeds, times, current, variation and deviation stated in the problem. It names no charted light, buoy, landmark or depth.

Mercator sailing with meridional parts on a sphere; one nautical mile is one minute of latitude; current set is the direction the water flows toward; easterly variation and deviation are positive.

The working areas are latitude and longitude boxes we drew over open water. The tool does not know the chart's features, so a generated track is a number exercise and never a safe route.

What a real module asks that this does not

The licensing exam's chart plot module is ten multiple choice questions with a 90 percent pass mark, worked on a training chart within three and a half hours. This practice uses those three figures and nothing else from it.

A real chart plot module also asks for bearings of charted lights, buoys and landmarks, fixes from three bearings on charted objects, course to steer allowing for leeway as well as current, and chart reading questions (bottom type, bridge clearance, tides and the Coast Pilot). This tool covers none of those; it drills the calculation and plotting skills underneath them.

Dead reckoning

Each leg is a course, a speed and the minutes between two clock times. Distance is speed times hours. The leg is a rhumb line: a straight line on the Mercator chart.

On paper you lay each course off the compass rose with parallel rules and step its distance off the latitude scale beside the leg. The engine works the same leg by calculation:

• distance = speed × minutes / 60
• DLat (minutes, north positive) = distance × cos C
• departure (miles, east positive) = distance × sin C
• DLo (minutes) = departure × DM / DLat, which is departure / cos L along a parallel

Course and distance by Mercator sailing

Meridional parts measure how far a latitude sits from the equator on a Mercator chart, in minutes of longitude. The engine takes them on a sphere. Against the WGS84 ellipsoid that moves a course by under 0.2 degree and a distance or position by under 0.1 mile across these working areas, inside the tolerances.

Courses near 090 and 270 are where DLat and DM both shrink toward zero. There the engine uses the parallel sailing limit, so nothing is ever divided by zero.

• M = (10800 / pi) × ln tan(45 + L / 2), in minutes
• DM = M(to) − M(from)
• tan C = DLo / DM
• distance = DLat / cos C, or DLo × cos L along a parallel

Speed, time and distance

Times are on a 24 hour clock. A run that passes midnight is counted to 2400 and then on from 0000, and the ETA is given on the clock as it will read, the next day.

• distance = speed × time
• time = distance / speed
• speed = distance / time

Current: course to steer, speed made good, set and drift

Set is the direction the water flows toward, in degrees true; drift is its speed in knots. To hold a track you steer into the part of the current that runs across it. On paper: draw one hour of current from the start, swing your speed through the water from its end to cut the track; that line is the course to steer and the cut is one hour of speed made good.

Set and drift come from a fix: the set is the direction from the DR position to the fix at the same time, and the drift is that distance divided by the hours run.

Course to steer is the hardest answer to get within 2 degrees by hand: a 1 degree slip with the parallel rules when you lay the track carries straight into the course to steer, on top of any error in the triangle.

• angle a = set − track
• sin b = −(drift × sin a) / speed through the water
• course to steer = track + b
• speed made good = speed × cos b + drift × cos a
• set = course from DR to fix; drift = distance from DR to fix / hours

Compass conversion

Variation and deviation are written with E or W; east is added going from compass toward true and taken off going the other way. Deviation depends on the ship's head, never on the bearing being converted.

Each problem prints its own deviation table, made up by us: whole degrees at every 30 degrees of ship's head magnetic. Every compass problem says the convention: interpolate between entries in a straight line and round only the final answer. A deviation taken from the nearest entry instead is accepted on the deviation part, because it is never more than 1 degree from the interpolated value in these tables; the heading or bearing is still graded within 1 degree. From compass to magnetic the table is entered with the compass heading first and then again with the magnetic heading that gives, until it settles.

In a bearing problem the bearing is chosen where the table gives a deviation at least 2 degrees different from the ship's head, wherever the table allows, so taking the deviation for the bearing instead of the head is marked wrong.

Two differences from the exam's own papers: the exam writes psc (per standard compass) where this site writes C, and the boxes accept either; and the exam's tables carry half degrees, where ours carry whole degrees.

• T = M + variation
• M = C + deviation
• east positive, west negative

Running fix on one point (an extra)

An extra: the public sample modules do not include a running fix, so it is not in the mock. It is kept because it is a core coastal plotting skill.

The point is only a named position, Point A, given by latitude and longitude. Draw the first line of position through it, advance that line by the run between the two bearings (the course and the distance run, with no current), and draw the second line of position. The running fix is where the advanced line crosses the second line.

The engine solves the same crossing on the rhumb lines of the Mercator chart. Every running fix it sets has the two bearings crossing at 30 to 150 degrees and both ranges from the point between 1.5 and 6 miles.

Tolerances, rounding and the mock

An answer is right when it is within the tolerance of the engine's unrounded answer, measured the same way from either side; angles and times are measured the short way round. A question with several parts counts only when every part is right. The ten question mock passes at 9 of 10, the 90 percent bar; its mix of the five calculation types is our own, and the running fix extra is not in it.

A real module is multiple choice, with answers a few tenths of a mile or a few degrees apart. The tolerances here are set to about that spacing: a position within 0.3 mile is about 7 mm on a 1:80,000 chart.

Rounding: Positions to 0.1 minute of latitude and longitude; Courses, bearings, headings and set to the whole degree; Distances to 0.1 mile, speeds and drift to 0.1 knot; Times on a 24 hour clock to the minute; Deviation read between table entries to 0.1 degree; Answers are worked from the givens as printed, and graded against the unrounded result.

• Position: within 0.3 mile of the answer
• Course or bearing read from the chart: within 2 degrees
• Compass conversion: within 1 degree
• Deviation: within 1 degree, named east or west correctly
• Set of the current: within 5 degrees
• Distance: within 0.2 mile
• Speed, speed made good or drift: within 0.2 knot
• Time of day: within 3 minutes

The working areas

Boxes we drew over open water and then checked against NOAA's electronic chart data: none touches land, and two were made smaller to keep clear of a restricted area, dumping grounds, traffic lanes and an island group. Where a box still crosses something, it says so below. They are not taken from any chart.

Each working area is a practice area defined by latitude and longitude only. The tool knows nothing of the chart's features inside it: depths, shoals, wrecks, channels, anchorages, traffic lanes, restricted areas, lights and buoys.

• Block Island Sound: 41 08.0 N to 41 15.0 N, 071 50.0 W to 071 40.0 W, between Montauk and Block Island, on a chart of Block Island Sound such as the training edition numbered 13205 TR. Variation used: 15° W, the figure the public sample modules for 13205 TR state, given in every compass problem.
• Eastern Long Island Sound: 41 07.0 N to 41 10.0 N, 072 47.0 W to 072 32.0 W, mid sound between the two shores, on a chart of eastern Long Island Sound such as the training edition numbered 12354 TR. It is a narrow box, three miles north to south, kept more than two miles from the Falkner Island group. It crosses a cable area. Variation used: 14° W, the figure the public sample modules for 12354 TR state, given in every compass problem.
• Rhode Island Sound: 41 16.0 N to 41 22.0 N, 071 20.0 W to 071 06.0 W, south of Newport and east of Point Judith, on a chart of Martha's Vineyard to Block Island such as the training edition numbered 1210 Tr. That chart is a common teaching chart; it is not one of the three charts used in the exam room. It crosses the Buzzards Bay traffic separation scheme and overlaps an anchorage ground off Brenton Point. Variation used: 15° W, our own round figure, given in every compass problem.
• Lower Chesapeake Bay: 37 07.0 N to 37 14.0 N, 076 12.0 W to 076 04.0 W, inside the bay, north of the bridge and tunnel crossing, on a chart of the Chesapeake Bay entrance such as the training edition numbered 12221 TR. It crosses ship channels (York Spit Channel and the York River entrance channel), an anchorage and a charted wreck. Variation used: 9° W, the figure the public sample modules for 12221 TR state, given in every compass problem.