Dealing with distortion

The difficulties of making flat charts of a round earth are best demonstrated by an orange. It is easy enough to take an orange and dress it up to look like the earth (figure 1a, facing page). The hard part is to force said orange into a flat plane, making a chart. The result (figure 1b) is always a mess, both literally and mathematically. All charts suffer from this problem. In squeezing a spherical earth onto a flat chart, some aspect of the real earth – angles, shapes, sizes, directions – has to be distorted. These distortions lurk on all charts, as do other approximations.

Whether printed on paper or displayed on a video monitor, all charts are used in similar ways, with similar strengths and weaknesses. Our goal here is to see why charts are useful representations of the seas around us. We’ll begin with a look at the shape of the earth, discuss the how and why of datums, examine how chart makers project the earth onto charts, and tie it all together by showing how all this affects the usefulness of charts. We’ll conclude with a brief explanation of why electronic charts won’t change any of the lessons we’ve learned. With these revelations, the cult of charts is exposed as nothing more than one more tool of safe – and fun – piloting.

Giving the world a shape

Don’t believe the grade school teacher who told you the earth is spherical. Sir Isaac Newton predicted in the 18th century that earth’s rotation should cause it to bulge out at the equator, into a shape called an ellipsoid. Though it took decades (filled with intrigue, adventure and national pride), he was eventually proved right. Ignoring the topography of its surface, earth is a flattened sphere, squashed at the poles by about 0.3%. Don’t be misled by such a seemingly small number. The Earth’s radius at the poles is a whopping 11.8 nautical miles (22 kilometers) less than its radius at the equator. This is a huge difference – after all, Mount Everest is only 4.75 nm (8.8 km) high! While they seem esoteric, ellipsoids are at the core of every map and chart used by the mariner. They provide a reference frame on which all locations on earth are measured, and from which all charts are plotted.

The ellipsoid isn’t real – it is an approximate mathematical description of earth’s shape, ignoring topography and other variations. For more than 150 years, cartographers have struggled to measure the exact shape and size of earth’s ellipsoid. Typically a country or region constructed an ellipsoid based on surveys in their own little corner of the planet. Australia, for example, used the Australian Ellipsoid, while the U.S. used the Clarke Ellipsoid. Unfortunately, these approximations of the earth were different beasts, with distinct shapes and sizes. As a result Australian charts are based on a different earth than American charts. In the 19th century and for most of the 20th century, these differences, which amounted to tens to a few hundreds of meters, were meaningless. No sextant-bearing mariner could ever hope to measure his or her position with that sort of precision. Technology changed that. Enter GPS.

With a $150 GPS receiver, anyone can measure their position with the sort of precision and accuracy that makes these different ellipsoids dangerously different. Each ellipsoid is the basis of a datum, a “zero point” from which locations and elevations are measured. The location of the exact same object in two different datums can, and generally will, be different. Buried deep inside the software of your favorite GPS unit is a model for the earth – an ellipsoid – that may be different than the one used on your favorite chart. Two different ellipsoids mean two different locations for the exact same spot on the earth. Your GPS unit uses, almost certainly, the World Geodetic System 1984 (known universally as WGS84) ellipsoid. It is based on measurements by satellites across the entire planet, not just a region. After all, the “G” in GPS does stand for “global.”

Alas, some of charts produced by NOS don’t use the WGS84, but still employ the North American Datum of 1927 (NAD 27). Hence the reason charts printed in the post-GPS era have a little note like the one shown in figure 2. More recent charts use the NAD 83 datum, which is the same as the WGS84 datum. It is important to understand, however, this form of chart distortion before you use your charts and your GPS together. Most GPS manufacturers allow a unit to be set to just about any datum, so if you are sailing in Tahiti, you can install the “Tahiti” datum into your unit.

Giving the world a height

The ellipsoid gives a reference frame for the horizontal location of objects, but doesn’t provide a basis for measuring the elevation of objects. We need then a vertical datum for measuring elevations. The obvious choice for centuries has been sea level, which is all well and good until one recognizes that sea level isn’t, well, level.

Ignore tides, ignore ocean currents, ignore storm surge and winds – ignore every dynamic factor affecting sea level, except for the spinning earth. The oceans would then assume some equilibrium shape, affected by only earth’s rotation and the strength of gravity. Another myth shattered – gravity’s strength varies from place to place across the globe; thus, sea level varies from the smooth shape you would otherwise expect. For example, gravity is so weak in the central Indian Ocean that sea level there is 328 feet (100 meters) lower than the ellipsoid would suggest, while sea level in the North Atlantic around Iceland is 213 feet (65 meters) higher.

Figure 3: This shows how the lines of latitude are slightly farther apart at the north end of a northern hemisphere Mercator chart.

The height of sea level relative to the ellipsoid is called the geoid, and provides a sort of horizontal datum for elevation measurement. The geoid changes gradually as one moves from place to place across the earth’s surface, but changes little over time at a fixed place. For most of history, geoid variations were utterly irrelevant to the mariner. Charts used a vertical datum based on local low water, which defined zero elevation. With the advent of GPS, geoid variations are still irrelevant, but only just. GPS elevation errors are so high that the differences between elevation measured relative to the ellipsoid (which is what the GPS gives) and geoid (which are “real”) can’t be resolved. At least for now. If history is any guide, new technologies are constantly crowding the boundaries of charting.

Getting the world on paper

Maps and charts are definitely an old technology. The oldest known maps date to 2,100 B.C., and there is excellent evidence of their widespread use by Egyptian, Babylonian, and Greek societies. As early as 200 B.C., the Greek philosopher (and all-around genius) Eratosthenes had determined the earth was spherical, and as early as 150 B.C. the Greeks knew that maps made of this round world are inherently wrong, because they are inherently distorted.

Figure 4: The latitude and longitude grid is based on angles formed at the center of the earth. For latitude, the angles are formed between the poles and the equator.

What distortions to minimize is one of the many decisions cartographers have to make when preparing a chart. For example, no chart can correctly represent both the relative area and shape of objects. Angles, directions, and distances are distorted on a chart as well. The map projection determines the balance of distortion between all of these features. A projection is a system for transferring earth’s features onto a flat plane. Conceptually, the process is simple: The chart maker chooses a developable surface – a shape that can be unrolled into a flat sheet – and wraps it around the earth. One projects the earth’s features onto the developable surface, and then unrolls the surface onto a flat plane. A developable cylinder is wrapped around the earth. Meridians of longitude are projected onto the cylinder as straight lines, while parallels of latitude are projected on as rings. When the cylinder is unrolled, we have a map. Conceptually is the operative word here – precious few projections in common use today use these simple geometric projections. Rather, the vast majority of charts in use today use mathematical projections, including all of the charts used by mariners.

Mercator’s famous projection is one of these mathematical projections. He started with a projection onto a cylinder, but then modified it extensively. He choose to wildly distort sizes, distances, and directions on his charts to minimize the distortion of rhumb lines. (A rhumb line is the constant direction course between two points, the easiest course to sail in a vessel equipped only with a compass.)

“Most NOAA charts use a Mercator projection,” noted NOAA’s Commander Steve Barnum. Since 1920, the various agencies responsible for U.S. marine charts have used this same venerable projection. On the Mercator projection, rhumb lines are straight lines, but distances, directions, and areas of features are distorted to various degrees. This distortion is relatively small for charts less than a few degrees wide, but easily observable when the chart exceeds 10 degrees of latitude or longitude. One can check this easily by measuring, with a pair of dividers, the distance between latitude lines at the north and south end of a chart. For charts of the northern hemisphere, the northern distance is always larger (figure 2). All charts have this balance of distortions built into them, a balance that can be used to advantage.

Giving the world a ruler

One of these advantages explains another of those cult-shrouded features of charts: why one can use the latitude scale, but not the longitude scale, to measure distances. Latitude and longitude are the solution to the vexing problem of how to locate a place on a sphere. A spinning sphere, like the earth, has only two unique points: the north and south poles. It seems natural to use the poles to define a plane midway between them – which provides us with the equator. A point’s (geocentric) latitude is the angle from the equator to the point, as measured at the center of the earth. Figure 4 shows that the angle between the poles and the equator, measured from the center of the earth, is 90°. Latitudes are always expressed as angles from 0° to 90° because one always uses the closest point on the equator to begin the measurement. Figure
4 also illustrates that lines of equal latitude are circles, parallel to the equator and each other. Hence the reason latitude lines are called parallels. Two given parallels are always the same distance apart.

Figure 5: National hydrographic offices have used different ellipsoids, or models of the earth, as the datum to draw their charts.

Ignoring for now earth’s deviation from a perfect sphere, the parallel nature of latitude lines means it can be used for measuring distances. Seen from the center of the earth, every degree of latitude maps out as same distance on earth’s surface. The original definition of a nautical mile was “the distance on earth’s surface equivalent to one minute of latitude.” This is why a chart’s latitude scale can be used to measure distances on the chart. A typical chart might show a full degree of latitude, which allows one to measure distances of up to 60 nautical miles easily and quickly. Depending on the age of the chart, the minutes of latitude will be divided into either arc seconds or tenths of a minute, the latter reflecting the increasing use of GPS on the water. Because the Mercator projection stretches the latitude scale out always use the part of the scale closest to your own parallel of latitude.

Parallels alone will not uniquely locate a point on the earth. A second coordinate is required, ideally one perpendicular to the latitude, fixed to the globe in some natural way with a naturally defined starting point. Longitude satisfies only the first two of these three requirements. Lines of longitude (called meridians) run from pole to pole, as shown in figure 3, and are everywhere perpendicular to parallels. Unlike parallels, meridians converge toward the poles. At the equator, one minute of longitude is one nautical mile wide. But beware! Meridians converge toward the poles! At the South Pole, for example, one can stand over the bronze pipe marking the pole and cover every line of longitude at once with one’s hand. Only at the equator is one minute of longitude equal to one mile, and that is why the longitude scale is never used to measure distances. This difference has nothing to do with the Mercator projection, but is inherent to the way meridians are defined.

Another difference between latitude and longitude is the starting point of the measurement. The equator is a natural starting point for latitude, but no natural starting point exists for longitude. One must be chosen arbitrarily. Or two, or three. Different map makers used different starting points, including the Canary Islands, Paris, Moscow, and Washington, D.C. It wasn’t until 1884 that an international conference in Washington choose the transit telescope at Greenwich Observatory, London, as the zero point, or Prime Meridian, for longitude. Longitude is the angle at the equator between the Prime Meridian and the meridian through the point. Unlike latitude, longitude runs from 0 degrees to 180 degrees, both East and West of Greenwich. The reason for the difference in scales is due to the difference in their definition. Viewed from the center of the earth, the angle between the equator and poles is 90 degrees, while the angle from the Prime Meridian to the International Date line is 180 degrees.

Giving the chart a ruler

The most significant distortion of a chart is perhaps the least noticed by users: the scale. Scale is the ratio of distance between two points on the chart and the corresponding distance in the real world. Because the scale is a ratio, both distances must be given in the same units of measurement. The unit is irrelevant – inches, centimeters, picas – but that unit must be used in both measurements. For example, assume two points are three inches apart on a chart, and 73,000 inches (one mile) apart in the real world. This chart has a scale of 3″/72,000″, or 1/24,000. One could measure the same distance on the same chart with the metric side of the ruler, and find a distance of 7.62 cm. Combined with the real-world separation of 182,880 cm, this gives a scale of 7.26 cm /182,880 cm or 1/24,000. The scale isn’t generally written as a fraction, but rather in the ratio notation some may remember from college-entrance examination, 1:24,000.

Figure 6: An ellipsoid is a model for the horizontal datum for a chart, while the geoid is used
to establish a chart’s vertical datum.

One of the more confusing facets of charts is the concept of large and small chart scales. A small-scale chart is one that shows a large area of the earth’s surface, and conversely a large-scale chart is one which shows a small area of the earth’s surface. The terminology derives from the scale. Written as a fraction (1/24,000 for the example above) this is a rather small one. Thus it is a small-scale chart. Another way of remembering the terminology is to picture the number of charts required, at a given scale, to represent the entire earth at that scale. A small-scale chart would require a small number of charts, while a large-scale chart would require a large number.

Despite the wonderful variety of charts, anyone who has spent more than 15 minutes on the water knows that a chart with the right scale, covering the right area, with the right detail, is simply not available. And it won’t be any time soon, even if one considers electronic charts. Avoiding the arguments about which type is better, electronic charts come in two varieties: raster and vector. Raster charts are nothing but digital representations of scanned paper charts. Except for the precious few “ENC” charts from the NOS, vector charts are simply latitude/longitude positions taken from the raster charts, and are not inherently more accurate than raster charts. All of the inherent approximations and distortions present on paper charts are thus present on their electronic cousins. Both types of e-charts have zoom abilities, which change the scale of presentation somewhat, but don’t change the information detail of the chart.

Nor do they make jumping from one chart to another any easier. Paper-based mariners will be familiar with the job of transferring their position from one chart to another, one with different scale, different detail and even different location of objects. This trying job is made necessary by the various distortions inherent in the chart-making process – even the less obvious distortions of paper, which swells and shrinks up to 1% because of humidity.

Electronic charts provide little relief from this. Raster charts reflect their paper ancestors, and hence suffer from the same limitations. Vector charts (again excluding the NOS’s products) have a more seamless appearance, but because their databases are derived from raster charts, they generally offer no more accuracy than their paper equivalents. Is there any hope for this situation to change in the future?

The NOS now has a print-on-demand service, which provides mariners with up-to-the-minute charts, corrected for recent changes. These charts are based entirely on the raster files of existing charts maintained by the NOS, and a user cannot modify the scale or boundaries of the chart, and certainly cannot do so while underway.

A new technology, geographic information systems (GIS), might allow future users more flexibility in ordering custom charts. A GIS stores all information digitally, referenced to the latitude-longitude position of a feature. Any feature on a chart – lighthouse location, light characteristics, water depths, bottom conditions, even labels – can be stored with its georeferenced position. It is possible, although no such system exists yet, that users could custom-design their own charts. One would, for example, enter the desired scale, latitude and longitude boundaries onto a web site. Then one would choose the “layers” of information wanted: water depths, tide information, current directions, even the location of gas stations or yacht club racing buoys could be used at the users’ discretion.

The chart could then be printed commercially, pieced out onto a users’ printer, or loaded onto a disk for use as an e-chart. The resulting chart would look quite like the current ones, except that they would provide exactly the information the user wanted most. This technology is not as far fetched as one may think – the new National Geographic Society’s map maker page allows users to do exactly this for a limited set of maps.

For the first time in the history of chart making, the average mariner can fix his or her location more accurately than charts allow. The National Ocean Service and its foreign counterparts are working hard to fix this embarrassment of precision, but for the time being it is worth remembering that the various distortions inherent to charts makes them valuable, but imperfect, aids. As always, it is best to remember the words of Nat Bowditch: the prudent mariner uses all of the tools the situation provides.

By Ocean Navigator