Choosing Projections and Scales for a Coastal County
A county planning office must produce two products from the same shoreline data: (1) a property-tax map that assigns every parcel in a 45 km coastal county an area accurate to within 1%, and (2) a navigation chart for small fishing boats entering an estuary with a narrow 120 m channel. The office has a GIS with a coastline digitized at 1:5,000, a 30 m digital elevation model, and aerial photography. A staff member proposes using a single web Mercator map at 1:50,000 for both products. A second proposes a conformal projection for the tax map and an equal-area projection for the chart. A third notes that the digitized shoreline, measured at 1:5,000, is 61 km long, while the same shoreline on a 1:250,000 topographic sheet measures 52 km. The office has one week and no budget for new survey work.
1. Which projection should be used for the tax-area map, and which for the navigation chart? Justify your choice with the properties of map projections and the specific requirements.
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The tax map needs an equal-area projection, such as an Albers or Lambert azimuthal equal-area projection, because the requirement is that each parcel's mapped area be accurate to within 1%. Equal-area projections preserve the ratio of areas, so parcel areas can be compared and taxed fairly; conformal projections preserve local angles and shapes but allow area distortion that grows away from the standard line. At county scale the distortion may be small, but an equal-area projection is the defensible choice. The navigation chart needs a conformal projection, normally Mercator, because it preserves angles and makes a constant bearing a straight line (rhumb line), which is what a small-boat navigator plots. Web Mercator is conformal but is a web-display projection, not a chart projection, and it is not equal-area. The second staff member's proposal reverses the correct logic: conformal for tax and equal-area for navigation would give the wrong property for each task.
2. Explain why the same shoreline is 61 km at 1:5,000 and 52 km at 1:250,000, and what the office should tell the public.
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Shoreline length depends on the resolution at which it is measured. At 1:5,000 the digitized line captures small cusps, inlets, rocks, and marsh edges; at 1:250,000 the cartographer generalizes those features into a smoother line, cutting off many small detours. Each detour adds length, so the finer measurement is longer. There is no single true length independent of scale; the length is a function of the measuring interval, a property often called the coastline paradox. The office should not present either number as the definitive shoreline length. It should state the dataset, scale, and method used, and choose one standard for legal or engineering purposes. For the tax map, parcel boundaries matter more than shoreline length; for the chart, the 1:5,000 source is more appropriate but still must be checked against hydrographic survey data.
3. At 1:50,000, how wide is the 120 m channel on paper, and what does that imply for the navigation chart?
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120 m divided by 50,000 equals 0.0024 m, or 2.4 mm. A 2.4 mm channel is about the width of a fine pen line; on a printed map it would be hard to distinguish from the shoreline symbol, and it cannot show channel edges, shoals, buoy positions, or depth soundings at a usable separation. Navigation requires a larger-scale chart, typically 1:10,000 to 1:25,000 for a narrow estuary, based on a hydrographic survey with measured depths and positions. The 1:50,000 map is suitable for regional context, not for piloting a boat through a 120 m channel. The office should produce a separate large-scale chart from the best available survey and mark it clearly as not for navigation until verified. Scale, symbol width, and positional accuracy all limit what can be safely shown.