Secant projection
Definition
A secant projection is a specific case of a map projection where the developable surface (such as a cylinder or cone) intersects, or "cuts," the globe along two lines (in a cylindrical or conic projection) or at a small circle (in an azimuthal projection), creating two standard lines of true scale where distortion is zero; this configuration reduces overall scale distortion across the mapped area by distributing it on either side of the secant lines, making it a more optimal choice for mapping regions with larger east-west or north-south extents compared to a tangent projection, which contacts the globe along a single line and often exhibits greater distortion away from that point of contact.
Application
A key practical application of a secant projection is in national and regional topographic mapping systems. For instance, the Lambert Conformal Conic projection with two standard parallels (a secant case) is the foundational coordinate system for official topographic map series in countries with large east-west extents, such as the United States (State Plane Coordinate System zones) and France (Lambert-93). By having the cone intersect the Earth along two chosen parallels, this secant configuration minimizes scale distortion across the entire mapped zone, ensuring that distance, shape, and directional measurements remain highly accurate for surveying, engineering, and land navigation purposes throughout the region, which would not be achievable with a single tangent line.
FAQ
What is the main difference between a tangent and a secant projection surface?
The key difference is the number of contact lines with the globe. A tangent projection touches the globe along a single line (a meridian or parallel) or at a single point, creating one line of true scale. A secant projection surface (a cylinder or cone) slices through the globe, intersecting it along two lines (small circles), creating two standard lines where scale is true. This secant case reduces overall distortion across the mapped area.
Why would a cartographer choose a secant projection over a tangent one?
A cartographer chooses a secant projection to minimize overall scale distortion across a wider region. Because distortion increases with distance from the line(s) of true scale, having two standard lines in a secant projection effectively "sandwiches" the distortion, keeping it lower and more balanced across the entire map sheet. This makes secant projections ideal for mapping larger areas, like entire states or countries, with greater accuracy.
How are the "standard lines" chosen in a secant projection?
The standard lines are strategically placed based on the geographic extent of the area to be mapped. A common rule of thumb is to position them at approximately 1/6th and 5/6th of the north-south extent of the region. For example, for a zone spanning from 30�N to 45�N, the standard parallels might be placed at 32�N and 43�N. The goal is to have the lines of zero scale error enclose the main area of interest.
What are some real-world examples of secant projection systems?
Secant projections form the basis of many official coordinate systems:
Lambert Conformal Conic (with 2 SP): Used in the U.S. State Plane Coordinate System for states with a predominant east-west extent (e.g., Tennessee, California zones).
Transverse Mercator (with 2 meridians): Used in the UTM (Universal Transverse Mercator) grid system, where each zone is a secant cylinder, reducing scale distortion to a maximum of 0.1%.
Lambert Azimuthal Equal-Area (secant case): Used for mapping continents or large countries in statistical atlases to preserve area accurately.
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