Understanding Plane Geometric Projections: A Comprehensive Guide
Plane Geometric Projections
Azimuthal Equidistant Projection
This projection maintains distances and angles from the center of projection. It is widely used in polar and skew aspects.
- Polar: Meridians converge at the pole, with parallel concentric circles separated by their true extent.
- Skew: Distances from any point (center of projection) remain accurate.
Lambert Azimuthal Equivalent Projection
This projection preserves surface areas, with rectilinear meridians and parallels separated to correspond to their actual surface areas.
- Polar: The radii of the parallels are the chords of the latitudes.
- Equatorial and Skew: Tables are used to determine specific values.
Polyhedric Projection
This projection decomposes the Earth’s surface into spherical trapezoids, which are then projected from the Earth’s center onto tangent planes at the midpoints of the trapezoids. It was used by the Spanish Military Geographic Institute (MTN) until 1965 and employs isosceles trapezoids with 10′ parallel separation and 20′ meridian separation. It is neither a conformal nor an equivalent projection.
Polyconic Projection
This projection decomposes the Earth’s surface into truncated cones with small heights. It was used for the 1:1 million scale World Map with a 4th parallel separation.
Equirectangular Projection
This projection features vertical meridians and horizontal parallels, both equidistant. A source parallel is divided into equal parts at true scale, and the meridians are divided to scale. If the source parallel is the Equator, the grid becomes square.
Cylindrical Projections
Mercator Projection (Gerhard Mercator Kremer, 1569)
This conformal, cylindrical projection preserves angles and is designed for navigation. The cylinder is tangent to the Equator, resulting in the following characteristics:
- Meridians are vertical lines at true scale, spaced at their true distance on the Equator.
- Parallels are horizontal lines spaced to maintain the true ratio of distances between parallels and meridians.
Since the secant of 90° is infinity, the poles cannot be represented in this projection.
Features:
- Conformal projection
- Rhumb lines are straight
- Scale varies with latitude, accurate only at the Equator
UTM Projection (Universal Transverse Mercator)
This is a Mercator projection where the cylinder is tangent to a meridian. Only the Equator and central meridian are straight lines; other parallels and meridians are complex curves.
- Universal: The globe is divided into 60 zones, each spanning 6° of longitude. The first zone’s origin is the antimeridian of Greenwich (180°), numbered 1 to 60 from west to east.
- Central Meridian: The central meridian of each zone has a scale factor of 1, making it an automeconic projection.
Coordinate System:
Each zone has an independent Cartesian coordinate system, with the x-axis representing the Equator and the y-axis parallel to the central meridian, shifted 500 km west.
Military Grid (UTM Grid):
Each zone is divided into 20 latitude bands of 8°, named C to X (excluding CH, I, LL, Ñ, O). Each band is further divided into 100 km squares, named A to Z (24 letters) for columns and A to V (20 letters) for rows.
Conic Projections
Lambert Conformal Conic Projection
This projection projects the Earth’s surface onto a cone tangent to a parallel (automeconic). Parallels are concentric circles centered at the cone’s vertex, while meridians are concurrent lines converging at the center.
Lambert Coordinates:
A grid system is used, with the OY-axis aligned with the Madrid meridian and the OX-axis perpendicular to the meridian at 40° latitude. The grid meridians are not parallel.
Convergence of Meridian:
The convergence angle (γ) is the angle between true north (geographic north) and grid north (Lambert north). It can be east or west, depending on the location relative to the central meridian.
Bonne Projection
This equivalent projection is used in the 1:200,000 scale military route map. It involves the development of a cone tangent to a parallel, with parallels represented by concentric circles at their true scale and meridians as non-concentric circles, spaced at their true scale along each parallel.
Sinusoidal Projection (Sanson-Flamsteed Projection)
This equivalent projection features horizontal, straight parallels spaced at their true scale and meridians as curves, spaced at their true scale along each parallel. The central meridian is a straight line.
Mollweide Projection (Homolographic Projection)
This equivalent projection maintains the true scale along the central meridian and Equator. Meridians are ellipses, spaced at their true scale along each parallel. Parallels are horizontal lines spaced to preserve area, resulting in a slightly denser spacing towards the Equator.
Goode Homolosine Projection
Similar to the Mollweide projection, but it uses multiple automeconic meridians to minimize distortion across different landmasses.