Definition of Projection
The surface of the Earth's ellipsoid is curved, while maps are typically drawn on flat surfaces. Therefore, cartography first requires unfolding the curved surface into a plane. However, a spherical surface is non-developable - direct flattening will cause cracks or wrinkles. Maps created with such imperfections are impractical, necessitating specialized methods to unfold the surface into seamless planes, which gave rise to map projection theory.
The fundamental principle involves representing positions on the sphere using latitude and longitude. During projection, intersections of selected meridians and parallels are plotted on a plane. Points sharing the same longitude form meridians, while those sharing latitude form parallels, creating a graticule network. Through this network, spherical points can be transferred to corresponding planar positions based on their coordinates, as illustrated:
 |
| Figure: Transferring Points from the Spherical Surface to the Plane Based on Latitude and Longitude |
Essence of Projection
Many analytical techniques and spatial data are designed for two-dimensional or planar coordinates. Converting three-dimensional GCS (Geographic Coordinate System) to planar coordinates through map projection becomes essential. Map projection mathematically transforms geographic coordinates (λ, φ) into planar coordinates (X, Y). As three-dimensional to two-dimensional conversion inherently causes distortion, projections minimize such deformations.
Map projection maintains geographical relationships and integrity during coordinate transformation, fulfilling fundamental cartographic requirements and enabling spatial operations/analysis. This makes projection crucial for GIS applications of geographic data.
Projection Methods
Two primary projection methods exist: Geometric Perspective Method and Mathematical Analytical Method.
The Geometric Perspective Method utilizes perspective relationships to project Earth's surface points onto planar surfaces (like cylindrical, conical, or planar surfaces), as shown:
 |
| Figure: Schematic Diagram of Perspective Projection |
The Mathematical Analytical Method establishes direct functional relationships between geographic coordinates on the ellipsoid and planar coordinates. This approach determines the correspondence between spherical coordinates and planar coordinates mathematically. Most modern projections employ this method.
- Geometric Perspective Method
- Mathematical Analytical Method
Projection Distortion
Flattening a non-developable Earth ellipsoid inevitably causes tearing or overlapping. To maintain continuity, stretched or compressed adjustments eliminate gaps/folds. These adjustments alter geometric properties of the graticule, resulting in projection distortion.
Map distortions include:
- Length Distortion
- Angular Distortion
- Area Distortion
- Shape Distortion
Length Distortion occurs when the scale factor (ratio of projected length to original length) deviates from 1. This fundamental distortion affects all projections and induces other deformations.
Angular Distortion refers to differences between angles measured on the projection and actual angles on the ellipsoid. It indicates shape deformation.
Area Distortion arises when the area scale (projected area vs original area) differs from 1, serving as a quantitative distortion measure.
Shape Distortion manifests as discrepancies between projected shapes and their real-world counterparts.
Related Topics
Reference System Conversion Methods