Ellipsoid and Geoid

In GIS and geodesy, the ellipsoid and geoid are fundamental models used to approximate the shape of the Earth. They serve as reference surfaces for geographic coordinate systems, positioning, and elevation measurements.

1. Ellipsoid (Spheroid)

An ellipsoid (or spheroid) is a mathematically defined smooth surface that approximates the Earth's shape. It is formed by rotating an ellipse around its minor axis, making the Earth slightly flattened at the poles and bulging at the equator.

Semi-Major Axis (a): The longest radius, measured along the equatorial plane.
Semi-Minor Axis (b): The shortest radius, measured from the center to the poles.
Flattening (f): The measure of how much the ellipsoid is compressed at the poles. Calculated as: f=a−baf = \frac{a - b}{a}
Reference Ellipsoid: A specific mathematical model used in geodetic calculations.

Examples of Ellipsoids

WGS84 (World Geodetic System 1984) – Used globally, including for GPS.
GRS80 (Geodetic Reference System 1980) – Used in North America.
Clarke 1866 – Older model used in North American mapping.

Practical Use Cases

GPS and Google Maps use the WGS84 ellipsoid for positioning.
GIS software relies on ellipsoidal coordinates to ensure accurate mapping.

2. Geoid

A geoid is a model of the Earth's shape based on gravity measurements. It represents the mean sea level (MSL) extended under landmasses, showing where gravity is constant. Unlike the smooth ellipsoid, the geoid has an irregular shape due to variations in Earth's gravity.

Geoidal Height (N): The difference between the geoid and the reference ellipsoid at any location. N=h−HN = h - H Where:
- h = Ellipsoidal height (from GPS)
- H = Orthometric height (real elevation above the geoid)
- N = Geoid undulation (difference between ellipsoid and geoid)

Examples of Geoid Models

EGM96 (Earth Gravitational Model 1996) – Used globally for precise height measurements.
EGM2008 – A more refined version for better accuracy.
NAVD88 (North American Vertical Datum 1988) – Used in the United States for elevation data.

Practical Use Cases

Flood risk mapping requires geoid-based elevations (orthometric heights).
Satellite altimetry uses geoid models to measure sea level changes.
Surveying and engineering projects use geoid corrections for precise elevation data.

3. Comparison

Feature	Ellipsoid	Geoid
Definition	A mathematically perfect surface approximating Earth's shape.	A physical model based on Earth's gravity field.
Surface Type	Smooth and regular.	Irregular, follows mean sea level.
Used For	Defining latitude, longitude, and ellipsoidal heights.	Defining mean sea level and orthometric heights.
Reference Models	WGS84, GRS80, Clarke 1866.	EGM96, EGM2008, NAVD88.
Key Application	GPS navigation, cartography.	Elevation modeling, sea level studies.

4. Importance in GIS

Accurate Positioning: GPS provides ellipsoidal height, but real-world applications require geoid-based heights.
Elevation Corrections: Engineers and surveyors use geoid models to adjust GPS height data.
Map Projections: Selecting the right reference ellipsoid ensures accurate geographic data representation.

The ellipsoid is a smooth, mathematical model used for latitude/longitude measurements.
The geoid is a gravity-based, irregular surface that represents mean sea level.
GIS applications often convert ellipsoidal heights (from GPS) to orthometric heights (using geoid models) for real-world accuracy.

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