### 1. Fitting an Ellipsoid

Because measurements are referenced to the geoid, an ellipsoid is selected and fit to it. The geoid is irregular so no single ellipsoid fits it perfectly. Fitting an ellipsoid requires some degree of compromise.

#### a. Regional Fit

One approach is to fit an ellipsoid to a specific region of the geoid. Figure C-1.

Figure C-1 Regional Ellipsoid Fit |

A centrally located point is used as the fit origin where the ellipsoid and geoid coincide. Parameters that are fixed:

- The point's geodetic latitude and longitude (see Section 3)
- Azimuth of a line from the origin point
- Two ellipsoid parameters

Either the minor axis is set parallel to the Earth's axis of rotation or the *deflection of the vertical* is specified. The deflection of the vertical, δ, is the angle between the vertical line and ellipsoid normal., Figure C-2. The ellipsoid normal lies in the meridian. Because the vertical line is defined by gravity, it is normal to the geoid so is not restricted to the meridian. δ can have two components- one in the meridian and the other perpendicular to it.

Figure C-2 Deflection of the Vertical |

Making the geoid and ellipsoid coincide assumes the two surfaces do not separate significantly throughout the area. This was a reasonable assumption for an early datum definition before a good mathematical geoid model was available. The error introduced wasn't significant for most surveys.

Different regions would use their own ellipsoid sizes and fits to meet their needs, Figure C-3.

Figure C-3 Multiple Fits |

Of course, what fit one area wouldn't fit well anywhere else. That was of little consequence before global measurements were possible.

#### b. Global fit

Evolution of global measurement systems required developing models that weren't limited to regions. A geocentric ellipsoid, Figure C-4, is used.

Figure C-4 Geocentric Fit |

Fitting parmeters are

- Location of the coordinate system origin (see Section 3)
- Orientation of the major and minor ellipsoid axes
- Two geometric ellipsoid attributes