# Geodetic Surveying: Aligning Local Astronomical Axes with Conventional Terrestrial Axes

This VRML module demonstrate the conversion of local vs. "universal" conventional terrestrial coordinates used in three-dimensional geodetic surveys.

## 3D Animation

You will need a VRML 2.0 viewer to see the illustration below. Please download the Cortona vrml plug-in.

Click here for the VRML Animation Within the animation you will see three buttons:

• Press the green cube labeled "1" to view the first rotation.
• Press the blue cube labeled "2" to animate the second rotation.
• Press the yellow cube labeled "R" to reset it to the original rotation.

## Conventional Terrestrial Coordinate System of the Earth Click on the image to enlarge.

When determining the position of a point on the Earth within a three-dimensional space, one method is to use the Conventional Terrestrial (CT) coordinate system: a Cartesian coordinate system with x, y, and z axes.

In this coordinate system,

• The origin (0,0,0) corresponds with the mass center of the earth.
• The X axis is parallel to the Equator and points through the Greenwich Meridian (0° longitude). The Greenwich Meridian is also known as the Prime Meridian.
• The Z axis is coincident to the conventional terrestrial pole (CTP) which was the mean position of the Earth's rotational axis between 1900 and 1905.
• The Y axis lies in the Earth's equatorial plane and is perpendicular to the X and Z axes and creates a right-handed Cartesian coordinate system.

The conventional terrestrial pole is commonly referred to as the Earth's North Pole.

## Local Astronomical Coordinates Click on the image to enlarge.

Surveyors generally use a three-dimensional Cartesian system called the Local Astronomical (LA) coordinates to describe positions in reference to their own location.

In this coordinate system:

• The origin (0,0,0) corresponds with location of the instrument used to make surveying measurements on the surface of the Earth: from now on called the observer's station.
• The x axis (N) points from the origin towards the CTP (north) and is a tangent with the curvature of the Earth.
• The z axis (U) points away from the surface of the Earth opposite the direction of gravity towards the observer's zenith. Its negative axis points in the direction of gravity and the observer's nadir.
• The y axis (E) creates a left-handed Cartesian coordinate system by being perpendicular to both the x and z axes and pointing east from the observer's station. This axis is tangent to the curvature of the Earth at the observer's station.

Note that unless the observer is at the North Pole, the direction of the U axis (local astronomical z axis) will not align with the Z axis in the CT coordinate system.

## Aligning the Two Coordinate Systems

To align local astronomical coordinates (LA) with a conventional terrestrial coordinate system you will need to perform mathematical conversions which "rotate" the LA coordinates around two axes.

The first rotation is around the E axis (LA y axis) and "pushes" the N axis (x-axis) down until the NE plane is parallel with the Earth's equator. The U axis (z axis) is now parallel with the CTP (North pole) and the N axis in the LA system is now pointing into the earth.

The second rotation is around the U axis in the LA system and aligns the once-rotated LA N axis with the CT x-axis in equatorial plane.

Because the LA coordinate system is a left-handed coordinate system and the CT coordinate system is a right-handed coordinate system, the East axis will be pointing 180° away from the CT y-axis (that is, west from the observer).

To make the systems truly align, a negative sign is introduced in the LA y-axis.