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A. Introduction

1. A Question of Dimensions

A formal reference system is one which covers an area large enough that the Earth's shape must be taken into account. It can be one- (1D),  two- (2D), or three-dimensional (3D). Generally, 1D systems are used for elevations, 2D for horizontal positioning, and both combined for 3D.  While small scale projects can use a local assumed system, those covering larger areas require more rigor as they must account for the Earth's shape and dynamics.

Most applications use a horizontal 2D grid coordinate system with elevations added if relief information is needed. This is largely because of:

• Surveying measurements - Horizontal (traversing) and vertical (leveling) measurements are done separately using different equipment and procedures.
• Mapping - Referenced to a horizontal base with separate depiction of elevations (eg, contours, TINs, DEMs, etc), even in digital environments.
• Specific characteristics - Horizontal and vertical positioning have unique issues that aren't resolved by combining them into a single 3D system

2. A Coordinated Attack

Going from the 3D Earth with its curvature, meridian convergence, and terrain variations to a 2D horizontal coordinate plane will introduce distortions. For some mapping applications, it might be possible to ignore the distortions if they are below the map's resolution. For most surveying applications, however, distortions could have significant impact on accuracy and have to be considered.

There are two ways to control distortions

• Minimize their effect in the first place. It's not possible to eliminate them entirely, but it is possible to design a system which keeps them with an acceptable tolerance.
• If they behave systematically, we can eliminate them by computation. To do this, we have to define and maintain mathematical rigor throughout the 3D to 2D process.

In this topic we will examine

• How grid coordinate systems are developed
• How to convert ground measurements to grid and vice versa.