### Coordinate Transformations

The position of a point can be expressed in different coordinate systems, which can be tied to different reference frames (datums). These coordinates may also change in time due to the Earth’s dynamics. Therefore, it is important to know how to apply transformations between coordinate systems, reference frames, and epochs in order to maintain compatibility amongst various datasets.

**Transformation between coordinate systems**: The mathematical formulas are exact and do not affect the precision of the coordinates. Geographic (*latitude*(*f*),*longitude*(*l*) and*height*(*H or h*)), Cartesian (*X*,*Y*,*Z*), and UTM, MTM and stereographic projections (*x*,*y*,*z*) are examples of coordinate systems.**Transformation between reference frames**: Helmert similarity transformations (using 7 or 14 parameters) are typically used for conversions between 3D geometric reference frames. While the estimated parameters have associated uncertainties, the transform provides exact coordinate shifts between the frames. When relating the coordinates of points from reference frames with different definitions, grid shift files are usually applied. They are built from the differences in coordinates at common points. The accuracy of the transformation depends significantly on the systematic distortion between the reference frames and the spatial distribution of the common points used to form the grid. NAD27, NAD83(Original), NAD83(CSRS), ITRF, CGVD28, and CGVD2013 are examples of reference frames.**Transformation between epochs**: The application of a transformation for the epoch is new. Until recently, crustal motion was generally neglected as most observation techniques were based on relative measurements (angle, distance and height differences) with relatively low accuracy at the continental scale. Nowadays, GNSS allows high accuracy at the global scale, allowing detection of horizontal and vertical crustal motion. As the crustal motion is not constant nation-wide, velocity models are developed as grid files.

Natural Resources Canada (NRCan) provides tools allowing users to apply the three types of transformations. These tools are available as on-line and desktop applications.

- TRX allows the transformation between coordinate systems, 3D geometric reference frames and epochs.
- NTv2 allows the transformation between horizontal datums using a grid shift file. Originally, it was developed to support the transformation between NAD27 and NAD83(Original). Now it also accepts several regional transformations developed by provincial agencies (e.g., NAD83(Original) to NAD83(CSRS), ATS77 to NAD83(Original)).
- GPS-H allows the transformation between two height components. It can be between an ellipsoidal height (e.g., NAD83(CSRS)) and a height with respect to a vertical datum (e.g., CGVD2013) or between two vertical datums. GPS-H can also apply transformation between coordinate systems.