# Coordinate Conversions

Depending upon the data and type of visualization, you may want to convert between normalized, data or device coordinates. This section details two-dimensional and three-dimensional coordinate system characteristics and provides resources for various coordinate conversions. See the following for details:

- Two-Dimensional Coordinate Conversion
- Three-Dimensional Coordinate Conversion
- Using Coordinate Conversions

## Two-Dimensional Coordinate Conversion

This section describes the formulae for conversions to and from each coordinate system. In the following discussion, Dx is a data coordinate, Nx is a normalized coordinate, and Rx is a raw device coordinate. Let Vx and Vy represent the size of the visible area of the currently selected display or drawing surface.

The field S is a two-element array of scaling factors used to convert X coordinates from data units to normalized units. S contains the parameters of the linear equation, converting data coordinates to normalized coordinates. S[0] is the intercept, and S[1] is the slope. Also, let Dx be the data coordinate, Nx the normalized coordinate, Rx the device coordinate, Vx the device X size (in device coordinates).

With the above variables defined, the linear two-dimensional coordinate conversions for the x coordinate can be written as follows:

Coordinate Conversion | Linear | Logarithmic
Data to normal |  |
Data to device |  |
Normal to device |  |
Normal to data |  |
Device to data |  |
Device to normal |  |

The y- and z-axis coordinates are converted in exactly the same manner, with the exception that there is no z device coordinate and that logarithmic z-axes are not permitted.

This coordinate conversion functionality is built into object graphics through the XCOORD_CONVERT and YCOORD_CONVERT properties or each type of visualization object. If you are working with a Direct Graphics display, you can use the CONVERT_COORD function.

## Three-Dimensional Coordinate Conversion

To convert from a three-dimensional coordinate to a two-dimensional coordinate, IDL follows these steps:

- Data coordinates are converted to three-dimensional normalized coordinates. To convert the x coordinate from data to normalized coordinates, use the formula Nx = X0 + X1Dx. The same process is used to convert the y and z coordinates using !Y.S and !Z.S.
- The three-dimensional normalized coordinate, P = (Nx, Ny, Nz), whose homogeneous representation is (Nx, Ny, Nz, 1), is multiplied by the concatenated transformation matrix !P.T:
- P′ = P•`!P.T`
- The vector P′ is scaled by dividing by w, and the normalized two-dimensional coordinates are extracted:
- N′x = P′x/P′w and N′y = P′y/P′w
- The normalized xy coordinate is converted to device coordinates.

## Using Coordinate Conversions

How coordinate conversions are defined depend upon the display type as follows:

- iTools — in an iTool display, the interactive nature of the tool makes coordinate conversions transparent. There is no need to programmatically configure the transformation matrices of the objects.
- Object Graphics — converting an object’s data coordinates into normalized coordinates for display is a common task.
- Direct Graphics — the IDL Direct Graphics system automatically positions and sizes static visualizations so there is no need to set up a transformation matrix. However, you can convert between the supported coordinate systems.