Theory: Coordinate systems

OrcaWave uses a number of frames of reference, each of which consists of a reference origin and a set of axes directions, to represent different coordinate systems. We have one global frame of reference and additionally a number of local coordinate systems, generally one for each body.

All the coordinate systems are right-handed, and positive rotations are clockwise when looking in the direction of the axis of rotation.

We denote the global frame of reference $\GXYZ$. It is a right-handed system with its $Z$-axis, $G\urm{Z}$, positive upwards and with its origin on the free surface of the water.

The local coordinate systems for each body are a mesh frame of reference $\Mxyz$ and a body frame of reference, $\Bxyz$. Each local system has its origin at a selected fixed point on the body and the axes are in particular fixed directions, such as the surge, sway and heave directions for a vessel.

Mesh coordinates

$\Mxyz$ is the coordinate system of a body mesh file. The location of the mesh origin in the global axes is given by the body data for mesh position: $X$, $Y$ and $Z$. The orientation of the mesh axes relative to the global axes is given by the body data for mesh orientation: heel, trim and heading.

Body coordinates

$\Bxyz$ differs from $\Mxyz$ only via data for vertical positioning of the body: $Z$ position, heel angle and trim angle:

With the body in its heeled and trimmed position, the body axis $B\urm{z}$ is vertical and positive upwards. The heel and trim angles define the rotation from $\Mxyz$ to $\Bxyz$, applied as pitch then roll about rotated axes.
The body origin is the mesh origin translated vertically by the body $Z$ position so that the body origin is always on the free surface of the water.

If the $Z$ position, heel angle and trim angle are all zero, then $\Bxyz$ coincides with $\Mxyz$.

By convention, we distinguish the global axes from the local axes by using upper case for global and lower case for local: the global directions are referred to as $X,Y,Z$ or $G\urm{X},G\urm{Y},G\urm{Z}$, and the local body directions as $x,y,z$ or $B\urm{x},B\urm{y},B\urm{z}$. Whenever data or results are coordinate-system dependent, they are referred to as being either relative to global (and are labelled with upper-case $X,Y,Z$ or $G\urm{X},G\urm{Y},G\urm{Z}$) or body-relative (and are labelled with lower-case $x,y,z$ or $B\urm{x},B\urm{y},B\urm{z}$).

We extend this notation to refer conveniently to planes defined in the various frames by pairs of axes. Thus, the $G\urm{XY}$ plane is that defined by the $G\urm{X}$ and $G\urm{Y}$ axes (and is, in fact, the horizontal plane); the $G\urm{XZ}$ plane is the vertical plane through the $G\urm{X}$ axis, and so on.

Some data and results are given relative to the global axes and some relative to local axes, as documented on each item's help page. Examples of data and results in global axes include

Examples of body-relative data and results include