Coordinate systems

OrcaFlex 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.

Firstly, we have one global frame of reference which we denote $\GXYZ$. The reference origin is the global origin and $G\urm{X}$, $G\urm{Y}$ and $G\urm{Z}$ are the global axes directions. There are then a number of local coordinate systems, generally one for each object in the model, denoted by $\Lxyz$ and with axes directions $L\urm{x}$, $L\urm{y}$ and $L\urm{z}$. Another coordinate system that we make widespread use of, $\Exyz$, is specifically for line end orientations.

All the coordinate systems are right-handed, and positive rotations are clockwise when looking in the direction of the axis of rotation. The following figure shows the global axes and a vessel with its own local vessel frame $\Vxyz$

The global frame of reference must be a right-handed system and its $Z$-axis $G\urm{Z}$ must be positive upwards, but otherwise it is chosen by the user. You can therefore choose the position of the global origin $G$ and the horizontal directions $G\urm{X}, G\urm{Y}$ to best suit your model.

The local coordinate systems for each type of object are described in the section about that object, but typically the origin is at a selected fixed point on the object and the axes are in particular fixed directions, such as the surge, sway and heave directions for a vessel. The seabed also has its own seabed origin and local axes, with respect to which the seabed shape is defined.

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 object directions as $x,y,z$ or $L\urm{x},L\urm{y},L\urm{z}$ (we may also use the body-specific form for local axes, such as the vessel frame above which has axes $V\urm{x},V\urm{y},V\urm{z}$). 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 object-relative (and are labelled with lower-case $x,y,z$ or $L\urm{x},L\urm{y},L\urm{z}$).

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. Planes in local frames can be defined similarly as $L\urm{yz}$ etc.

You can ask OrcaFlex to draw the local axes on the 3D view. This enables you to see the local axes and check that they are as wanted.

Data and results are usually given relative and with respect to the global axes, including

• data defining the sea, such as the mean sea surface $Z$ coordinate and the current and wave directions
• positions of objects, for example the position of the vessel is defined by giving the global coordinates of the vessel origin $V$.

The most common object-relative items are

• the coordinates of points which move with the object, such as the vertices of a vessel or the connection point of an object connected to a buoy
• the direction-dependent properties of objects, such as drag and added mass coefficients, moments of inertia, etc.