## 2. Coordinate transformations

B2000++ requires two types of coordinate transformations, i.e. translations and rotations (change of bases). In what follows, we will refer to two coordinate systems, namely the master system m and the slave system s.

The transformation will go from s to m. A translation is quite evident: Consider a translation vector ri. A vector xi is then translated from s to m by:

yi = xi + ri

The change of base needs some further explanation. Consider the covariant coordinate systems m and s. The transformation from s to m is defined by the scalar products of the (normalized) base vectors:

tij = mi · sj

A vector xi is rotated from s to m by the rotation matrix tij:

yi = tij · xj

Note that the transformation is orthogonal. As an example, consider the transformation from the branch coordinate system to the global coordinate system:

rg = r0 + tij · rj

rg designating the global (artesian) coordinate vector and r0 the translation vector from the global coordinate system to the branch coordinate system. tij is the change of base according to scalar product

tij = ei · bj

where ei are the base vectors of the global coordinate system and bj the base vectors of the branch coordinate system relative to the global coordinate system.

### 2.1. Local coordinate system transformations

All node related properties, i.e natural boundary conditions, DOF's, or essential boundary conditions are formulated with respect to the node local coordinate system whose orientation is determined by a local base relative to the branch coordinate system. Any node with a node local coordinate system will have the transformation attribute specified by the transformation option of the coor MDL input specification and contained in the third column of the NODA dataset. By default, i.e. if not specified otherwise, the node local coordinate system is identical to the branch coordinate system. Any node i which has a local coordinate system will have an index k > 0 in the NODA table. The index k then points to the k-th row of the NLCS array.