Linear Algebra

These classes are meant to provide a usable representation of a vectorspace. At the moment, the capabilities are limited to vectors and their transformations in coordinate frames with orthonormal basis vectors, but allows for arbitrary interrelations of them.

The Data Model

Every vector and frame class is a subclass of numpy.ndarray. The way of inheritance is the mixture of both approaches suggested in NumPy’s docs. A data class is inherited by directly subclassing numpy.ndarray, and is responsible to store coordinates in the actual coordinate frame of the vector. This is the representation of a vector in a frame. The class for vectors is implemented the other way arounds, and its instances actually wrap an instance of the first class. The purpose of the second class is to handle frames and transformations on the stored array. This creates a frontend-backend architecture, where a frontend can be served by several kinds of backends. An example for this is the TopologyArray class, which operates on either a numpy.ndarray or an awkward.Array (depending on the arguments at object creation), yet the frontend provides a unified interface.

The Direction Cosine Matrix

The notion of the Direction Cosine Matrix (DCM) is meant to unify the direction of relative transformation between two frames.

Note

Click here to read the extended version of this brief intro as a pdf document.

If a vector \(\mathbf{v}\) is given in frames \(\mathbf{A}\) and \(\mathbf{B}\) as

\begin{equation} \mathbf{v} = \alpha_1 \mathbf{a}_{1} + \alpha_2 \mathbf{a}_{2} = \beta_1 \mathbf{b}_{1} + \beta_2 \mathbf{b}_{2}, \end{equation}

then the matrix \(^{A}\mathbf{R}^{B}\) is called the DCM from A to B. It transforms the components as

\begin{equation} \left[ \boldsymbol{\beta} \right] = \left[ ^{A}\mathbf{R}^{B} \right] \left[ \boldsymbol{\alpha}\right], \quad \left[ \boldsymbol{\alpha} \right] = \left[ ^{A}\mathbf{R}^{B} \right]^{-1} \left[ \boldsymbol{\beta}\right] = \left[ ^{B}\mathbf{R}^{A} \right] \left[ \boldsymbol{\beta}\right] \end{equation}

and the base vectors as

\begin{equation} \left[ \mathbf{b}_i \right] = \left[ ^{B}\mathbf{R}^{A} \right] \left[ \mathbf{a}_i \right], \quad (i=1,2,3,...). \end{equation}

Vectors

Frames

Arrays