2. 6. 3 Geometric Examples

In some cases it is necessary to determine the volume of a cell. For instance, many formulae for distinct integrals lead to a formulation which contains the original volume. In the following case the calculation of the volume is shown for a tetrahedron. A typical formulation of the volume of a general simplex can be defined as follows.

$\displaystyle \frac{ \Big\vert \det \Bigl[ \begin{array}{ccc} x_4-x_1 & y_4-y_1...
...& z_3-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ \end{array} \Bigr] \Big\vert } {3!}$ (2.60)

For the determination of this determinant value it is necessary to obtain one definite vertex from the set of incident vertices $ CV$ . This can be easily provided by a first vertex function $ FCV$ which returns only one vertex of the cell. Furthermore, it is necessary to remove the respective vertex from the set of vertices $ CV$ . Using the $ \bigsqcup$ operator, a matrix can be provided. The vector value quantity $ \mathbf{x}$ contains the coordinate of the vertices.

$\displaystyle \mathrm{V}(\mathbf{c}) = \bigwedge^{\underline{v} := FCV(\bullet)...
...ine{v}} [ \mathbf{x}_{\underline{v}} - \mathbf{x} ]] \mid } {[\bigoplus_{CV}]!}$ (2.61)

The expression $ \mathbf{x}_{\underline{v}}$ evaluates the coordinate of the first of the incident vertices passed by the $ FCV$ function. The accumulation $ \bigsqcup_{CV \setminus \underline{v}}$ forms a vector in which the resulting vectors of the subtractions $ x_n - x_1$ are inserted.

Michael 2008-01-16