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.

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

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