Consider a rigid body in contact with the ground plane. Let the ground plane normal be \(n\). Let various contact forces \(F_i\) act on the rigid body at positions \(r_i\). The center of pressure is then defined as \begin{equation} x_c = \frac{\sum_i (F_i \cdot n) r_i}{\sum_i F_i \cdot n} \end{equation} where the generalization from finitely many forces to a force field should be obvious. We generally assume that the normal component of the contact forces, \(F_i \cdot n\), is greater or equal to 0 (i.e. that the contact forces are non-sticky). This implies that, for \begin{equation} \alpha_i := \frac{F_i \cdot n}{\sum_j F_j \cdot n} \end{equation} we have \(0 \le \alpha_i \le 1\). The center of pressure, as defined above, is thus a convex sum and must lie within the convex hull of the contact points \(r_i\).

The value of the center of pressure concept comes from the fact that we can calculate it without having to know the contact points \(r_i\) and the corresponding forces \(F_i\). Let us assume we are given only the total force and torque resulting from the contact, \begin{eqnarray} F &=& \sum_i F_i\\ T &=& \sum_i r_i \times F_i \end{eqnarray} To calculate the center of pressure from these vectors, we note that \begin{eqnarray} n \times T &=& n \times \left( \sum_i r_i \times F_i \right) = \sum_i (n \times (r_i \times F_i))\\ &=& \sum_i \left( r_i (n \cdot F_i) - F_i (n \cdot r_i) \right) \end{eqnarray} As the contact points all lie in the plane, \(n \cdot r_i\) is equal for all \(i\). Let \(n \cdot r_i =: h\). Then \begin{equation} n \times T = \sum_i r_i (n \cdot F_i) - h \sum_i F_i = \sum_i r_i (n \cdot F_i) - h F \end{equation} Thus \begin{equation} x_c = \frac{n \times T + h F}{F \cdot n} \end{equation}