Introduction
Creating a reproducible position for a component on three contact points is a common engineering case. Choosing a preload to force the component in place must be done carefully; when the line of action of this force is not chosen correctly, friction in two contact points prevents contact with the third contact point. This results in the position of the component not being reproducible, possibly impairing the functionality of the design.
Idealized model: no friction
In the figure below, a body is supported on three points, labelled 1, 2 and 3 respectively. A normal is drawn through each contact point. The three intersections P12, P13 and P23 represent rotation poles. An infinitesimal rotation of the component around a pole breaks the contact in one point. The direction of the rotation around each pole that assures contact in the opposing contact point is indicated in each pole.

Based on the pole rotations, a force can be found that preloads the body against all three contact points. The line of action of this force satisfies the correct direction of rotation around each pole. Some lines of action are not suitable. It can be found that such lines pass through certain areas. These are marked in red. In case all pole rotations have the same direction, torque should be applied to preload the body.
Detailed model: friction included
Additionally to the idealized model above, friction has to be taken into consideration, as the friction in two points can prevent contact in the third during assembly. The definition of the ‘friction angle’ is important for this: The friction angle is defined as the maximum angle ($\alpha$) between the line of action of an applied force and the normal, beyond which a body begins to slip. This angle is determined by the coefficient of friction ($c_f$) between the body and the surface:
$$\alpha=\arctan{c_f}$$
The friction angle needs to be included to make sure that slip is possible at the contact points and contact at all three points is guaranteed. Forces with lines of action that satisfy this can be found in the following way: For each contact point, the friction angles are drawn in the figure below. The areas where the friction angles overlap are marked in yellow. The line of action of the preload may not pass through such an area; avoiding the marked areas ensures slip in each contact point.

The areas marked in red from the idealized model are updated in the figure below. The areas marked in yellow (overlapping friction angles) are now included in the area marked in red. Using these marked areas as a guide, a line of action for the preload may now be chosen. A possible solution is drawn in the figure below.
Sensitivity analysis
It is important to keep in mind that friction between two components is hard to accurately predict. By increasing the friction angle, the overlapping areas become larger. It is wise to find out for which value of the coefficient of friction no suitable line of action can be chosen anymore. This gives an impression of the sensitivity of the design to deviations in the coefficient of friction. In the figure below the friction angle has been increased. The previously chosen line of action is no longer a viable option.