For a rigid object in contact with a fixed environment and acted upon by gravity in the vertical direction, its support polygon is a horizontal region over which the center of mass must lie to achieve static stability. For example, for an object resting on a horizontal surface (e.g. a table), the support polygon is the convex hull of its "footprint" on the table.
The support polygon succinctly represents the conditions necessary for an object to be at equilibrium under gravity. That is, if the object's center of mass lies over the support polygon, then there exist a set of forces over the region of contact that exactly counteracts the forces of gravity. Note that this is a "necessary" condition for stability, but "not a sufficient" one.
Derivation.
Let the object be in contact at a finite number of points formula_1. At each point formula_2, let formula_3 be the set of forces that can be applied on the object at that point. Here, formula_3 is known as the "friction cone", and for the Coulomb model of friction, is actually a cone with apex at the origin, extending to infinity in the normal direction of the contact.
Let formula_5 be the (unspecified) forces at the contact points. To balance the object in static equilibrium, the following Newton-Euler equations must be met on formula_5:
where formula_11 is the force of gravity on the object, and formula_12 is its center of mass. The first two equations are the Newton-Euler equations, and the third requires all forces to be valid. If there is no set of forces formula_5 that meet all these conditions, the object will not be in equilibrium.
The second equation has no dependence on the vertical component of the center of mass, and thus if a solution exists for one formula_12, the same solution works for all formula_15. Therefore, the set of all formula_12 that have solutions to the above conditions is a set that extends infinitely in the up and down directions. The support polygon is simply the projection of this set on the horizontal plane.
These results can easily be extended to different friction models and an infinite number of contact points (i.e. a region of contact).
Properties.
Even though the word "polygon" is used to describe this region, in general it can be any convex shape with curved edges. The support polygon is invariant under translations and rotations about the gravity vector (that is, if the contact points and friction cones were translated and rotated about the gravity vector, the support polygon is simply translated and rotated).
If the friction cones are convex cones (as they typically are), the support polygon is always a convex region. It is also invariant to the mass of the object (provided it is nonzero).
If all contacts lie on a (not necessarily horizontal) plane, and the friction cones at all contacts contain the negative gravity vector formula_17, then the support polygon is the convex hull of the contact points projected onto the horizontal plane.
References.
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