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.
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.
Let the object be in contact at a finite number of points
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.
To balance the object in static equilibrium, the following Newton-Euler equations must be met on
The second equation has no dependence on the vertical component of the center of mass, and thus if a solution exists for one
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.
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).
It is also invariant to the mass of the object (provided it is nonzero).
, then the support polygon is the convex hull of the contact points projected onto the horizontal plane.