= Static tipping margin
{title2=$d=\operatorname{dist}(c,\partial P)$}
A <static tipping margin> is the distance of an \i[interior] <center of mass> projection $c$ from the boundary of a horizontal <support polygon> $P$. It quantifies the horizontal displacement reserve in the gravity-only rigid-contact model. For a convex polygon written as $P=\{x:a_i\mathbin\cdot x\le b_i\}$, with nonzero edge normals $a_i$, the margin is
$$
d=\min_i\frac{b_i-a_i\mathbin\cdot c}{\|a_i\|},\qquad c\in P.
$$
This is the radius of the largest Euclidean ball about $c$ contained in $P$: each normalized edge inequality bounds how far the projection can move toward that edge. The admissible contact-reaction argument is proved in <support polygon> and in the primary equilibrium model.<Robotic Systems by Kris Hauser>[Chapter 12, Equilibrium conditions, support polygon]{ref}
For an original rectangular footprint, let $P=[-0.30,0.30]\times[-0.20,0.20]$ in metres and $c=(0.10,0.05)\,\mathrm m$. Its four edge distances are $0.20$, $0.40$, $0.15$, and $0.25\,\mathrm m$, so $d=0.15\,\mathrm m$. Any horizontal projection perturbation of norm at most $0.05\,\mathrm m$ leaves a margin of at least $0.10\,\mathrm m$. This follows because each normalized edge gap changes by at most the perturbation norm.
Unsigned distance alone must not classify an exterior point as stable. At $c=(0.40,0.05)\,\mathrm m$, distance to the polygon boundary is still positive, namely $0.10\,\mathrm m$, but gravity-only balance is impossible. The minimum signed edge gap is then negative. For an exterior point, that minimum is a useful feasibility indicator; its magnitude is not generally the Euclidean distance to the polygon, especially beyond a corner.
A <directional tipping margin> can exceed the nearest-edge margin when the allowed displacement direction is restricted. The scalar $d$ protects against every horizontal direction, so it is the appropriate geometric reserve only for an isotropic displacement bound. It omits <center of mass> height, <static friction>, applied moments, and contact compliance. Under a specified horizontal load, <tipping of a rigid body> instead compares the load's overturning <torque> with the restoring gravitational <torque>.
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