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A directional tipping margin is the distance an interior projection of the center of mass can move along one specified direction before reaching a boundary of its support polygon. It differs from the nearest-edge static tipping margin, which protects against movement in every horizontal direction. For a bounded polygon $P=\{x:a_i\cdot x\le b_i\}$, interior point $c$, and unit direction $u$, the ray $c+su$ stays inside exactly while all edge inequalities hold. HenceEdges with nonpositive denominator do not limit positive motion along that ray. The underlying gravity-only feasibility criterion is the nonnegative-reaction convex-hull condition in support polygon.^[ref]

For an original triangular footprint, take $x\ge 0$, $y\ge 0$, $x+y\le 1$ in metres and $c=(0.20,0.30)\mathrm{m}$. Moving right gives $s_{\max }=0.50\mathrm{m}$; moving downward gives $0.30\mathrm{m}$; moving along $u=(1,1)/\sqrt{2}$ gives $0.50/\sqrt{2}\approx 0.3536\mathrm{m}$. The all-direction margin is only $0.20\mathrm{m}$, set by the left edge. A displacement allowance restricted to the right can therefore be larger than the isotropic allowance without contradiction.

The same geometry can enter a planar load calculation. A horizontal force $Fu$ applied at height $h$ shifts the effective support-resultant point by $(Fh/mg)u$ under the corresponding static torque balance. The directional threshold is $F_{\mathrm{tip}}=mg{s}_{\max }/h$, subject also to friction and the other contact assumptions. With $m=10\mathrm{kg}$ and $h=0.50\mathrm{m}$, the rightward example gives $98.1\mathrm{N}$, whereas the nearest-edge worst-direction value is $39.24\mathrm{N}$.

A measured pull direction and contact layout are therefore part of the specification. Changes in contact set, applied couples, or slipping require the fuller analysis in tipping of a rigid body; this geometric directional reserve alone is not a dynamical stability certificate.

A static tipping margin is the distance of an 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 $P=\{x:a_i\cdot x\le b_i\}$, with nonzero edge normals $a_i$, the margin isThis 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.^[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.