= Directional tipping margin
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. Hence
$$
s_{\max}(c,u)=\min_{i:a_i\cdot u>0}\frac{b_i-a_i\cdot c}{a_i\cdot u}.
$$
Edges 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>.<Robotic Systems by Kris Hauser>[Chapter 12, Equilibrium conditions]{ref}
For an original triangular footprint, take $x\ge0$, $y\ge0$, $x+y\le1$ 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 $u=(1,1)/\sqrt2$ gives $0.50/\sqrt2\approx0.3536\,\mathrm m$. 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> $F u$ applied at height $h$ shifts the effective support-resultant point by $(Fh/mg)u$ under the corresponding static <torque> balance. The directional threshold is $F_{\rm tip}=mg s_{\max}/h$, 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.
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