= Affine equivariance of a geometric basis
{title2=$T(\sum_i b_iP_i)=\sum_i b_iT(P_i),\quad\sum_i b_i=1$}
A geometry representation with scalar <partition of unity> basis functions commutes with every <affine map> of its <control points>. For $T(x)=Ax+b$, linearity moves $A$ through the sum and the sum-to-one identity supplies the translation $b$. Positivity is needed for a <convex hull> property, but not for this equivariance. The same argument applies to linear <subdivision matrices> whose rows sum to one. General <projective transformations> require homogeneous <control points> and rational dehomogenization rather than transforming Euclidean controls with unchanged weights.
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