Affine equivariance of a geometric basis ($T({\sum }_i{b}_i{P}_i)={\sum }_i{b}_{i}T(P_i),{\sum }_i{b}_i=1$)

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.