# Triangle mesh surface area after affine transformation

Let $\mathbb{M}$ be the set of all 3D triangle meshes.

Let $a:\mathbb{M} \rightarrow \mathbb{R}$ be a function that computes surface area of the mesh.

Let $\mathbb{T}$ be the set of 3D affine transformation matrices.

Let $t:\mathbb{M} \times \mathbb{T} \rightarrow \mathbb{M}$ be a function that transforms mesh $M$ with matrix $T$.

Let $\mathbb{C}$ be a set of values of unknown (at the moment) structure, with the following properties:

• There is a function $c:\mathbb{M} \rightarrow \mathbb{C}$ that maps meshes to these values.
• There is a function $a:\mathbb{C} \rightarrow \mathbb{R}$, such that $(\forall M \in \mathbb{M})(a(c(M)) = a(M))$, i.e. computes surface area of the mesh indirectly, via first mapping it to $\mathbb{C}$.
• There is a function $t:\mathbb{C} \times \mathbb{T} \rightarrow \mathbb{C}$ such that $(\forall M \in \mathbb{M})(\forall T \in \mathbb{T})(t(c(M), T) = c(t(M, T)))$, i.e. $C$s can be transformed with the same effect on computed surface area as if transformation was applied to the mesh itself.

So, is there such $\mathbb{C}$? Obviously I don't want $\mathbb{C} = \mathbb{M}$, and prefer $a(C)$ and $t(C)$ to be of constant asymptotic complexity with respect to amount of triangles in the mesh.