Isotopic piecewise affine approximation of algebraic or $C^1$ varieties

We propose a novel sufficient condition establishing that a piecewise affine variety has the same topology as a variety of the sphere ${\mathbb{S}}^n$ defined by positively homogeneous $C^1$ functions. This covers the case of $C^1$ varieties in the projective space ${\mathbb{P}}^n$. We prove that this condition is sufficient in the case of codimension one and arbitrary dimension. We describe an implementation working for homogeneous polynomials in arbitrary dimension and codimension and give experimental evidences that our condition might still be sufficient in codimension greater than one.