Define for a smooth compact hypersurface $M^n$ of ${\mathbf{R}}^{n+1}$ its crumpleness $\kappa \left(M^n\right)$ as the ratio ${diam}_{{\mathbf{R}}^{n+1}}(M^n)/r(M^n)$, where $r\left(M^n\right)$ is the distance from $M^n$ to its central set. (In other words, $r\left(M^n\right)$ is the maximal radius of an open non-selfintersecting tube around $M^n$ in ${\mathbf{R}}^{n+1}.)$We prove that any $n$-dimensional non-singular compact algebraic hypersurface of degree $d$ is rigidly isotopic to an algebraic hypersurface of degree $d$ and of crumpleness $\le exp\left(c\right(n\left)d^{\alpha \left(n\right)d^{n+1}}\right)$. Here $c\left(n\right)$, $\alpha \left(n\right)$ depend only on $n$, and rigid isotopy means an isotopy passing only through hypersurfaces of degree...

Let $M$ be an oriented cusped hyperbolic 3-manifold and let $\tau$ be a topological ideal triangulation of $M$. We give a characterization for $\tau$ to be isotopic to an ideal geodesic triangulation; moreover we give a characterization for $\tau$ to flatten into a partially flat triangulation. Finally we prove that straightening combinatorially equivalent topological ideal cell decompositions gives the same geodesic decomposition, up to isometry.