If we take the tangent plane (say, $\mathsf{T}$) to a convex surface $\mathcal{S}$ associated with a given surface normal vector $\mathtt{n}$, I assume the surface will have a fixed rate of change along a disk defined on $\mathsf{T}$, centered at the base of $\mathtt{n}$ (point $\mathtt{p}$, say, considered as a point in $\mathsf{T}$). For any $\mathsf{T}$-direction emanating from $\mathtt{p}$ $-$ perhaps designated as an angle $\alpha$ from an arbitrarily selected $\alpha = 0$ direction $-$ I assume the rate at which $\mathcal{S}$ diverges from $\mathsf{T}$ defines a continuous function (say, $\varsigma$) on $\alpha$ dependent on $\mathcal{S}$ and $\mathtt{n}$. The nature of $\varsigma$ seems obvious for a few cases $-$ everywhere constant for a sphere; near-zero for an almost flat surface $-$ and I'd be interested in calculating (what I'm calling) $\varsigma$ for other surfaces (maybe there's a recognized name for this function). For mesh geometry, I assume $\varsigma$ can be calculated at any mesh point by examining the other mesh points connected to it, partitioning the disk around $\mathtt{p}$ into sectors. But what I'm mostly interested in is the opposite direction $-$ is there a way to construct a triangulation/convex hull, or a NURBS surface, by asserting several normal vectors, and then for each normal present the desired $\varsigma$ function either sampled at particular $\alpha$s or as a closed function? Is this a recognized method for defining surfaces in Computer-Aided-Design?

Information I've found on this topic seems to address $\mathcal{S}$ curvature as a field on curves in the neighborhood of $\mathtt{p}$ (e.g. Geodesic Torsion) but I have not seen a statement of $\varsigma$ as a one-place function on $\alpha$ (for each $\mathcal{S}$ and $\mathtt{n}$). It seems as if this $\varsigma$ might be a limit of the Geodesic Torsion on the space of all closed curves that wind once around $\mathtt{p}$ in $\mathtt{p}$-disks as their radius shrinks to zero $-$ or maybe equivalently just the torsion on circles of radius $\rho$ as $\rho \rightarrow 0$ $-$ but maybe I'm picturing this wrong.


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