# Partial derivatives of surface curvature relative to a tangent plane

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.