Partial derivatives of surface curvature relative to a tangent plane
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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.
differential-geometry partial-derivative 3d surfaces computer-vision
asked Dec 25 '18 at 15:15
Nathaniel ChristenNathaniel Christen
11
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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.
differential-geometry partial-derivative 3d surfaces computer-vision
asked Dec 25 '18 at 15:15
Nathaniel ChristenNathaniel Christen
11
$endgroup$
add a comment |
$begingroup$
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.
differential-geometry partial-derivative 3d surfaces computer-vision
asked Dec 25 '18 at 15:15
Nathaniel ChristenNathaniel Christen
11
$endgroup$
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.
differential-geometry partial-derivative 3d surfaces computer-vision
differential-geometry partial-derivative 3d surfaces computer-vision
asked Dec 25 '18 at 15:15
Nathaniel ChristenNathaniel Christen
11
asked Dec 25 '18 at 15:15
Nathaniel ChristenNathaniel Christen
11
asked Dec 25 '18 at 15:15
Nathaniel ChristenNathaniel Christen
11
asked Dec 25 '18 at 15:15
Nathaniel ChristenNathaniel Christen
11
asked Dec 25 '18 at 15:15
Nathaniel ChristenNathaniel Christen
11
11
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