Hi, All —
I would like to compute the average curvature of a closed 3D surface for which I know the inside-outside function, F(x,y,z)=0. (FWIW, centered on the origin and symmetric about all three Cartesian axes.)
I have implemented a couple of different approaches for Gaussian and mean curvature using divergence, the Hessian, the first and second fundamental forms, and other approaches of which I have a similarly weak grasp. I am generally able to get things to work for degenrate cases (e.g., a sphere), but if I increase the complexity of my object, I am tending to get answers that imply I am flirting with the precipice of numerical instability (e.g., imaginary results where Re(x)/Im(x)~10^8).
I like the idea of this approach because it "feels clean": (1) use continuous function; (2) apply magic; (3) integrate in spherical coordinates; and (4) profit. However, I am beginning to think that I need to tune out the sires sweetly singing and go for a more practical (i.e., tractable) approach.
My other idea is to start by using the inside-outside function to generate [X,Y,Z] similar to what is used by surf(). I can then use surfature() from the FEX to get the curvatures at every point on my [X,Y,Z] mesh. Finally, I could convert everything to spherical coordinates and integrate the curvature arrays to get average values.
My concern here is that reregularly-spaced [X,Y,Z] does not translate to regularly-spaced [r,theta,phi], so the calculation is somehow pathologically biased. I am also unsure about how to do the actual integration after converting my array to spherical coordinates.
Apologies for such a long text-dense question. I thought about trying to illustrate this better with a couple of code snippets, but the algebra is really messy and not all that informative.
Thanks, in advance.Hi, All —
I would like to compute the average curvature of a closed 3D surface for which I know the inside-outside function, F(x,y,z)=0. (FWIW, centered on the origin and symmetric about all three Cartesian axes.)
I have implemented a couple of different approaches for Gaussian and mean curvature using divergence, the Hessian, the first and second fundamental forms, and other approaches of which I have a similarly weak grasp. I am generally able to get things to work for degenrate cases (e.g., a sphere), but if I increase the complexity of my object, I am tending to get answers that imply I am flirting with the precipice of numerical instability (e.g., imaginary results where Re(x)/Im(x)~10^8).
I like the idea of this approach because it "feels clean": (1) use continuous function; (2) apply magic; (3) integrate in spherical coordinates; and (4) profit. However, I am beginning to think that I need to tune out the sires sweetly singing and go for a more practical (i.e., tractable) approach.
My other idea is to start by using the inside-outside function to generate [X,Y,Z] similar to what is used by surf(). I can then use surfature() from the FEX to get the curvatures at every point on my [X,Y,Z] mesh. Finally, I could convert everything to spherical coordinates and integrate the curvature arrays to get average values.
My concern here is that reregularly-spaced [X,Y,Z] does not translate to regularly-spaced [r,theta,phi], so the calculation is somehow pathologically biased. I am also unsure about how to do the actual integration after converting my array to spherical coordinates.
Apologies for such a long text-dense question. I thought about trying to illustrate this better with a couple of code snippets, but the algebra is really messy and not all that informative.
Thanks, in advance. Hi, All —
I would like to compute the average curvature of a closed 3D surface for which I know the inside-outside function, F(x,y,z)=0. (FWIW, centered on the origin and symmetric about all three Cartesian axes.)
I have implemented a couple of different approaches for Gaussian and mean curvature using divergence, the Hessian, the first and second fundamental forms, and other approaches of which I have a similarly weak grasp. I am generally able to get things to work for degenrate cases (e.g., a sphere), but if I increase the complexity of my object, I am tending to get answers that imply I am flirting with the precipice of numerical instability (e.g., imaginary results where Re(x)/Im(x)~10^8).
I like the idea of this approach because it "feels clean": (1) use continuous function; (2) apply magic; (3) integrate in spherical coordinates; and (4) profit. However, I am beginning to think that I need to tune out the sires sweetly singing and go for a more practical (i.e., tractable) approach.
My other idea is to start by using the inside-outside function to generate [X,Y,Z] similar to what is used by surf(). I can then use surfature() from the FEX to get the curvatures at every point on my [X,Y,Z] mesh. Finally, I could convert everything to spherical coordinates and integrate the curvature arrays to get average values.
My concern here is that reregularly-spaced [X,Y,Z] does not translate to regularly-spaced [r,theta,phi], so the calculation is somehow pathologically biased. I am also unsure about how to do the actual integration after converting my array to spherical coordinates.
Apologies for such a long text-dense question. I thought about trying to illustrate this better with a couple of code snippets, but the algebra is really messy and not all that informative.
Thanks, in advance. differential geometry, curvature, spherical averaging MATLAB Answers — New Questions

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