Finding all valid surfaces that go through a vector field

[GabrielCoriiu]

Hi,

I'm trying to find all the valid surfaces that go through a vector field so that the normal of the surface at any point is equal with the vector from the vector field at the same point.

The vector field is defined by the function:
$$ \hat N(p) = \hat L(p) \cos \theta + \hat R(p) \sin\theta $$ where $p$ is a 3D point in the vector field.

I've tried breaking $\hat N$ into the $x(p), y(p), z(p)$ components, and integrate them, but it didn't seem to be the right solution.

 

[andrewkirk]

What sort of a vector field is it? Unless it is fairly smooth - probably at least continuously differentiable - there will be no such surfaces, at least not everywhere.
Many vector fields in physics are conservative fields, meaning they can represent the gradient of a scalar field. Potential fields like gravitational and electrostatic, are conservative fields. For conservative fields, the normal surfaces are the surfaces of constant value of the scalar field - isosurfaces, or isolines if the space is 2D.

 

[GabrielCoriiu]

The vector field represents what the normal should be at any point so that a refracted ray of light focuses on a given point

I've managed to integrate the gradient at a given depth within the field, which gives a height map
Integration technique https://math.stackexchange.com/ques...ically-calculate-a-function-from-its-gradient

Look at this beauty ❤

 

[WWGD]

I wonder if you can extend the idea of integral curves to integral surfaces, or maybe Morse theory somehow?

 

[GabrielCoriiu]

I'm sure that integral curves can be applied to integral surfaces given that a surface is just a collection of curves, in the same way a curve is just a collection of points. As for the Morse theory, I'm not familiar with it, but as far as I understand it's mainly used for reconstructing a surface, whereas I needed to find some valid points within a vector field. I guess it could be applied when discretizing the field, in order to generate more accurate, but also messier topology as opposed to sampling on a grid.

 

[WWGD]

It seems if you have an inner product you can define a normal vector field as the kernel < v,w> for w in the tangent space; the normal field would be the ortho complement of the tangent space?

 

[GabrielCoriiu]

Yes, the normal is always orthogonal to the tangent plane, where the gradient lies. However, there are infinitely many tangent spaces at every point (all the possible rotations of a plane around the normal axis). One of them could be defined by $\vec z_t = \vec N, \vec y_t=\vec G, \vec x_t = \vec N \times \vec G$ where $\vec N$ is the normal and $\vec G$ is the gradient

 

[WWGD]

Sorry for my incomplete replies Gabriel, my PC does not work well, dies out at random and I cannot fully read your ( nor any other) post. My latest idea is to see if one can define a vector field as the kernel of a 2-form; contact forms are described , at least locally, as the kernels of 1-forms. I think the definition can be made global under minor conditions.

 

[WWGD]

Maybe @lavinia can chime in here with some bundle advice, a sort of ortho bundle? And see traits a surface must have to allow for such bundles?

 

[wrobel]

The problem looks like https://en.m.wikipedia.org/wiki/Frobenius_theorem_(differential_topology)