Need a Functional Analysis Pro - Haaalp!!

[bloodninja]

Given the coordinates (f(u,v),g(u,v),h(u,v)) (note the conspicious presence of 3 functions of 2 'parameters') consider the level sets



g = const.



h = const.



in the presence of the inverse function theorem (which kr alludes to); what do we have then? There is a cyclic property here and we can choose f/g const or f/h const etc etc; which goes to show that coordinates typically do not account for fundamental things about surfaces [instead, coordinates are used to compute things about surfaces....some of which end up being fundamental to the surface].



The difference between notating the problem in terms of f,g,h and f,g,h(f,g) is sometimes called a 'graph'; but this is verbiage.



What would help is if we were told what the problem is, and what functions of theta and phi you have in mind:



How 'bout



x = r(theta, phi) sin theta sin phi



y = r(theta, phi) cos theta sin phi



z = r(theta, phi) cos phi

[zer0snr]

Thanks for all your thoughts.....must confess that some it escapes me but some has seeped

through....some I'll have to re-read to fully comprehend what you'll suggest....



I'm beginning to see the importance of how the surface coordinates are defined. You'll

mention the fundamental form of the surface,



ds^2 = Edu^2 + Fdudv + Gdv^2



and kr also mentions the Jacobian of the transformation (which is related to the

coefficients E,F,G of the fundamental form). How the surface is parameterized is important,

or loosely the shape of the Jacobian is important. Mapping a uniform grid (u,v) in

such a way so that the Jacobian is "singular" at some point (an artifact of sorts) could

be the culprit in the "performance" of any series expansion or approximation by basis sets

defined in the (u,v)-space. I.e., given w(x,y,z) on a surface, and [x,y,z] =

[f(u,v), g(u,v), h(u,v)], we effectively have w=w(u,v), and thus if the (x,y,z)->(u,v)

mapping introduces singularities (or near singularities), then derivatives of w(u,v) w.r.t.

u & v will be sharp in some regions, and any series expansion of w in terms of some basis

set defined on (u,v) will most likely have slower convergence. To begin with, if w(x,y,z)

were very smooth, but your mapping pathological, one's unnecessarily introduced problems.



I think I'm beginning to see why my original question was confusing....



chiral3, will look into RBF. bloddninja, was looking into filtering and came across a

paper on spherical harmonic filters (some geophysical application), which made me wonder

more generally for arbitrary surfaces. Couldn't quite figure it out and was getting worked

up, and tossed the question. kr, thanks for mentioning the Jacobian, made me realize the

connx. between the Jacobian and the surface fundamental forms.



I think I need to repose my question .... will get back in a couple of days since I've

got to move out of my place tomorrow.....wouldn't want to waste your time (I probably

already have), so let me think about it and repose it intelligently (hopefully!!) ...

Once again thanks all.

[chiral3]

Mr Bloodninja, MD wrote out my parameterized x's a bit nicer than I did Cool

[bloodninja]

alright dudes, well... when you get back maybe we can talk about L2 error etc

[Nonius]

still want to know what the problem is.  and yes, coordinatizing a patch via a "graph" is verbology, but, it does seperate the distinction between (x,y,f(x,y)) and (f(x,y),g(x,y),h(x,y)) AND it makes clear that you cannot always coordinatize an arbitrary patch with a graph.  I find that people can handle the immediacy of a graph and, with a little background, they can handle spherical coordinates....but, the general case does not render any nice representation.  This is why I think that WHATEVER the question is, it depends on the generality of the posing of the question.