Observing Geometric Properties

[Xittenn]

How would I use Lie Algebras, observing infinitesimal transformations over a smooth manifold(non-complex Lie Group,) to gain insight into the geometric properties of some shape?

 

What thoughts could I be observing when looking into the mathematics that this involves?

 

I had looked at the concept of Lie Groups a number of years ago but abandoned it completely as it made no sense when I had. Just looking at the Wiki, most of it is pretty plain English now but Wiki leaves much to be said about proper application. I am beginning a study on topology and will be returning to group theory after I complete this study. I would like to have my thoughts in my mind focused on Lie Algebras and their practical application to the observation of shapes as I go through the material.

 

Am I off in how I have established my preconceived ideas of what is going to develop through these concepts? I wish to do work in the areas of spacial and shape recognition and analysis, machine vision.

[ajb]

By a shape you mean a "geometric figure" or a "Rigid body" in a Euclidean space?

 

If so, you have the Euclidean group [math]E(n)[/math], which is the group of isometries of the Euclidean metric.

 

Infinitesimally, the isometries are described by Killing vectors. These form a Lie algebra. I guess these are what you are interested in.

 

Exactly how this helps you with your problem I do not know.

[Xittenn]

Yes, I mean more so geometric figure here. I would like to take the gradients created by shadows and use these gradients to define geometric shapes and utilize this data in analysis.

 

Thank you for pointing out the 'Killing Vector,' at first glance I thought you were saying to kill vectors. I guess I would be looking for specific groups under analysis incorporating 'Killing Vectors' and by process of elimination. I'm sure more will be understood by me as I approach nearer my goals.

[DrRocket]

How would I use Lie Algebras, observing infinitesimal transformations over a smooth manifold(non-complex Lie Group,) to gain insight into the geometric properties of some shape?

 

What thoughts could I be observing when looking into the mathematics that this involves?

 

I had looked at the concept of Lie Groups a number of years ago but abandoned it completely as it made no sense when I had. Just looking at the Wiki, most of it is pretty plain English now but Wiki leaves much to be said about proper application. I am beginning a study on topology and will be returning to group theory after I complete this study. I would like to have my thoughts in my mind focused on Lie Algebras and their practical application to the observation of shapes as I go through the material.

 

Am I off in how I have established my preconceived ideas of what is going to develop through these concepts? I wish to do work in the areas of spacial and shape recognition and analysis, machine vision.

 

I think you need to complete some of your planned studies of topology and group theory, plus some analysis on manifolds before you tackle Lie Algebras. A Lie group is a manifold that is also a group and for which the group operation is smooth. The Lie algebra of a Lie group is the tangent space at the identity with a non-associative multiplication that comes from viewing the tangent space as differential operators. if this seems a bit abstract, I am, not surprised, but it will make more sense after you have learned more mathematics. Don't get ahead of yourself.

 

I am not at all sure that Lie algebras are an appropriate tool for your end application. I suspect not.

[Xittenn]

Actually it's pretty understandable though it is still ahead of me! I don't understand why you both state that it might not be relevant but it is something I have kept in mind that maybe I just don't understand it well enough to see exactly why this is, this was kind of why I asked.

 

I guess I figured if I could assemble a graph of a relation that embedded such supporting mathematics I could walk on sets of geometries in the form of groups, thereby enabling the representation of complex geometric figures as sets of simpler manifold identities. :/

 

I'll get there soon enough!

[DrRocket]

I guess I figured if I could assemble a graph of a relation that embedded such supporting mathematics I could walk on sets of geometries in the form of groups, thereby enabling the representation of complex geometric figures as sets of simpler manifold identities. :/

 

 

huh ?

[Xittenn]

Using analysis I would like to decompose an image and identify individual smooth manifolds. If I can describe a portion of a geometric figure as a group that is a smooth manifold then I can create a set where each elements value is a group that represents a given smooth manifold. I could then use the set and an algorithm or set of algorithms that essentially establishes a graph of a relation on the set. Then another algorithm could be established that walks the graph as a matrix and identifies the object that is being viewed.

 

Something like that yes.

 

I guess, looking over what constitutes an Lie Group, it is sufficient to restrict any future statements to [math] E_n® [/math].

 

Which ajb mentioned!