Transformation of one shape into another

[IwillBeGood]

I have been out of class for really long, I don't remember anything,
I have lately become interested in transformation of one shape into another. Is this also about topology ?

If so, I would like to know how you can define such a beautiful transformation ? It is just so strange to me, true!, how a star can turn into a circle with some sort of computation.

-Forgive and Forget
Thank you

 

[Phrak]

Well, yeah. That's topology. Informally known as rubber sheet topology. A donut and a ceramic coffee cup are topologically objects of equivalence. A sphere and a cube with rounded edges and corners are equivalent objects, etc...

 

[IwillBeGood]

Thankyou That's interesting, But if I want to turn a sphere into make a cube, how can I be able to do it ?

 

[Phrak]

Thankyou That's interesting, But if I want to turn a sphere into make a cube, how can I be able to do it ?

I don't know the mathematical machinery. It's not my fault--I only care for the physics! But that's a topological no-no. Sharp corners are out, just as the hole in a donut distinguishes it from a sphere, the sharp points are distinguishing features that distinguish one shape from another. If it has places where differentiation gives you infinite values it's not a manifold--I think.

We'll both have to wait for the mathematical geniuses to show up, to say why.

 

[maze]

You can find a conformal map from any open polygon to the open unit disk using the Schwarz Christoffel mapping,
http://www.math.udel.edu/~driscoll/research/conformal.html

The key thing being that they are open (boundary not included), so that the corners are not a problem.