Re: manifold learning - how do you back caluclate from low D to high D

hi greg..
i can explain u what is charting of a manifold.
Earth is an example of a manifold.
It can be embedded in a 3D vector space, which is intuitively quite
clear.
Now in neighborhood of every point on it, we can think of it as a flat
plane, thereby mapping a 3d space to a 2d space. Such a mapping is
called a chart or patch. This is done in the world atlas. The atlas is
made up on paper, but it refers to the earth, which is a 3d space. I
hope this example clarifies your doubt. Moreover collection of all
such mappings is also called atlas in mathematics. For good
introduction to manifolds(surfaces) i would refer the book:
Elementary Differential Geometry
Andrew Pressley
Springer
-----------

.

Relevant Pages

  • Re: manifold learning - how do you back caluclate from low D to high D
    ... Earth is an example of a manifold. ... thereby mapping a 3d space to a 2d space. ... This is done in the world atlas. ...
    (sci.image.processing)
  • Re: Misinterpretation of the radial parameter in the Schwarzschild solution?
    ... The fact is that by cutting up the original manifold and introducing ... The Schwarzschild chart is smooth over the Schwarzchild surface ... f was supposed to be neither an atlas nor be defined on the entire ... atlas on R, spanning the point where f is singular, cannot be ...
    (sci.physics.relativity)
  • Re: GR1916, available online.
    ... Let's consider a 4-d spacetime manifold as a model of some region of the ... means that it lies within 45 degrees of the time direction of the model ... Let's imagine the earth is in the region of the model, ... Yes, I understand this visualisation. ...
    (sci.physics.relativity)
  • Re: GR1916, available online.
    ... For simplicity, lets omit one of the spatial dimensions, and imagine that the 3-d spacetime manifold is embedded in our 3-d space. ... Let's imagine the earth is in the region of the model, and is scaled down to 1 mm in radius; the moon is much smaller, and orbits the earth about 35 mm away from the earth; ignore all other objects. ...
    (sci.physics.relativity)
  • Re: GR1916, available online.
    ... down to 1 mm in radius; the moon is much smaller, and orbits the earth ... about 35 mm away from the earth; ... The earth is not a line, its TRAJECTORY is a line. ... d, i.e., confined to the cylindrical manifold, making the orbit ...
    (sci.physics.relativity)
Quantcast