How do you just inverse a shape?

[SamBridge]

So I know most shapes have an inverse, some of which do and don't have defanite edges, I know that you can program a transformation of a shape to its inverse, but I don't get what you do exactly especially for 3-D shapes. If I have a square, the "inverse" of a square seems be some kind of infinitely stretched thing with an open hole in the middle with arc lengths whos endpoints correspond to the vertices of a square. How would I see what it looks like for say...an inverse of a tube?

[imatfaal]

Could you explain more - perhaps with a few links exactly what you mean by inverse shape. Do you mean some form of inverse (that in parallel to algebra) operates through some form of addition on the original to create the identity (and what is the identity (a single point, the origin)

 

[latex]Shape \oplus Shape^{-1} = Identity[/latex]

 

I am using the term identity like you would use it in matrices

 

[latex]\mathbf{A}\mathbf{A}^{-1}=\mathbf{I}[/latex]

[ajb]

I think the OP mean inversive geometry, which is the study of properties of geometric figures under generalisations of the Euclidean group in 2d.

 

I think the situation in higher dimensions is much more complicated.

[Acme]

So I know most shapes have an inverse, some of which do and don't have defanite edges, I know that you can program a transformation of a shape to its inverse, but I don't get what you do exactly especially for 3-D shapes. If I have a square, the "inverse" of a square seems be some kind of infinitely stretched thing with an open hole in the middle with arc lengths whos endpoints correspond to the vertices of a square. How would I see what it looks like for say...an inverse of a tube?

 

Perhaps you mean as in a "dual"? Try this...

 

Duality_(projective_geometry)

 

...

Three dimensions

In a polarity of real projective 3-space, PG(3,R), points correspond to planes, and lines correspond to lines. By restriction the duality of polyhedra in solid geometry is obtained, where points are dual to faces, and sides are dual to sides, so that the icosahedron is dual to the dodecahedron, and the cube is dual to the octahedron. ...

 

 

PS Also check out the Peaucellier–Lipkin linkage. >> Peaucellier-Lipkin_linkage

 

Inversive geometry

Thus, by the properties of inversive geometry, since the figure traced by point D is the inverse of the figure traced by point B, if B traces a circle passing through the center of inversion O, then D is constrained to trace a straight line. But if B traces a straight line not passing through O, then D must trace an arc of a circle passing through O. Q.E.D.

...

 

Edited February 12, 2013 by Acme

[SamBridge]

What I was talking about more was topology. Like you can transform a mobius strip into a klein bottle, stuff like that.