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What is the difference between an isometric deformation and a conformal deformation? In fact, even the definition of both deformations is still unclear clear for me.

Is it possible to define an isometric deformation from $\mathrm{R}^2$ to $\mathrm{R}^3$ ?

So I would like to ask if anyone has a reference in mind that can help me understand each concept. Any help will be appreciated, but please note that I am not a geometry student, but a mathematics student working in applied mathematics. Honestly, I am working in computer vision, and precisely in the 3D reconstruction domain. There is an idea saying that one can reconstruct a 3D object after being isometricaly defomed, just using its initial 3D form and a 2D image of the deformed object. So, in order to comprehensively understand the problem it is important for me to understand what "deformation" (isometric) refers to.

I already searched for some references, but isometry is usually considered from an algebraic point of view, as isometric transfomation.

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    $\begingroup$ Please provide us with one reference, even from those you mentioned that are algebraic. Without a proper definition of what you call an isometric deformation, we might not be able to help you. The only notions that I can think of are not compatible with your question about deforming a two dimensional region into a three dimensional region. $\endgroup$
    – CvZ
    Sep 17, 2015 at 18:54
  • $\begingroup$ It is unclear what you are asking. Can you provide a reference to where you see those terminologies used, specifically concerning going from $\mathbb{R}^2$ to $\mathbb{R}^3$? $\endgroup$
    – Willie Wong
    Sep 18, 2015 at 3:09
  • $\begingroup$ Sorry for being unclear, I dited my quesion, would you please have a look agian @CvZ $\endgroup$
    – Nizar
    Sep 18, 2015 at 8:26
  • $\begingroup$ We still need at least one reference, it can be from the computer vision literature. It is going to be hard to help you without it, unless someone working with computer vision is actually able to clarify your question (but that person needs to see your question here...). $\endgroup$
    – CvZ
    Sep 18, 2015 at 8:51
  • $\begingroup$ It's possible that you are asking about what's known as "isometric embeddings" and "conformal embeddings". $\endgroup$
    – user147263
    Sep 18, 2015 at 20:59

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In the context of 3D shapes:

An isometric deformation is a deformation of the shape the preserves geodesic distances on the shape. https://en.wikipedia.org/wiki/Isometry

A conformal deformation is a deformation of the shape that preserves angles on the surface of the shape. https://en.wikipedia.org/wiki/Conformal_map

These definitions can be used on smooth manifolds or discrete manifold approximations like triangle meshes.

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