Stereographic Projection for general surfaces

In summary, the conversation discusses the use of stereographic projection in computer vision for generating a reflectance map of an illuminated surface. The method is used for a sphere but can be generalized to other surfaces, although the details of this generalization are not explicitly stated. The book mentioned, "Computer Vision" by Michael Brady and H. G. Barrow, uses the projection to parameterize the sphere and induce local coordinate systems for other surfaces.
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Deltinu
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Stereographic Projection for "general" surfaces

First off, sorry if this is in the wrong forum. I came across this while studying computer vision, but it's of a somewhat mathematical nature. Please move it if it's in the wrong place.

In the book I'm reading*, stereographic projection is used to generate a reflectance map of an illuminated surface. A similar approach can be found here: http://books.google.com/books?id=XEOsAAAAIAAJ&pg=PA117&source=gbs_toc_r&cad=0_0#PPA154,M1 (that link should get you to page 154 of a book called "Computer Vision" by Michael Brady and H. G. Barrow)

My problem is this: the stereograpic projection is put forward for a sphere, then (apparently) generalised to more general surfaces. However this generalisation isn't made explicit. What is it? Am I missing something really simple? EDIT: From further reading, I've come across the Gaussian sphere, which seems to play a role here...but the details are still a little hazy to me.

Many thanks in advance!

*"Artificial Intelligence" 2 ed. by Patrick Henry Wilson, picked up at a local charity shop for £2.99 :-)
 
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Um, it looks like the book you linked to was just using the projection to parameterize the sphere, and thus the set of surface orientations. Of course, this parameterization will induce local coordinate systems for most places on suitably well-behaved surfaces (is this the ultimate goal? Google wouldn't let me read very far). What did the other book say that makes you think that you need to generalize directly.
 

FAQ: Stereographic Projection for general surfaces

What is stereographic projection for general surfaces?

Stereographic projection is a mathematical technique used to map a three-dimensional surface onto a two-dimensional plane. It involves projecting points on the surface onto a plane, resulting in a distorted representation of the surface.

What is the purpose of stereographic projection?

The purpose of stereographic projection is to provide a visual representation of a complex three-dimensional surface in a two-dimensional format. This allows for easier analysis and understanding of the surface's properties and characteristics.

How does stereographic projection work?

Stereographic projection involves projecting points on the surface onto a plane, using a specific mathematical formula. This projection can be done from any point on the surface, resulting in a unique representation of the surface on the plane.

What types of surfaces can be represented using stereographic projection?

Stereographic projection can be used for general surfaces, meaning any three-dimensional surface can be represented. This includes surfaces with irregular shapes, curved surfaces, and even surfaces with holes or openings.

What are some applications of stereographic projection in science?

Stereographic projection has various applications in science, including mapping geological structures, representing crystallographic patterns in materials, and visualizing complex mathematical surfaces. It is also used in astronomy to map celestial bodies and in fluid mechanics to model fluid flow over surfaces.

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