Conformal geometry vs. projective geometry

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In summary, the conversation discusses the relationship between conformal geometry and projective geometry, specifically how the stereographic projection is related to conformal geometry. The person also asks for book recommendations that explain the relationships among different geometries, with suggestions including books by Schouten and Bill Burke.
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Lapidus
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How are those two geometries realeted?

Conformal geometry is a metric geometry. Projective geometry is not. But the stereographic projection is related to the conformal geometry.

Or does someone know a book/ notes where the individual geometries (affine, projective, euclidean, hyperbolic, conformal) are listed and the relationships among them is (neatly and well) explained?
 
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Related to Conformal geometry vs. projective geometry

1. What is the main difference between conformal geometry and projective geometry?

Conformal geometry studies the properties of shapes and figures that are preserved under conformal transformations, which involve scaling and rotation. Projective geometry, on the other hand, studies properties that are preserved under projective transformations, which involve perspective and collinearity.

2. How are conformal and projective geometries related?

Conformal geometry is actually a subset of projective geometry, as all conformal transformations are also projective transformations. This means that all conformal geometric properties are also projective geometric properties, but not vice versa.

3. What practical applications do conformal and projective geometries have?

Conformal geometry is often used in cartography and map-making, as conformal transformations preserve angles and therefore allow for accurate representation of shapes on a flat surface. Projective geometry has applications in computer graphics and computer vision, as it can be used to represent and manipulate 3D objects and scenes.

4. Are there any famous theorems or results in conformal or projective geometry?

Yes, some well-known theorems in conformal geometry include the Riemann mapping theorem, which states that any simply connected open subset of the complex plane can be conformally mapped to the unit disk, and the uniformization theorem, which states that any simply connected Riemann surface can be conformally mapped to the plane, the sphere, or the unit disk. In projective geometry, the fundamental theorem of projective geometry states that any two projective planes of the same order are isomorphic.

5. Can conformal and projective geometries be applied to non-Euclidean spaces?

Yes, both conformal and projective geometry can be extended to non-Euclidean spaces, such as hyperbolic and elliptic spaces. In fact, the Riemann mapping theorem mentioned earlier applies to non-Euclidean spaces as well, and projective geometry is particularly well-suited for studying these types of spaces.

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