In projective geometry, the sphere can be thought of as the complex projective ... 2 As a complex manifold; 3 As the complex projective line; 4 As a sphere; 5 Metric ..... by Douglas N. Arnold and Jonathan Rogness (a video by two University of ...

Projective 3-sphere

In mechanics and geometry, the 3D rotation group is the group of all rotations about the origin (mathematics) of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (such as it is an isometry), and orientation (i.e. handedness of space). A distance-preserving transformation which reverses orientation is an improper rotation, that is a reflection or, in the general position, a rotoreflection. The origin in Euclidean space establishes certain one-to-one correspondence between points and their coordinate vectors. Rotations about the origin can be thought of as magnitude-preserving linear transformations of Euclidean 3-dimensional vectors (whose vector space is also denoted as R3).

Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties (along with the associative property, which rotations obey), the set of all rotations is a group under composition. Moreover, the rotation group has a natural manifold structure for which the group operations are smooth; so it is in fact a Lie group. The rotation group is often denoted SO(3) (or, less ambiguously, SO(3, R)) for reasons explained below.

This is an excerpt from the article Projective 3-sphere from the Wikipedia free encyclopedia. A list of authors is available at Wikipedia.

Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties (along with the associative property, which rotations obey), the set of all rotations is a group under composition. Moreover, the rotation group has a natural manifold structure for which the group operations are smooth; so it is in fact a Lie group. The rotation group is often denoted SO(3) (or, less ambiguously, SO(3, R)) for reasons explained below.

This is an excerpt from the article Projective 3-sphere from the Wikipedia free encyclopedia. A list of authors is available at Wikipedia.

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Projective space - Wikipedia, the free encyclopedia

1 Introduction; 2 Definition of projective space; 3 Projective space as a manifold ... Every such line meets the sphere of radius one centered in the origin exactly ...

en.wikipedia.org/wiki/Projective_space

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HARMONIC TORI IN QUATERNIONIC PROJECTIVE 3-SPACES ...

in spheres and complex projective spaces. In this paper, we shall classify con- formal non-superminimal harmonic two-tori in a 2- or 3-dimensional quater- ...

www.med.nihon-u.ac.jp/home/sudagawa/ProcAMS.pdf

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HARMONIC TORI IN QUATERNIONIC PROJECTIVE 3-SPACES ...

in spheres and complex projective spaces. In this paper, we shall classify con- formal non-superminimal harmonic two-tori in a 2- or 3-dimensional quater- ...

www.ams.org/proc/1997-125-01/S0002-9939-97-03638-1/S0002-9939-97-03638-1.pdf

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Spheres and Projective Spaces - University of Waterloo

Jul 7, 2010 ... Complex Projective Spaces. Quaternionic Projective Spaces. Octonionic Projective Spaces. The Hopf Fibrations. 3 Spheres and Almost ...

www.math.uwaterloo.ca/~karigian/talks/CUMC-2010.pdf

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Möbius and projective 3-manifolds - MathOverflow

May 5, 2010 ... A M\"obius 3-manifold is defined similarly, with the projective space replaced with the standard 3-sphere and projective transformations ...

mathoverflow.net/questions/23567/mobius-and-projective-3-manifolds

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PICTURES OF THE PROJECTIVE PLANE

The Montana Mathematics Enthusiast, ISSN 1551-3440, Monograph 3, pp.3-16 ... antipodal points of the sphere, the model gives us a picture of the projective ...

www.math.umt.edu/tmme/Monograph3/Artmann_Monograph3_pp.3_16.pdf

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Projective 3-sphere | Facebook

Projective 3-sphere. 2 likes · 1 talking about this. ... Projective 3-sphere. Projective 3-space was merged with this page · Interest ...

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Are there 3D tilings of a 3D projective hyperplane or 3-sphere?

I noticed that pentagons tile the projective plane (a spherical dodecahedron). Something they do not do on a flat euclidean plane. Is there analogous 3D tilings ...

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Projective modules over the real algebraic sphere - EPFL

Using this, we show that all the finitely generated projective modules over the algebraic real 3-sphere are free. Contents. 1. Introduction. 1. 1.1. Conventions. 3 ...

alg-geo.epfl.ch/~jfasel/Cancel_rev.pdf

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Trends in Mathematical Physics: Proceedings of the Conference on ... - Google Books Result

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Projective 3-sphere in science

Riemann sphere - Wikipedia, the free encyclopedia

Projective space - Wikipedia, the free encyclopedia

1 Introduction; 2 Definition of projective space; 3 Projective space as a ... Every such line meets the sphere of radius one centered in the origin exactly twice, say in ..... Cambridge Studies in Advanced Mathematics, Cambridge University Press, ...

[PDF]Spheres and Projective Spaces - University of Waterloo

Jul 7, 2010 ... Complex Projective Spaces. Quaternionic Projective Spaces. Octonionic Projective Spaces. The Hopf Fibrations. 3 Spheres and Almost ...

Dr J Bolton - Durham University

Durham University ... 'From surfaces in the 5-sphere to 3-manifolds in complex projective 3-space'. 'Bull. ... Linearly full harmonic 2-spheres in S^4 of area 20\pi.

[PDF]IMMERSIONS OF THE PROJECTIVE PLANE WITH ONE TRIPLE ...

UNIVERSITY OF SOUTH CAROLINA, COLUMBIA. Abstract. We consider C∞ generic immersions of the projective plane into the. 3-sphere. Pinkall has shown ...

[PDF]Invariant differential operators on the sphere - Australian National ...

Third CoE Talk at the University of Tokyo – p. 1/21 ... The projective circle: {(x, y) ∼ λ(x, y) for λ > 0}. E. T .... The contact Projective 3-sphere = Sp(4,R)/P. • The CR ...

W2012 - University of Chicago Geometry/Topology Seminar

Thursdays (and sometimes Tuesdays) 3-4pm, in Eckhart 308 ... Rice University; Polynomial Pick forms for affine spheres and real projective polygons; Abstract: ...

[PDF]Introduction to Topology - The Department of Mathematics at ...

4.2.2 The Projective Plane as a Quotient Space of the Sphere . . . . . . . . . . 65. 4.2.3 The Projective Plane as an identification space of a disc . . . . . . . . . . 66 ...

Understanding Projective Geometry - University of Toronto

Sep 10, 1996 ... Okay, I'm just wondering about the applicability of projective and affine .... Now, if you put a sphere with its centre at the origin in 3-d space, then ...

[PDF]HARMONIC TORI IN QUATERNIONIC PROJECTIVE 3-SPACES ...

formal non-superminimal harmonic two-tori in a 2- or 3-dimensional quater- ... The harmonic two-spheres in n-dimensional quaternionic projective space HPn ...... Department of Mathematics, School of Medicine, Nihon University, Itabashi, ...

Books on the term Projective 3-sphere

Trends in Mathematical Physics: Proceedings of the Conference on ...

The labels on the 3-orbifolds of Figure 4 denote the characteristic invariant lattice complexes (letters) and singular set ... 3-disk, 3-sphere, real projective 3- plane, real projective 2-plane plus a 3-disk half boundary, and double real projective ...

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Seifert and Threlfall, A textbook of topology

In fact, one cannot even completely classify the topologically distinct 3- dimensional spaces. The higher ... These are projective 3-space and the 3-sphere. Projeclive 3-space arises from the requirement that one completes Euclidean 3- ...

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Handbook of Geometric Topology

Having analyzed essential 2-spheres, we now turn our attention to essential projective planes. By the previous step, we can restrict the analysis to 3- manifolds that contain no essential 2-spheres. It turns out that we then have a much stronger ...

Algorithmic Foundations of Robotics VIII: Selected Contributions ...

that the 3-sphere is a double covering of RP3, since one point of the projective space has two corresponding points on the 3-sphere. • Haar Measure on SO(3). Up to a scalar multiple, there exists a unique measure on SO(3) that is invariant ...

How surfaces intersect in space: an introduction to topology

The projective plane is substantial in projective space because the core disk of the 2-handle is pierced by a transverse arc ... Identifying Points on the 3-sphere Consider the possibility of identifying diametrically opposite points on the 3-sphere.

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Projective 3-sphere

Projective 3-sphere

Visualizing High Dimensional Data: Azhimutal projection for RPM on projective spaces

Among those manifolds used by relational perspective map (RPM), the 2-dimensional real projective plane (P2) has been one of my favorite space. As a visualization space for high dimensional data, P2 works similarly as the 2-dimensional flat torus (T2) used initially by RPM.

jamesxli.blogspot.com/2013/04/azhimutal-projection-for-rpm-on.html
Real & Complex Projective Space (Part 3) | isaacsolomonmath

General mathematical distractions for people with too much free time

isaacsolomonmath.wordpress.com/2011/12/29/real-complex-projective-part-3/
The Universe | Complex Projective 4-Space

Where exciting things happen

cp4space.wordpress.com/2013/05/26/the-universe/
Interesection Numbers in RP^3

Interesection Numbers in RP^3 in Differential Geometry is being discussed at Physics Forums

www.physicsforums.com/showthread.php?t=473333
Rolling Circles and Balls (Part 5) | Azimuth

Last time I promised to show you how the problem of a little ball rolling on a big stationary ball can be described using an 8-dimensional number system called the split octonions… if the big ball has a radius that’s 3 times the radius of the little one!
So, let’s get started.
First, I must admit that I lied.

johncarlosbaez.wordpress.com/2013/01/02/rolling-circles-and-balls-part-5/
S0(3)/SO(2) is isomorphic to the projective plane

S0(3)/SO(2) is isomorphic to the projective plane in Linear & Abstract Algebra is being discussed at Physics Forums

www.physicsforums.com/showthread.php?t=161908
Altitudes of a triangle and the Jacobi identity | Mathematics under the Microscope

Atomic objects, structures and concepts of mathematics

micromath.wordpress.com/2008/04/07/altitudes-of-a-triangle-and-the-jacobi-identity/
On the Topology of Isolated Singularities in Analytic Spaces ebook @ evenmoreo2的部落格 :: 痞客邦 PIXNET ::

Download On the Topology of Isolated Singularities in Analytic Spaces Author: Jos? Seade Type: eBoo

evenmoreo2.pixnet.net/blog/post/55479712-on-the-topology-of-isolated-singularities-in-analytic-spaces
Syntropy: CJ Fearnley’s Blog » Models of Projective Geometry
blog.cjfearnley.com/2012/07/24/models-of-projective-geometry-2/

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More cyclopaedia entries

11. Projective cover

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32. Projective Plane

33. Projective Space