Euclidean plane

In order to give a physical interpretation of special relativity it is necessary to understand the implications of the Lorentz rotation. Within Galilean relativity the three-dimensional line element of euclidean space (r2 = r r) is an invariant and the transformation corresponds to a rotation in three-dimensional space. The fact that this line element is not Lorentz invariant shows that world space has more dimensions than three. When rotated in four-dimensional space the physical invariance of the line element is either masked by the appearance of a fourth coordinate in its definition, or else destroyed if the four-space is not euclidean. An illustration of the second possibility is the geographical surface of the earth, which appears to be euclidean at short range, although on a larger scale it is known to curve out of the euclidean plane. 

A vector with n components is called an n-vector. Thus (x, y, z) is a 3-vector, and vectors representing points in the Euclidean plane, (x,y), are 2-vectors. 

Figure 7.2 The norm of the vector, invariant to rotation in the Euclidean plane, seems to contract when rotated into a third unobserved dimension.

A projective plane may be generated by adding to the Euclidean plane a line at infinity. The Euclidean plane itself is equivalent to the gnomonic projection of a sphere on a plane, a shown in figure 12. Each point P in the... 

Figure 7.12 Gnomonic projection of point Pi on P in the Euclidean plane a. The line m maps the stippled great circle.

An example of topological analysis is shown in figure 13. A rectangular rubber sheet is transformed into a torus by the identification of pairs of opposite sides, first to form an open cylinder which transforms into a torus by joining the two open ends. In technical language, the Euclidean plane is... 

This simple form of the Young-Laplace equation shows that if the radius of the sphere increases, AP decreases, and when sph—> °°, AP —> 0, so that when the curvature vanishes and transforms into a flat Euclidean plane, there will be no pressure difference, and the two phases will be in hydrostatic equilibrium as stated above. 

A furtherexample is the buckminsterfullerene molecule of Figure 1 as would have a complex identified to a spherical surface. Indeed for any of the fullerenes (Cfio, C70, C76, C7S, Cs4, and so on) the associated complex is homeomorphic to a sphere. The graphite (or honeycomb) lattice has a complex corresponding to the Euclidean plane El. 

This important theorem refers to two- and three-dimensional structures. It expresses the fact that tiling of the Euclidean plane by regular polygons can be achieved only with the triangle, the square and the hexagon. A four-dimensional periodic structure can allow other symmetry operations. 

Fig. 2.10. Tiling of two-dinnensional Euclidean planes (a) arbitrary lattice, twofold axes 2 (b) rectangles, reflection lines nn (c) diannonds, rectangular centered cell, reflection lines m and glide lines g (d) squares, fourfold axes 4 (e) triangles, threefold axes 3 (f) hexagons, sixfold axes 6, same type of cell as (e)...

Groups made up entirely or in part by translations are of infinite order these groups may also contain rotations and reflections with or without a translation component as well as rotoinversions. In the Euclidean plane they are called plane groups and in three-dimensional space, space groups ... 

In this section we will describe the two-dimensional space groups, or plane groups, which will serve as an introduction to the 230 space groups. The symmetry elements in the Euclidean plane are ... 

The discussion of the plane groups presented in Sections 2.7.1 and 2.7.2 contains all of the ideas necessary to understand space groups. The transition from the Euclidean plane to three-dimensional space requires no new concepts. However, because of the large number of space groups, we will look at only a limited number of examples. The International Tables for Crystallography assemble all the information for these groups. It is thus important to know how to use this compilation correctly and efficiently. 

The previous conclusions are put into perspective by reference to Figure 3.6. All points on I are elements of the Euclidean plane R. The radial lines in that connect the Euclidean points to the point of projection (P) are points in the projective plane P, which in addition contains the points at inhnity, i.e. 

The second possibility b in which opposite sides of the plane rectangle are joined together, always in the opposite sense, leaving two vertices distinct, is equivalent to a real projective plane. Recall that the real projective plane is obtained by adding a line at inhnity to the Euclidean plane. 

A procedure to map space-time events to a Euclidean plane is described by Synge (1950). The Euclidean plane has orthogonal cartesian axes —oo < u,v < oo, and a new variable u — v ) is introduced to replace the radial parameter, r, in construction of the Schwarzscild line element. By using this coordinate system Kruskal (1960) defined the line element... 

The mapping (8) of the five-dimensional space to the tangent space hence implies some sort of perpendicular projection of the Euclidean space with metric 7 /3 on the Euclidean plane with metric 5. ... 

The simplest example in a Euclidean plane without a point (we discard the origin 0), we consider a vector field... 

Let us identify R (/iC) with R (c), then the equation K e) = C2 transforms for each fixed e into an equation of two-dimensional Euclidean plane orthogonal to the vector e and moved away from the origin by one and the same quantity (independent of e). See Fig. 9. 

Suppose that the metric determined by the kinetic energy T is Euclidean at infinity. This means that in Af a certain neighbourhood of each of the infinitely remote points of Kt M is isometric to the neighbourhood of infinity on a Euclidean plane. 

Affine Transformations Transformations on the Euclidean plane include translation, scaling, rotation, and reflection. Affine transformations can be used to convert one geometric shape into another. 

The underlying assumption is that the studied systems are indirectly observable in the sense that the relevant phenomena which are responsible for the data variatirMi/pattems are hidden and not directly measurable/observ-able. This explains the term latent variables. Once uncovered, latent variables (PCs) may be represented by scatter plots in a Euclidean plane. 

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