EUCLID’S GEOMETRY

Prepared by Ajay Chetri

What Is Euclidean Geometry ?The geometry with which we are

most familiar is called Euclidean geometry.

Euclidean geometry deals with points, lines and planes & how they interact to make complex figures.

We are considering physical/ tensorial quantities that exist in space and are coordinate independent, it behaves us to take a closer look at this distinction.

A line is an example of a Euclidean 1-space. It has one dimension, extends to ± ∞, and has ametric (e.g.: the unit interval). Can be extend end-to-end along the entire line in

both directions from the starting point. A line thus marked, with numbers added for

reference, is called Real Number line.

A Euclidean structure will allow us to deal with metric notions such as orthogonality and length (or distance).A plane is an example of a Euclidean 2-space. It has two dimensions, extends to

infinity in all directions. It has a linear metric (the unit interval)

and an areal metric (the unit square).

In the geometry of Euclid, The characteristics are Objects, such as triangles, squares, or circles, can

be moved about in the plane without deformation.

Compared to one another using such relationships as similarity or congruence.

In the plane, the coordinate system of choice is the Cartesian system comprising two real number lines that meet at right angles.

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Physical significance of Euclidian space

Such a space is a region in which material objects and/or beams of light

can be moved about without deformation. But since gravity permeates all space and

time, no such region exists in the universe at large.

Thus it was that Einstein abandoned Euclidean spaces a basis for his General Relativity and adopted a differentially metric non-Euclidean space

Non Euclidian space

Spherical geometry is the geometry of the two dimensional surface of sphere. It is an example

of a geometry that is not Euclidean. Unlike the plane, the sphere does

not extend to infinity; the sphereis a closed, finite surface. The sphere has a differential linear metric and a differential areal metric. It also has a geometry, though one

quite different from that of Euclid.

Introduction of concept of differential metric system

A differential metric is used wherever a unit metric is hard to deal with.

Use differential quantities that, in the limit of ‘smallness,’ behave as though they were Euclidean. So that a simple algebraic metric can be written for

differential quantities. In the plane, the algebraic metric is Pythagoras’

theorem: s2 = x2 + y2, describing the relationship between the length of the hypotenuse, s, and the two sides, x and y, of a right triangle.

In a sphere, the corresponding relationship would have additional terms: s2 = αx2 + βy2 + γxy. Such a metric is certainly approachable, but in the limit of smallness. Pythagoras theorem reappears: ds2 = dx2 + dy2, where ds, dx, and dy are differential lengths. This a sufficiently small portion of a sphere that we can consider flat to any accuracy wedesire.For example:Navigators use just this type of geometry when traveling across the face of our earth. For them,

Introduction to type of system in sphere Here are no parallels in the sphere, however, because there are no Euclidean straight lines All pairs of curves that approximate lines always meet at two antipodal points. Neither is there a Cartesian coordinate system in the sphere. Coordinate systems in the sphere can beconstructed using great circles, but these systems have no unique origin.Great circles: Circle whose radii equal that of the sphere itself.

In each space, different kinds of coordinate systems are possible. In the plane, wespoke of the Cartesian system; but there is also the polar system, the triangular system, and soon. All of these systems can be used to map the same plane; yet, all are different. Final result drawn from the studyPhysical quantities existing in the plane must be independent of the particular coordinate systemchosen. These quantities are not necessarily independent of the space that contains them, however. The same idea applies to all other spaces and coordinate systems as well.E.g.. In triangle, Sum of interior angle=180. While , In sphere, Sum of interior angle>180.

Benefit and advantage of using Tensor analysis

Takes account of coordinate independence of the strange habit of different kinds of spaces in one analysis. Its formalisms are structurally the same regardless of the space involved, the number of dimensions etc. The systems themselves may not be easy to solve, but they are usually obtainedwith more convenience and practical approach. Tensors are very effective tools for setting up systems of equations in “physics or engineeringapplications

The sum of the angles of a triangle are greater than 180°

The main difference between Euclidean & non-Euclidean geometry is that instead of describing a plane as a flat surface, a plane is a sphere.

A line on the sphere is a great circle which is any circle on the sphere that has the same center as the sphere.

Points are exactly theSame, just on a sphere.

EUCLID’S GEOMETRYVS.

NON-EUCLIDEAN GEOMETRY

The main difference between Euclidean and non-Euclidean geometry is with parallel lines.

Two lines are parallel if they never meet. However, on a sphere any two great circles will intersect in two points. This means that it is not possible to draw parallel lines on a sphere, which also eliminates all parallelograms and even squares and rectangles.

In developing Non-Euclidean geometry, we will rely heavily on our knowledge of Euclidean geometry for ideas, methods, and intuition.

Coordinate Curves and Coordinate SurfacesCo-ordinate System: In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space. 2-Dimensional Cartesian system The system consists of anx- and a y-axis that are orthogonal. These two axes determine a unique point of intersection. This point is designated the origin of the system and is given the special label x = 0, y = 0.

In a 3-dimensional Cartesian system: There are three orthogonal axes (x, y, and z) and three coordinate planes (xy, xz, and yz). Any point P is uniquely specified by the number triple P = (x, y, z).In an n-dimensional Cartesian system, by extension, there are n orthogonal axes and (n-1)! coordinate planes. Any point P is uniquely specified by a number n-tuple P = (x1, x2, x3, ... , xn)where the change to subscripted notation in necessitated for purposes of generality.

From Cartesian system,some specific characteristics:• The coordinate axes are straight lines defined to intersect at a single point, the origin.• The coordinate axes are mutually orthogonal.• The coordinate planes are completely determined by the axes.Suppose we were to relax these conditions, Then• The coordinate axes are general curves defined to intersect at least once, The point of origin..• The coordinate axes are not necessarily mutually orthogonal.• Pairs of coordinate axes uniquely determine curvilinear coordinate surfaces as productspaces.

Introduction to different Co-Ordinate systemsa) The cylindrical co-ordinate system: A cylindrical

coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis.

Spherical Co-Ordinate system: In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection

To define spherical coordinates, we take an axis (the polar axis) and a perpendicular plane (the equatorial plane), on which we choose a ray (the initial ray) originating at the intersection of the plane and the axis (the origin O). The coordinates of a point P are: the distance ρ from P to the origin; the angleϕ (zenith) between the line OP and the positive polar axis; and the angle θ (azimuth) between the initial ray and the projection of OP to the equatorial plane. See Figure 1. As in the case of polar and cylindrical coordinates, is only defined up to multiples of 360°, and likewise . Usually is assigned a value between 0 and 180°, but values of between 180° and 360° can also be used; the triples (,,) and (, 360°-, 180°+) represent the same point. Similarly, one can extend to negative values; the triples (,,) and (-, 180°-, 180°+) represent the same point.

Figure 1: A set of spherical coordinates for P is (,,)=(10,60°,30°).

CoVariance and Contra Variance:In 3D Plane geometry, choose three unit vectors at P such that each vector is tangent to one of the axes. Such a triple is usually designated (i, j, k). Any vector V at P can then be written V = αi + βj + γkwhere α, β, and γ are the usual x, y, and z scalar components of the vector. Now suppose that we had chosen unit vectors perpendicular to each of the planes rather than tangent to each of the coordinate axes. Let’s do so and call the resulting triple (i*, j*, k*). Again,any vector V at P can be writtenV = α*i* + β*j* + γ*k*where α*, β*, and γ* are the scalar components of the vector referred to the i*, j*, k* triple.

There is nothing surprising in what we have just done, and our representation is satisfactoryprovided we ensure thatαi + βj + γk = α*i* + β*j* + γ*k*.But, we might find that what we have done is trivial since it is apparent from geometry that thetwo unit vector triples comprise the same set; i.e., thati = i*j = j*k = k*.Still, we used two distinct approaches to defining a unit vector triple at P.But these approaches donot produce a valid result in all case. So we define a Covariant and Contra variant Vector approach