DOCSLIB.ORG

3D Geometrical Transformations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
3D Geometrical Transformations 3D Geometrical Transformations 3D Geometrical Transformations Foley & Van Dam, Chapter 5 • 3D point representation • Translation • Scaling, reflection • Shearing • Rotations about x, y and z axis • Composition of rotations • Rotation about an arbitrary axis • Transforming planes 3D Coordinate Systems Reminder: Cross Product UxV Right-handed coordinate system: z V T U y VxU x Left-handed coordinate system: U u V nˆ U V sin T y xˆ yˆ zˆ ª u y v z u z v y º « » U u V u x u y u z « u z v x u x v z » v v v « u v u v » z x y z ¬ x y y x ¼ x 3D Point Representation 3D Geometrical Transformations • A 3D point P is represented in homogeneous • In homogeneous coordinates, 3D affine coordinates by a 4-dimensional vector: transformations are represented by 4x4 matrices: x ª º ª a b c t x º « » y « d e f t » P « » « y » « z » « » « g h i t z » ¬ 1 ¼ « » ¬ 0 0 0 1 ¼ •As for 2D points: •A point transformation is performed: ª x º ª D x º « y » « y » ª x ' º ª a b c t x º ª x º « » « D » p { « y '» « d e f t » « y » « z » « D z » « » « y » « » « » « » 1 « z ' » « g h i t z » « z » ¬ ¼ ¬ D ¼ « » « » « » ¬ 1 ¼ ¬ 0 0 0 1 ¼ ¬ 1 ¼ 3D Translation 3D Scaling ª a 0 0 0 º ª x º ª x ' º ª ax º P in translated to P' by: « 0 b 0 0 » « y » « y '» « by » « » « » « » « » « » « » « » « » ª 1 0 0 t x º ª x º ª x ' º ª x t x º 0 0 c 0 z z ' cz « » « » « » « » « » « » « » « » 0 1 0 t y y y ' y t y ¬ 0 0 0 1 ¼ ¬ 1 ¼ ¬ 1 ¼ ¬ 1 ¼ « » « » « » « » « 0 0 1 t z » « z » « z ' » « z t z » « » « » « » « » ¬ 0 0 0 1 ¼ ¬ 1 ¼ ¬ 1 ¼ ¬ 1 ¼ Or S P P ' Or: T P P ' z z z y y y x x x 1 1 S P ' P Inverse: T P ' P 3D Reflection 3D Shearing • A reflection through the xy plane: • Shearing: ª 1 a b 0 º ª x º ª x ay bz º « c 1 d 0 » « y » « cx y dz » ª 1 0 0 0 º ª x º ª x º « » « » « » « e f 1 0 » « z » « ex fy z » « 0 1 0 0 » « y » « y » « » « » « » « » « » « » ¬ 0 0 0 1 ¼ ¬ 1 ¼ ¬ 1 ¼ « 0 0 1 0 » « z » « z » « » « » « » ¬ 0 0 0 1 ¼ ¬ 1 ¼ ¬ 1 ¼ • Change in each coordinate is a linear combination of all three • Transforms a cube into a general parallelepiped •Reflections through the xz and the yz planes are defined similarly 3D Rotation 3D Rotation • To generate a rotation in 3D we have to • Counterclockwise rotation about x-axis specify: ª x ' º ª 1 0 0 0 º ª x º « y '» « 0 cos T sin T 0 » « y » – axis of rotation (2 d.o.f.) « » « » « » « z ' » « 0 sin T cos T 0 » « z » « » « » « » – amount of rotation (1 d.o.f.) ¬ 1 ¼ ¬ 0 0 0 1 ¼ ¬ 1 ¼ •Note, the axis passes through the origin p ' R x (T ) p y z z y x x 3D Rotation 3D Rotation • Counterclockwise rotation about y-axis • Counterclockwise rotation about z-axis ª x ' º ª cos T 0 sin T 0 º ª x º ª x ' º ª cos ș sin ș 0 0 º ª x º « » « » « » y ' sin cos 0 0 y y ' 0 1 0 0 y « » « ș ș » « » « » « » « » « z ' » 0 0 1 0 « z » « z ' » « sin T 0 cos T 0 » « z » « » « » « » « » ¬« 1 ¼» ¬« 0 0 0 1 ¼» ¬« 1 ¼» ¬ 1 ¼ ¬ 0 0 0 1 ¼ ¬ 1 ¼ p ' R z (T ) p p ' R y (T ) p z z y y x x Composite Rotation Change of Coordinates • Rx, Ry, and Rz, can perform any rotation • Problem: Given the XYZ orthonormal coordinate about an axis passing through the origin system, find a transformation M, that maps a representation in XYZ into a representation in the orthonormal system UVW, with the same origin •Inverse rotation: •The matrix M transforms the UVW vectors to the XYZ vectors z p R 1 (T ) p' R(T ) p' R T (T ) u=(ux,uy,uz) y x v=(vx,vy,vz) Change of Coordinates Change of Coordinates • Solution: M is rotation matrix whose rows • Let's check the transformation of U under M: are U,V, and W: ª u x u y u z 0 º ª u x º ªºuuuxyz0 « v v v 0 » « u » MU « x y z » y «»vvv0 w w w 0 « u » M «»xyz « x y z » « z » « » « 1 » «»wwwxyz0 ¬ 0 0 0 1 ¼ ¬ ¼ «» ¬¼0001 2 2 2 ª u x u y u z º ª 1 º • Note: the inverse transformation is the « » 0 0 « » X transpose: « 0 » « 0 » « » « 1 » ª u x v x w x 0 º ¬ 1 ¼ ¬ ¼ 1 T « u y v y w y 0 » M M « » • Similarly, V goes into Y, and W goes into Z « u z v z w z 0 » ¬« 0 0 0 1 ¼» Change of Coordinates Rotation About an Arbitrary Axis • Let's check the transformation of the X axis • Axis of rotation can be located at any point: -1 under M : 6 d.o.f. (we must specify 2 points p1 and p2) u v w 0 u • The idea: make the axis coincident with ª 1 º ª x x x º ª 1 º ª x º « » « » 1 « 0 » u y v y w y 0 « 0 » u y M 0 « » 0 « » U one of the coordinate axes (z axis), rotate by « » u z v z w z 0 « » u z « 1 » « » « 1 » « » ¬ ¼ ¬« 0 0 0 1 ¼» ¬ ¼ ¬« 1 ¼» T, and then transform back z p p 2 • Similarly, Y goes into V, and Z goes into W 1 y x Rotation About an Arbitrary Axis Rotation About an Arbitrary Axis § 1 0 0 x 1 · y ¨ 0 1 0 y ¸ y Step1: T ¨ 1 ¸ step 3 • 0 0 1 z p p ¨ 1 ¸ 1 2 p p1 ¨ 0 0 0 1 ¸ 2 x © ¹ x § u x u y u z 0 · z Initial Position z ¨ ¸ Rotate the Object v x v y v z 0 • Step 2: M ¨ ¸ Around the z Axis ¨ w x w y w z 0 ¸ ¨ ¸ y y © 0 0 0 1 ¹ p1 step 1 p1 step 4 § cos ș sin ș 0 0 · p2 ¨ ¸ p2 • Step 3: R sin ș cos ș 0 0 ¨ 0 0 1 0 ¸ x x ¨ ¸ z z © 0 0 0 1 ¹ Translate p to the Origin Rotate the Axis to the 1 § u x v x w x 0 · original Orientation ¨ u v w 0 ¸ • Step 4: M 1 ¨ y y y ¸ ¨ u v w 0 ¸ ¨ z z z ¸ y y © 0 0 0 1 ¹ § 1 0 0 x · step 2 p1 step 5 p ¨ 1 ¸ p1 2 1 0 1 0 y • Step 5: T ¨ 1 ¸ ¨ 0 0 1 z 1 ¸ x ¨ ¸ z p2 x © 0 0 0 1 ¹ z Translate to the 1 1 Rotate p2 onto the z Axis • Composition: Original Position P c T M R M T P Rotation About Arbitrary Axis Transforming Planes • Constructing an orthonormal system along the rotation axis: • Plane representation: – A vector W parallel to the rotation axis: – By three non-collinear points s – By implicit equation: ªºx s p 2 p 1 ; w s «»y Ax By Cz D >@ A B C D «» 0 – A vector V perpendicular to W: «»z «» § 0 · ¬¼1 ¨ ¸ ª A B C º a « 1» a w u ¨ 0 ¸ ; v ¬ D D D ¼ ¨ ¸ a © 1 ¹ P1 – A vector U forming a right-handed orthogonal system P2 with W and V: u v w >@x y z 1 u P0 Transforming Planes Transforming Planes • One way to transform a plane is by transforming • Note that: ªºx any three non-collinear points on the plane «»y >@ABC DTT 1 «» 0 «»z • Another way is to transform the plane equation: «» Given a transformation T such that ¬¼1 T [x, y, z, 1] = [x', y', z', 1] • Thus, the transformation that we should apply to the plane equation is: find [A',B',C',D'], such that: ªºx ' «» 1 y ' ABCDcccc ABCDT >@ABC''' D ' «» 0 > @ > @ «»z ' «» ¬¼1.
Load more

Recommended publications
  • Vectors, Matrices and Coordinate Transformations
    S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Since we will making extensive use of vectors in Dynamics, we will summarize some of their important properties. Vectors For our purposes we will think of a vector as a mathematical representation of a physical entity which has both magnitude and direction in a 3D space. Examples of physical vectors are forces, moments, and velocities. Geometrically, a vector can be represented as arrows. The length of the arrow represents its magnitude. Unless indicated otherwise, we shall assume that parallel translation does not change a vector, and we shall call the vectors satisfying this property, free vectors. Thus, two vectors are equal if and only if they are parallel, point in the same direction, and have equal length. Vectors are usually typed in boldface and scalar quantities appear in lightface italic type, e.g. the vector quantity A has magnitude, or modulus, A = |A|. In handwritten text, vectors are often expressed using the −→ arrow, or underbar notation, e.g. A , A. Vector Algebra Here, we introduce a few useful operations which are defined for free vectors. Multiplication by a scalar If we multiply a vector A by a scalar α, the result is a vector B = αA, which has magnitude B = |α|A. The vector B, is parallel to A and points in the same direction if α > 0.
    [Show full text]
  • Chapter 5 ANGULAR MOMENTUM and ROTATIONS
    Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another.
    [Show full text]
  • Solving the Geodesic Equation
    Solving the Geodesic Equation Jeremy Atkins December 12, 2018 Abstract We find the general form of the geodesic equation and discuss the closed form relation to find Christoffel symbols. We then show how to use metric independence to find Killing vector fields, which allow us to solve the geodesic equation when there are helpful symmetries. We also discuss a more general way to find Killing vector fields, and some of their properties as a Lie algebra. 1 The Variational Method We will exploit the following variational principle to characterize motion in general relativity: The world line of a free test particle between two timelike separated points extremizes the proper time between them. where a test particle is one that is not a significant source of spacetime cur- vature, and a free particles is one that is only under the influence of curved spacetime. Similarly to classical Lagrangian mechanics, we can use this to de- duce the equations of motion for a metric. The proper time along a timeline worldline between point A and point B for the metric gµν is given by Z B Z B µ ν 1=2 τAB = dτ = (−gµν (x)dx dx ) (1) A A using the Einstein summation notation, and µ, ν = 0; 1; 2; 3. We can parame- terize the four coordinates with the parameter σ where σ = 0 at A and σ = 1 at B. This gives us the following equation for the proper time: Z 1 dxµ dxν 1=2 τAB = dσ −gµν (x) (2) 0 dσ dσ We can treat the integrand as a Lagrangian, dxµ dxν 1=2 L = −gµν (x) (3) dσ dσ and it's clear that the world lines extremizing proper time are those that satisfy the Euler-Lagrange equation: @L d @L − = 0 (4) @xµ dσ @(dxµ/dσ) 1 These four equations together give the equation for the worldline extremizing the proper time.
    [Show full text]
  • Geodetic Position Computations
    GEODETIC POSITION COMPUTATIONS E. J. KRAKIWSKY D. B. THOMSON February 1974 TECHNICALLECTURE NOTES REPORT NO.NO. 21739 PREFACE In order to make our extensive series of lecture notes more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The images have not been converted to searchable text. GEODETIC POSITION COMPUTATIONS E.J. Krakiwsky D.B. Thomson Department of Geodesy and Geomatics Engineering University of New Brunswick P.O. Box 4400 Fredericton. N .B. Canada E3B5A3 February 197 4 Latest Reprinting December 1995 PREFACE The purpose of these notes is to give the theory and use of some methods of computing the geodetic positions of points on a reference ellipsoid and on the terrain. Justification for the first three sections o{ these lecture notes, which are concerned with the classical problem of "cCDputation of geodetic positions on the surface of an ellipsoid" is not easy to come by. It can onl.y be stated that the attempt has been to produce a self contained package , cont8.i.ning the complete development of same representative methods that exist in the literature. The last section is an introduction to three dimensional computation methods , and is offered as an alternative to the classical approach. Several problems, and their respective solutions, are presented. The approach t~en herein is to perform complete derivations, thus stqing awrq f'rcm the practice of giving a list of for11111lae to use in the solution of' a problem.
    [Show full text]
  • Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
    Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation
    [Show full text]
  • Coordinate Transformation
    Coordinate Transformation Coordinate Transformations In this chapter, we explore mappings – where a mapping is a function that "maps" one set to another, usually in a way that preserves at least some of the underlyign geometry of the sets. For example, a 2-dimensional coordinate transformation is a mapping of the form T (u; v) = x (u; v) ; y (u; v) h i The functions x (u; v) and y (u; v) are called the components of the transforma- tion. Moreover, the transformation T maps a set S in the uv-plane to a set T (S) in the xy-plane: If S is a region, then we use the components x = f (u; v) and y = g (u; v) to …nd the image of S under T (u; v) : EXAMPLE 1 Find T (S) when T (u; v) = uv; u2 v2 and S is the unit square in the uv-plane (i.e., S = [0; 1] [0; 1]). Solution: To do so, let’s determine the boundary of T (S) in the xy-plane. We use x = uv and y = u2 v2 to …nd the image of the lines bounding the unit square: Side of Square Result of T (u; v) Image in xy-plane v = 0; u in [0; 1] x = 0; y = u2; u in [0; 1] y-axis for 0 y 1 u = 1; v in [0; 1] x = v; y = 1 v2; v in [0; 1] y = 1 x2; x in[0; 1] v = 1; u in [0; 1] x = u; y = u2 1; u in [0; 1] y = x2 1; x in [0; 1] u = 0; u in [0; 1] x = 0; y = v2; v in [0; 1] y-axis for 1 y 0 1 As a result, T (S) is the region in the xy-plane bounded by x = 0; y = x2 1; and y = 1 x2: Linear transformations are coordinate transformations of the form T (u; v) = au + bv; cu + dv h i where a; b; c; and d are constants.
    [Show full text]
  • Unit 5: Change of Coordinates
    LINEAR ALGEBRA AND VECTOR ANALYSIS MATH 22B Unit 5: Change of Coordinates Lecture 5.1. Given a basis B in a linear space X, we can write an element v in X in a unique 3 way as a sum of basis elements. For example, if v = is a vector in X = 2 and 4 R 1 1 2 B = fv = ; v = g, then v = 2v + v . We say that are the B 1 −1 2 6 1 2 1 B 3 coordinates of v. The standard coordinates are v = are assumed if no other 4 basis is specified. This means v = 3e1 + 4e2. n 5.2. If B = fv1; v2; ··· ; vng is a basis of R , then the matrix S which contains the vectors vk as column vectors is called the coordinate change matrix. Theorem: If S is the matrix of B, then S−1v are the B coordinates of v. 1 1 6 −1 5.3. In the above example, S = has the inverse S−1 = =7. We −1 6 1 1 compute S−1[3; 4]T = [2; 1]T . Proof. If [v]B = [a1; : : : ; an] are the new coordinates of v, this means v = a1v1 + ··· + −1 anvn. But that means v = S[v]B. Since B is a basis, S is invertible and [v]B = S v. Theorem: If T (x) = Ax is a linear map and S is the matrix from a basis change, then B = S−1AS is the matrix of T in the new basis B. Proof. Let y = Ax. The statement [y]B = B[x]B can be written using the last theorem as S−1y = BS−1x so that y = SBS−1x.
    [Show full text]
  • COORDINATE TRANSFORMATIONS Members of a Structural System Are Typically Oriented in Differing Directions, E.G., Fig
    COORDINATE TRANSFORMATIONS Members of a structural system are typically oriented in differing directions, e.g., Fig. 17.1. In order to perform an analysis, the ele- ment stiffness equations need to be expressed in a common coor- dinate system – typically the global coordinate system. Once the element equations are expressed in a common coordinate system, the equations for each element comprising the structure can be Figure 17.1 – Frame Structure 1 (Kassimali, 1999) 2 assembled. Coordinate Transformations: Frame Elements Consider frame element m of Fig. 17.7. Y e x y m b X (a) Frame Q , u 6 6 Q , u m 4 4 Q3, u3 Q , u Q5, u5 1 1 Figure 17.7: Local – Global Q2, u2 Coordinate Relationships (b) Local Coordinate End Forces and Displacements 3 4 1 Figure 17.7 shows that the local – QcosFsinFxXY global coordinate transformations QsinFcosFyXY (17.9) can be expressed as QFzZ x = cos X + sin Y y = -sin X + cos Y where x, X = 1 or 4; y, Y = 2 or 5; and z, Z = 3 or 6. and since z and Z are parallel, this coordinate transformation is Utilizing (17.9) for all six member expressed as force components and expressing z = Z the resulting transformations in Using the above coordinate matrix form gives transformations, the end force and QFb [t] [0] b displacement transformations can QFee[0] [t] be expressed as or 5 {Q} = [T] {F} (17.11) T {Q} Q Q Q T where b 123 = member; {Q} = <<Q>b <Q>e> = beginning node local coordinate element local coordinate force T T force vector; {Q}e = <Q4 Q5 Q6> = vector; {F} = <<F>b <F>e> = end node local coordinate force element global coordinate force T vector; {F}b = <F1 F2 F3> = [t] [0] beginning node global coordinate vector; and [T] = = T [0] [t] force vector; {F}e = <F4 F5 F6> = end node global coordinate force element local to global coordinate cos sin 0 transformation matrix.
    [Show full text]
  • 1 Vectors & Tensors
    1 Vectors & Tensors The mathematical modeling of the physical world requires knowledge of quite a few different mathematics subjects, such as Calculus, Differential Equations and Linear Algebra. These topics are usually encountered in fundamental mathematics courses. However, in a more thorough and in-depth treatment of mechanics, it is essential to describe the physical world using the concept of the tensor, and so we begin this book with a comprehensive chapter on the tensor. The chapter is divided into three parts. The first part covers vectors (§1.1-1.7). The second part is concerned with second, and higher-order, tensors (§1.8-1.15). The second part covers much of the same ground as done in the first part, mainly generalizing the vector concepts and expressions to tensors. The final part (§1.16-1.19) (not required in the vast majority of applications) is concerned with generalizing the earlier work to curvilinear coordinate systems. The first part comprises basic vector algebra, such as the dot product and the cross product; the mathematics of how the components of a vector transform between different coordinate systems; the symbolic, index and matrix notations for vectors; the differentiation of vectors, including the gradient, the divergence and the curl; the integration of vectors, including line, double, surface and volume integrals, and the integral theorems. The second part comprises the definition of the tensor (and a re-definition of the vector); dyads and dyadics; the manipulation of tensors; properties of tensors, such as the trace, transpose, norm, determinant and principal values; special tensors, such as the spherical, identity and orthogonal tensors; the transformation of tensor components between different coordinate systems; the calculus of tensors, including the gradient of vectors and higher order tensors and the divergence of higher order tensors and special fourth order tensors.
    [Show full text]
  • Lecture 34: Principal Axes of Inertia • We’Ve Spent the Last Few Lectures Deriving the General
    Lecture 34: Principal Axes of Inertia • We’ve spent the last few lectures deriving the general expressions for L and Trot in terms of the inertia tensor • Both expressions would be a great deal simpler if the inertia tensor was diagonal. That is, if: = δ Iij Ii ij or I1 0 0 … Ÿ I = 0 I 0 … 2 Ÿ …0 0 I3 ⁄Ÿ • Then we could write = ω = δ ω = ω Li ƒ Iij j ƒ Ii ij j Ii i j j = 1 ω ω = 1 δ ω ω = 1 ω 2 Trot ƒ Iij i j ƒ Ii ij i j ƒ Ii i 2 i, j 2 i, j 2 i • We’ve already seen that the elements of the inertia tensor transform under rotations • So perhaps we can rotate to a set of axes for which the tensor (for a given rigid body) is diagonal – These are called the principal axes of the body – All the rotational problems you did in first-year physics dealt with rotation about a principal axis – that’s why the equations looked simpler. • If a body is rotating solely about a principal axis (call it the i axis) then: = ω = Li Ii i , or L Ii • If we can find a set of principal axes for a body, we call the three non-zero inertia tensor elements the principal moments of inertia Finding the Principal Moments • In general, it’s easiest to first determine the principal moments, and then find the principal axes • We know that if we’re rotating about a principal axis, we have: L = I A principal moment = ω • But the general relation L i I i j j also holds.
    [Show full text]
  • Coordinate Systems in De Sitter Spacetime Bachelor Thesis A.C
    Coordinate systems in De Sitter spacetime Bachelor thesis A.C. Ripken July 19, 2013 Radboud University Nijmegen Radboud Honours Academy Abstract The De Sitter metric is a solution for the Einstein equation with positive cosmological constant, modelling an expanding universe. The De Sitter metric has a coordinate sin- gularity corresponding to an event horizon. The physical properties of this horizon are studied. The Klein-Gordon equation is generalized for curved spacetime, and solved in various coordinate systems in De Sitter space. Contact Name Chris Ripken Email [email protected] Student number 4049489 Study Physicsandmathematics Supervisor Name prof.dr.R.H.PKleiss Email [email protected] Department Theoretical High Energy Physics Adress Heyendaalseweg 135, Nijmegen Cover illustration Projection of two-dimensional De Sitter spacetime embedded in Minkowski space. Three coordinate systems are used: global coordinates, static coordinates, and static coordinates rotated in the (x1,x2)-plane. Contents Preface 3 1 Introduction 4 1.1 TheEinsteinfieldequations . 4 1.2 Thegeodesicequations. 6 1.3 DeSitterspace ................................. 7 1.4 TheKlein-Gordonequation . 10 2 Coordinate transformations 12 2.1 Transformations to non-singular coordinates . ......... 12 2.2 Transformationsofthestaticmetric . ..... 15 2.3 Atranslationoftheorigin . 22 3 Geodesics 25 4 The Klein-Gordon equation 28 4.1 Flatspace.................................... 28 4.2 Staticcoordinates ............................... 30 4.3 Flatslicingcoordinates. 32 5 Conclusion 39 Bibliography 40 A Maple code 41 A.1 Theprocedure‘riemann’. 41 A.2 Flatslicingcoordinates. 50 A.3 Transformationsofthestaticmetric . ..... 50 A.4 Geodesics .................................... 51 A.5 TheKleinGordonequation . 52 1 Preface For ages, people have studied the motion of objects. These objects could be close to home, like marbles on a table, or far away, like planets and stars.
    [Show full text]
  • 2 Review of Stress, Linear Strain and Elastic Stress- Strain Relations
    2 Review of Stress, Linear Strain and Elastic Stress- Strain Relations 2.1 Introduction In metal forming and machining processes, the work piece is subjected to external forces in order to achieve a certain desired shape. Under the action of these forces, the work piece undergoes displacements and deformation and develops internal forces. A measure of deformation is defined as strain. The intensity of internal forces is called as stress. The displacements, strains and stresses in a deformable body are interlinked. Additionally, they all depend on the geometry and material of the work piece, external forces and supports. Therefore, to estimate the external forces required for achieving the desired shape, one needs to determine the displacements, strains and stresses in the work piece. This involves solving the following set of governing equations : (i) strain-displacement relations, (ii) stress- strain relations and (iii) equations of motion. In this chapter, we develop the governing equations for the case of small deformation of linearly elastic materials. While developing these equations, we disregard the molecular structure of the material and assume the body to be a continuum. This enables us to define the displacements, strains and stresses at every point of the body. We begin our discussion on governing equations with the concept of stress at a point. Then, we carry out the analysis of stress at a point to develop the ideas of stress invariants, principal stresses, maximum shear stress, octahedral stresses and the hydrostatic and deviatoric parts of stress. These ideas will be used in the next chapter to develop the theory of plasticity.
    [Show full text]