DOCSLIB.ORG

Geodesic Equations and Their Numerical Solutions in Geodetic and Cartesian Coordinates on an Oblate Spheroid

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geodesic Equations and Their Numerical Solutions in Geodetic and Cartesian Coordinates on an Oblate Spheroid J. Geod. Sci. 2017; 7:31–42 Research Article Open Access G. Panou* and R. Korakitis Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid DOI 10.1515/jogs-2017-0004 given two points P0 and P1 on an oblate spheroid, deter- Received December 5, 2016; accepted February 17, 2017 mine the geodesic distance s01 between them and the az- imuths α , α at the end points. Abstract: The direct geodesic problem on an oblate 0 1 These problems have a very long history and several spheroid is described as an initial value problem and different methods of solving them have been proposed by is solved numerically using both geodetic and Cartesian many researchers, as they are reported in a comprehensive coordinates. The geodesic equations are formulated by list by Karney (2016) in GeographicLib. Also, some of the means of the theory of differential geometry. The initial existing methods have been presented by Rapp (1993) and value problem under consideration is reduced to a sys- Deakin and Hunter (2010). tem of first-order ordinary differential equations, which is The methods of solving the above problems can be solved using a numerical method. The solution provides divided into two general categories: (i) using an auxil- the coordinates and the azimuths at any point along the iary sphere, e.g., Bessel (1826), Rainsford (1955), Robbins geodesic. The Clairaut constant is not used for the solu- (1962), Sodano (1965), Saito (1970), Vincenty (1975), Saito tion but it is computed, allowing to check the precision of (1979), Bowring (1983), Karney (2013) and (ii) without us- the method. An extensive data set of geodesics is used, in ing an auxiliary sphere, e.g., Kivioja (1971), Holmstrom order to evaluate the performance of the method in each (1976), Jank and Kivioja (1980), Thomas and Featherstone coordinate system. The results for the direct geodesic prob- (2005), Panou (2013), Panou et al. (2013), Tseng (2014). The lem are validated by comparison to Karney’s method. We methods which use an auxiliary sphere are based on the conclude that a complete, stable, precise, accurate and classical work of Bessel (1826) and its modifications. On fast solution of the problem in Cartesian coordinates is ac- the other hand, the methods without using an auxiliary complished. sphere are attacking the problems directly on an oblate Keywords: Clairaut’s constant, Direct geodesic prob- spheroid. They are conceptually simpler and can be gener- lem, Geometrical geodesy, Karney’s method, Numerical alized in the case of a triaxial ellipsoid, as has been already method presented by Holmstrom (1976) and Panou (2013). The solution of the geodesic problems, with one of the above two methods, includes evaluating elliptic inte- 1 Introduction grals or solving differential equations using: (i) approxi- mate analytical methods, e.g., Vincenty (1975), Holmstrom (1976), Pittman (1986), Mai (2010), Karney (2013) or (ii) nu- In geodesy, there are two traditional problems concern- merical methods, e.g., Saito (1979), Rollins (2010), Sjöberg ing geodesics on an oblate spheroid (ellipsoid of revolu- (2012), Sjöberg and Shirazian (2012), Panou et al. (2013). tion): (i) the direct problem: given a point P0 on an oblate The approximate analytical methods are usually based on spheroid, together with the azimuth α0 and the geodesic the fact that an oblate spheroid deviates slightly from a distance s01 to a point P1, determine the point P1 and sphere, so these methods essentially involve a truncated the azimuth α1 at this point, and (ii) the inverse problem: series expansion. On the other hand, numerical methods can be used for ellipsoids of arbitrary flattening. In addi- tion, they do not require a change in the theoretical back- *Corresponding Author: G. Panou: Department of Surveying Engi- ground with a modification of the computational environ- neering, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece, E-mail: [email protected] ment. However, they may suffer from computational er- R. Korakitis: Department of Surveying Engineering, National Tech- rors, which are reduced with the improvements in modern nical University of Athens, Zografou Campus, 15780 Athens, Greece © 2017 G. Panou and R. Korakitis, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. 32 Ë G. Panou and R. Korakitis computational systems. Furthermore, since the numerical precision of the numerical integration. We should mention methods perform computations at many points along the that other methods for the solution of the direct geodesic geodesic, they can be used as a convenient and efficient problem, such as Rollins (2010) and Sjöberg and Shirazian approach to trace the geodesic. If tracing is not needed, an (2012), use the Clairaut constant as an a priori known analytical method may be sufficient to give the results of quantity in the equations and they employ iterative tech- the geodesic problems. niques. From the vast literature on geodesic problems, one no- Inevitably, in curvilinear coordinates there are two tices that the evaluation of the performance of the geodesic poles (singularities) and hence all of the above methods, algorithms is based on the aspects of stability, accuracy without modifications, can be ill-behaved. This does not and computational speed. Of course, the execution time happen in Cartesian coordinates and the algorithms based (︀ )︀ depends strongly on the programming environment and on them are insensitive to singularities, such as tan π/2 . the computing platform used. Today, with the broad avail- Also, the Cartesian coordinate system can be easily related ability of high speed computers, the execution time does to other curvilinear systems, many formulas in this system not longer play an essential role. With regard to stability, are simpler, without numerical difficulties and computa- the algorithms should be stable in the domain of use, i.e. tions do not demand the use of trigonometric functions, without limitations to the input data of the problem. Fi- which can make computer processing slow, as pointed out nally, they should provide results with high accuracy, de- by Felski (2011). pending on the demands of the application. Using the calculus of variations, the geodesic equa- In this work, the direct geodesic problem and the cor- tions in Cartesian coordinates were derived on a sphere responding equations, with independent variable s, are by Fox (1987) and on a triaxial ellipsoid by Holmstrom numerically solved as an initial value problem, directly on (1976). Although the approximate analytical solution given an oblate spheroid, using two coordinate systems: geode- by Holmstrom (1976) can be applied in the degenerate case tic and Cartesian. We note that, Panou et al. (2013) have nu- of an oblate spheroid, it is of low precision, since the pre- merically solved the inverse geodesic problem and the cor- cision was not his primary consideration. responding equations, with independent variable λ, as a The method proposed in this work introduces a new boundary value problem. The method presented here can approach to the geodesic initial value problem, both in be generalized in the case of a triaxial ellipsoid and can be terms of its theoretical basis and its numerical solution. used for arbitrary flattening. However, in order to evalu- Part of the solution constitutes the solution of the direct ate the method, we limit the numerical applications to the geodesic problem, where the position and the azimuth are case of the WGS84 oblate spheroid. determined only at the end point of the geodesic. In addi- In general, there are several numerical methods for tion, the geodesic initial value problem directly provides solving an initial value problem (see Hildebrand 1974). In the trace of the full path of the geodesic. this study, we will use a relatively simple and commonly applied fourth-order Runge-Kutta method (see Butcher 1987), which has been applied successfully to the geodesic 2 Geodesics in geodetic boundary value problem (Panou 2013, Panou et al. 2013). coordinates In addition, we examine the performance (stability, pre- cision, execution time) of the method in both coordinate The geodesic initial value problem, expressed in geodetic systems. coordinates on an oblate spheroid, consists of determin- Kivioja (1971) has solved the geodesic initial value ing a geodesic, parametrized by its arc length s, φ = φ(s), problem by numerical integration of a system of two dif- λ = λ(s), with azimuths α = α(s) along it, which passes ferential equations. Thomas and Featherstone (2005) im- through a given point P0 (φ (0) , λ (0)) in a known direction proved Kivioja’s method by altering the system of the two (given azimuth α0 = α (0)) and has a certain length s01. differential equations along the geodesic, in order to avoid the singularity when the geodesic passes through the ver- tices. Alternatively, in this study we propose the numeri- 2.1 Geodesic equations cal integration of a system of four differential equations of a geodesic in geodetic coordinates, which is free from We consider an oblate spheroid which, in geodetic coordi- this singularity. Although there are more equations, from nates (φ, λ), is described parametrically by the solution we can determine the Clairaut constant, at any point along the geodesic, and thus we are able to check the x = N cos φ cos λ (1) Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates Ë 33 1 2 2 y = N cos φ sin λ (2) Γ12 = Γ11 = Γ22 = 0 (15) Therefore, the geodesic equations, expressed in geodetic (︁ )︁ z = N 1 − e2 sin φ (3) coordinates on an oblate spheroid, are given by (Struik 1961, p. 132) where φ (–π/2 ≤ φ ≤ +π/2) is the geodetic latitude, λ (–π 2 (︂ )︂2 (︂ )︂2 < λ ≤ +π) is the geodetic longitude and d φ 1 dφ 1 dλ + Γ11 + Γ22 = 0 (16) a ds2 ds ds N = 1/2 (4) (︁ 2 )︁ 1 − e2 sin φ d2λ dφ dλ + 2Γ2 = 0 (17) ds2 12 ds ds is the radius of curvature in the prime vertical normal sec- The initial conditions associated with these equations are tion, with a the major semiaxis and e the first eccentricity.
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]
  • 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
    [Show full text]