GEODETIC POSITION COMPUTATIONS

1 GEODETIC POSITIONCOMPUTATIONSE. J. KRAKIWSKYD. B. THOMSONF ebruary 1974 TECHNICAL REPORT NO. 217 lecture NOTESNO. 39 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 Krakiwsky Thomson Department of Geodesy and Geomatics Engineering University of New Brunswick 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.

2 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 be stated that the attempt has been to produce a self contained package , 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.}

3 It is hoped that this approach Yill give the reader an appreciation for the foundation upon which the f' are based, and in the end, the formulae themselves. The notes evolved out of lecture notes prepared by Krakiwsky and from research work performed by Thomson over recent years at The authors acknowledge the use of ideas, contained in the lecture notes , of Professors Urho A. Uotila and Richard H. Rapp of the Department of GEODETIC Science, The Ohio State University, Columbus, Ohio. Other sources used for important details are referenced within the text. The authors wish to acknowledge the contribution made by the Surveying Engineering undergraduate class of 1975 to improving these i notes by finding typographical errors.

4 Mr. c. Chamber~ain is particular=b' acltnow~edged for his constructive criticism, and assistance in preparing the manuscript for publication. Kraltiwsky Thomson February ~4, 1974 ii TABLE OF CONTENTS PREFACE .. LIST OF ILLUSTRATIONS .. nmtODUCTIOB .. SECTION I: ELLIPSOIDAL Gl!DMETRY .. 1. The Ellipsoid of Rotation .. f. Ellipsoidal Parameters 1. 2 Radii o~ CUrvature Meridian Radius o~ Curvature Prime Vertical Radius of Curvature Radius of CUrvature in A:J:q Azimuth CUrves on the Surface of an Ellipsoid The Normal Section The Geodesic .. Page i v 1 2 2 2 "5 5 10 13 19 19 24 SEGI'ION II: REDUCTION OF TERRESTRIAL GEODETIC OBSERVATIONS 31 2.

5 Reduction to the Surface of the Reference Ellipsoid .. Reduction of Horizontal Directions (or Angles) Geometric Effects Gravimetric Effects 2. 2 Zenith Distances 2. 3 Spatial Distances Reduction of Computed GEODETIC Quantities to the i'e:rz-Un .. SECTION III: COMPUTA!riON OF GJDDEriC POSITIONS. ON THE REFERENCE ELLIPSOID .. 3. Puissant's Formula -Short Lines .. Introduction 3. 2 Direct Probl~ Inverse Problem . 3. 4 Summary of Equations for the Solution of the Direct and Inverse Probl-ems Using Puissant ' s Formulae 3.

6 5 The Gauss Mid-Latitude Other Short Line Formulae 4. Bessel's Formulae -Long Lines Introduction .. Fundamental Relationships Solution of the Elliptic Integral 4. 4 Direct Problem 4. 5 Inverse Problem Other Long Line Formulae 31 31 32 37 31 38 41 43 43 43 43 49 52 53 54 55 55 55 62 72 75 77 TABLE OF CONTENTS { Cont' d) SECTION IV: COMPUTATION OF GEODEriC POSITIONS IN THREE DIMENSIONS .. 5 Direct and Inverse Prob~ems in Three Dimensions . ~ .. Direct Problem. Inverse Prob~em .. ~ . 6. Intersection Problems in Three Dimensions Azimuth Ihtersection Spatial Distance Intersection 7. Conc~uding Remarks REFERENCES iv.}

7 Page 78 79 79 83 85 85 92 96 98 Figure No. 1 2 3 4 5 6 1 8 9 10 11 12 13 14 15 16 11 18 19 20 2la and 2lb 22 LIST OF ILLUSTRAfiO:RS The Ellipsoid of Rotation Meridian Normal Section Showing the Meridian Radius of Curvature (M) Prime Vertical Normal Section Showing the Prime Vertical Radius of CUrvature (l'l) Meridian Radius of Curvature (M) Prime Vertical Radius of Curvature (:N) Normal Section at Any Azimuth a Indicatrix for Solution of R a Section Along PP' (a) for Solution of R a Solution of Z for Solution of R a Reciprocal Normal Sections .Reciprocal Normal Section Triangle : Angular Separation Between Reciprocal Normal Sections Geodesic Differential Equation of a Geodesic on the SUrface of an Ellipsoid of Rotation Separation Between Normal Section and Geodesic Skew Correction Deflection of the Vertical Correction Spatial Distance Reduction Puissant's Formula.

8 For Direct Problem Puissant's for Inverse Problem Fundamental Relationships for the Development of Bessel's Formulae Reduced Sphere and Ellipsoid v Page 3 6 1 9 11 14 15 16 18 20 22 23 25 28 30 33 36 39 44 50 57 51 59 LIST OF ILLUSTRATIONS (Cont'd} Figure No. Page 23 Solution of ds/da 63 24 So1ution of dl 70 25 So1ution of Arc Length a 74 26 Direct Prob1em (Local GEODETIC ) 80 27 Direct Prob1em (Local Astronomic). 82 28 Unit Vectors in the Local GEODETIC System 86 29 Azimuth Intersection in Three Dimensions 88 30 Spatial Distance Intersection in Three Dimensions 93 Vi INTRODUCTION The first three sections ot these notes deal with the com-putation of GEODETIC positions on an ellipsoid. In chapter one, a review of ellipsoidal geometry is gi-ven in order that the development of further can be understood ~.)

9 Camnon to all of the classical ellipsoidal COMPUTATIONS is the necessity to reduce GEODETIC observations onto the ellipsoid, thus an entire chapter is devoted to this topic. Two classical geometric GEODETIC computation problems are treated; they are called the direct and inverse GEODETIC problems. There are various approaches that can be adopted for solving these problems. Generally, they are classified in terms of "short" , "medium" , and "long" line formulae. Each of' them involve different apprax:!ma-tions which tend to restrict the interstation distance over which some formulae are useful for a given accuracy. The last section of' the notes deals with the camputation of GEODETIC positions in three dimensions. First, the direct and inverse problems are developed, then two special problems -those of azimuth and spatial distance intersections -are dealt vith.

10 These solutions offer an alternative to the classical approach of GEODETIC POSITION COMPUTATIONS . 1 SECTION I: ELLIPSOIDAL GEOMETRY ~~ The\ Rotation Since an ellipsoid of rotation (reference ellipsoid) is generallY considered as the best approximation to the size and shape of the earth, it . is used as the surfa~e upon which to perform ten-estrial GEODETIC COMPUTATIONS ~ Immediate~; we study several. geometric properties o:f an ellipsoid of rotation that are of special interest to geodesists. In particular, the radii of curvature of points on the suf'ace of the ellipsoid, and some curves on that surface, are described. ~. ~ Ellipsoidal Parameters Figure ~ shovs an ellipsoid of rotation. The parameters of a reference ellipsoid, which describe its size and shape, are: i) the semi-major axis, a, U) the semi-minor axis, b.