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E-raamat: Map of the World: An Introduction to Mathematical Geodesy [Taylor & Francis e-raamat]

(Guandong Technion - Israel Institute of Technology),
  • Formaat: 273 pages, 9 Tables, black and white; 79 Line drawings, black and white; 10 Halftones, black and white; 89 Illustrations, black and white
  • Ilmumisaeg: 21-Aug-2019
  • Kirjastus: CRC Press
  • ISBN-13: 9780429265990
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  • Taylor & Francis e-raamat
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Map of the World: An Introduction to Mathematical GeodesySuurem pilt
  • Formaat: 273 pages, 9 Tables, black and white; 79 Line drawings, black and white; 10 Halftones, black and white; 89 Illustrations, black and white
  • Ilmumisaeg: 21-Aug-2019
  • Kirjastus: CRC Press
  • ISBN-13: 9780429265990
Teised raamatud teemal:
Taylor & Francis e-raamatudRohkem infot Taylor & Francis e-raamatute kohta
Raamatu kodulehekülg: https://www.taylorfrancis.com/books/9780429265990
Carl Friedrich Gauss, the "foremost of mathematicians," was a land surveyor. Measuring and calculating geodetic networks on the curved Earth was the inspiration for some of his greatest mathematical discoveries. This is just one example of how mathematics and geodesy, the science and art of measuring and mapping our world, have evolved together throughout history.

This text is for students and professionals in geodesy, land surveying, and geospatial science who need to understand the mathematics of describing the Earth and capturing her in maps and geospatial data: the discipline known as mathematical geodesy. Map of the World: An Introduction to Mathematical Geodesy aims to provide an accessible introduction to this area, presenting and developing the mathematics relating to maps, mapping, and the production of geospatial data. Described are the theory and its fundamental concepts, its application for processing, analyzing, transforming, and projecting geospatial data, and how these are used in producing charts and atlases. Also touched upon are the multitude of cross-overs into other sciences sharing in the adventure of discovering what our world really looks like.

FEATURES

Written in a fluid and accessible style, replete with exercises; adaptable for courses on different levels.

Suitable for students and professionals in the mapping sciences, but also for lovers of maps and map making.
Preface xi
Acknowledgments xiii
Acronyms xv
1 A brief history of mapping
1(16)
1.1 Map and scale
2(1)
1.2 Gerardits Mercator and Mercator s map of the world
3(2)
1.3 Construction of the Mercator projection
5(2)
1.4 Loxodromes or rhumb lines
7(1)
1.5 A political map of the world?
8(1)
1.6 Towards the precise figure of the Earth
9(5)
1.6.1 Rotation and flattening of the Earth
10(3)
1.6.2 The big project of the French Academy of Sciences
13(1)
Exercises
14(3)
2 Popular conformal map projections
17(12)
2.1 The Lambert projection
19(3)
2.2 About the isometric latitude
22(1)
2.3 The Mercator projection
23(1)
2.4 The stereographic projection
24(4)
Exercises
28(1)
3 The complex plane and conformal mappings
29(14)
3.1 Complex numbers
29(2)
3.2 The polar representation of complex numbers
31(2)
3.3 On the nature of the exponential function
33(2)
3.4 Euler's formula
35(1)
3.5 De Moivre's formula
36(1)
3.6 The logarithm function
37(2)
Exercises
39(4)
4 Complex analysis
43(10)
4.1 Limit and continuity of a complex function
43(2)
4.2 The complex derivative and analytic functions
45(1)
4.3 The Cauchy--Riemann conditions
46(2)
4.4 Series expansions
48(1)
4.5 An analytic function as a power series
49(2)
4.6 Complex Taylor series
51(1)
Exercises
52(1)
5 Confermal mappings
53(14)
5.1 Helmert mappings or transformations
54(3)
5.2 Mobius transformations
57(5)
5.3 Mobius transformations and the stereographic projection
62(2)
5.4 Numerical conformal mappings
64(1)
Exercises
65(2)
6 Transversal Mercator projections
67(10)
6.1 Map projections used in Finland
67(5)
6.1.1 The historical Gauss--Kruger and KKJ
67(3)
6.1.2 Modern map projections Gauss--Kruger and UTM
70(2)
6.2 Gauss--Kruger projection computations
72(5)
6.2.1 Analyticity and the Cauchy--Riemann conditions
73(1)
6.2.2 Progress of the computation
74(3)
7 Spherical trigonometry
77(14)
7.1 Spherical excess
77(2)
7.2 Surface area of a triangle on a sphere
79(1)
7.3 Rectangular spherical triangle
80(1)
7.4 General spherical triangle
80(2)
7.5 Deriving the equations using vectors in space
82(2)
7.6 Polarization
84(2)
7.7 Solving the spherical triangle with the method of additaments
86(1)
7.8 Solving the spherical triangle using Legendre's method
87(1)
7.9 The half-angle cosine rule
87(1)
7.10 The geodetic forward problem on the sphere
88(1)
7.11 The geodetic inverse problem on the sphere
89(1)
Exercises
89(2)
8 The geometry of the ellipsoid of revolution
91(18)
8.1 The geodesic as a solution of a system of differential equations
92(1)
8.2 An interesting invariant
93(1)
8.3 The geodetic forward problem
94(1)
8.4 The geodetic inverse problem
95(1)
8.5 Representations of sphere and ellipsoid
96(1)
8.6 Different types of latitude and relations between them
97(2)
8.7 Different measures for the flattening
99(1)
8.8 Co-ordinates on the meridian ellipse
99(1)
8.9 Three-dimensional rectangular co-ordinates on the reference ellipsoid
100(1)
8.10 Computing geodetic co-ordinates from rectangular ones
101(2)
8.11 Length of the meridian arc
103(1)
8.12 The reference ellipsoid and the gravity field
103(4)
Exercises
107(2)
9 Three-dimensional co-ordinates and transformations
109(18)
9.1 About rotations and rotation matrices
110(1)
9.2 Chaining matrices in three dimensions
111(1)
9.3 Orthogonal matrices
112(2)
9.4 Topocentric systems
114(4)
9.4.1 Geocentric to topocentric and back
115(3)
9.5 Solving the parameters of a three-dimensional Helmert transformation
118(4)
9.6 Variants of the Helmert transformation
122(1)
9.7 The geodetic forward or inverse problem with rotation matrices
123(3)
9.7.1 The geodetic forward problem
124(1)
9.7.2 The geodetic inverse problem
125(1)
9.7.3 Comparison with the ellipsoidal surface solution
125(1)
Exercises
126(1)
10 Co-ordinate reference systems
127(16)
10.1 History
127(1)
10.2 Parameters of the GRS80 system
128(2)
10.3 GRS80 and modern co-ordinate reference systems
130(2)
10.4 Co-ordinate reference systems and their realizations
132(4)
10.4.1 Terminology
132(1)
10.4.2 The practical viewpoint
132(2)
10.4.3 The ITRS/ETRS systems and their realizations
134(2)
10.5 Transformations between different ITRFs
136(1)
10.6 The Finnish ETRS89 realization, EUREF-FIN
137(1)
10.7 A triangle-wise affine transformation for the Finnish territory
138(4)
Exercises
142(1)
11 Co-ordinates of heaven and Earth
143(18)
11.1 Orientation of the Earth
145(1)
11.2 Polar motion
146(1)
11.3 Geocentric systems
147(5)
11.3.1 Geocentric terrestrial systems
148(1)
11.3.2 The conventional terrestrial system
149(1)
11.3.3 Instantaneous terrestrial system
150(1)
11.3.4 The geocentric quasi-inertial system
150(2)
11.4 Sidereal time
152(2)
11.5 Celestial trigonometry
154(3)
11.6 Rotation matrices and celestial co-ordinate systems
157(1)
Exercises
158(3)
12 The orbital motion of satellites
161(20)
12.1 Kepler's laws
161(7)
12.2 The circular orbit approximation in manual computation
168(1)
12.3 Crossing a certain latitude in the inertial system
169(2)
12.4 Topocentric co-ordinates of the satellite
171(2)
12.5 Latitude crossing in the terrestrial system
173(1)
12.6 Determining the orbit from observations
173(6)
Exercises
179(2)
13 The surface theory of Gauss
181(22)
13.1 A curve in space
182(2)
13.2 The first fundamental form (metric)
184(2)
13.3 The second fundamental form
186(2)
13.4 Principal curvatures of a surface
188(4)
13.5 A curve on a surface
192(3)
13.5.1 Tangent vector
192(1)
13.5.2 Curvature vector
192(3)
13.6 The geodesic
195(3)
13.6.1 Describing the geodesic interiorly, in surface co-ordinates
196(1)
13.6.2 Describing the geodesic exteriorly, rising vectors in space
196(2)
13.7 More generally on surface theory
198(1)
Exercises
199(4)
14 Riemann surfaces and charts
203(24)
14.1 What is a tensor?
205(5)
14.1.1 Vectors
205(1)
14.1.2 Tensors
206(1)
14.1.3 The geometric presentation of a tensor and its invariants
207(1)
14.1.4 Tensors in general co-ordinates
208(1)
14.1.5 Trivial tensors
209(1)
14.2 The metric tensor
210(3)
14.3 The inverse metric tensor
213(1)
14.3.1 Raising or lowering tensor indices
213(1)
14.3.2 The eigenvalues and eigenvectors of a tensor
214(1)
14.3.3 Visualization of a tensor
214(1)
14.4 Parallellity of vectors and parallel transport
214(1)
14.5 The Christoffel symbols
215(3)
14.6 Alternative equations for the geodesic
218(1)
14.7 The curvature tensor
219(3)
Exercises
222(5)
15 Map projections in light of surface theory
227(18)
15.1 The Gauss total curvature and spherical excess
227(5)
15.2 Curvature in quasi-Cartesian co-ordinates
232(2)
15.3 Map projections and scale
234(5)
15.3.1 On the Earth's surface
234(1)
15.3.2 In the map plane
235(1)
15.3.3 The scale
235(3)
15.3.4 The Tissot indicatrix
238(1)
15.4 Curvature of the Earth's surface and scale
239(2)
Exercises
241(4)
A The stereographic projection maps circles into circles 245(2)
B Isometric latitude on the ellipsoid 247(2)
C Useful relations between the principal radii of curvature 249(2)
D Christoffel's symbols from the metric 251(2)
E The Riemann tensor from the Christoffel symbols 253(2)
F Conformal mappings in spaces of any dimension 255(2)
References 257(8)
Index 265
Martin Vermeer is professor of geodesy in the Department of the Built Environment, Aalto University, Finland.

Antti Rasila is associate professor of mathematics at Guangdong Technion Israel Institute of Technology, Guangdong, China.
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