Reference Coordinate Systems (3/3) José E. Sanguino I Instituto de elecomunicações Instituto Superior écnico, v. Rovisco Pais, 049-00 Lisboa, Portugal el: +35.848468, Fax: +35.84847, Email: sanguino@lx.it.pt Reference Coordinate Systems Coordinate System WGS-84 and PZ-90 Frames Geodetic Coordinates Geodetic to Cartesian Coordinates Cartesian to Geodetic Coordinates he Geoid Reference Ellipsoids Datum ransformations Molodensky ransformation Rotation Matrices ENU Coordinates zimuth & Elevation Orthodromes Loxodromes Local Geodetic Datums Earth-Centered, Earth-Fixed WGS World Geodetic System PZ Parametry Zemli (Russian) ENU East-North-Up
Orthodromes are the shortest path, on the surface of sphere, between two points; Orthodromes follow great circles; Great circles share the centre with the sphere; Heading changes along orthodromes; Orthodromes are straight lines in a gnomonic projection. (latitude:, longitude: λ ) (latitude:, longitude: λ ) z x λ y λ r orthodrome length r Earth radius
(latitude:, longitude: λ ) (latitude:, longitude: λ ) z z y x λ y x λ (λ )Ref. (latitude:, longitude: λ ) (latitude:, longitude: λ ) z z x z x λ y y y x λ (λ ) (- )Ref. 3
(latitude:, longitude: λ ) (latitude:, longitude: λ ) x 3 x z z z 3 x λ y 3 y λ (λ ) (- ) (- )Ref.3 (latitude:, longitude: λ ) (latitude:, longitude: λ ) x 3 z x 4 x λ y y 4 y 3 (λ ) (- ) (- ) (- )Ref.4 λ 4
(latitude:, longitude: λ ) (latitude:, longitude: λ ) z z 5 x 5 x 4 z 4 x λ y (λ ) (- ) (- ) (- ) ( ) Ref.5 λ (latitude:, longitude: λ ) (latitude:, longitude: λ ) z 6 z x 5 y 6 y 5 x λ y λ x 6 (λ ) (- ) (- ) (- ) ( ) ( )Ref.6 5
(latitude:, longitude: λ ) (latitude:, longitude: λ ) z 6 z x λ y λ x 6 (λ ) (- ) (- ) (- ) ( ) ( ) (-λ ) (latitude:, longitude: λ ) (latitude:, longitude: λ ) (λ ) (x,y,z ) (- ) (x,y,z ) (- ) (x 3,y 3,z 3 ) (-λ ) (x 6,y 6,z 6 ) ( ) (x 5,y 5,z 5 ) ( ) (x 4,y 4,z 4 ) (- ) (λ ) (- ) (- ) (- ) ( ) ( ) (-λ ) 6
(latitude:, longitude: λ ) (latitude:, longitude: λ ) (λ ) (x,y,z ) (- ) (x,y,z ) (- ) (x 3,y 3,z 3 ) (-λ ) (x 6,y 6,z 6 ) ( ) (x 5,y 5,z 5 ) ( ) (x 4,y 4,z 4 ) (- ) (λ ) (- ) (- ) (- ) ( ) ( ) (-λ ) (latitude:, longitude: λ ) (latitude:, longitude: λ ) (λ -λ ) (x,y,z ) (- ) (x,y,z ) (- ) (x 6,y 6,z 6 ) (x 3,y 3,z 3 ) ( ) (x 5,y 5,z 5 ) ( ) (x 4,y 4,z 4 ) (- ) 7
(latitude:, longitude: λ ) (latitude:, longitude: λ ) (λ -λ ) (x,y,z ) (- ) (x,y,z ) (- ) (x 6,y 6,z 6 ) (x 3,y 3,z 3 ) ( ) (x 5,y 5,z 5 ) ( ) (x 4,y 4,z 4 ) (- ) (latitude:, longitude: λ ) (latitude:, longitude: λ ) (λ -λ ) (x,y,z ) (- ) (x,y,z ) (- ) (x 6,y 6,z 6 ) (x 3,y 3,z 3 ) ( ) (x 5,y 5,z 5 ) ( ) (x 4,y 4,z 4 ) (- ) 8
(latitude:, longitude: λ ) (latitude:, longitude: λ ) (λ -λ ) (x,y,z ) (- ) (x,y,z ) (- ) (x 6,y 6,z 6 ) (x 3,y 3,z 3 ) ( ) (x 5,y 5,z 5 ) ( ) (x 4,y 4,z 4 ) (- ) (latitude:, longitude: λ ) (latitude:, longitude: λ ) Rot M Rot Rot Rot Rot I x y x y z Rot y M, N,,,,, N,, I M,, N,, M,, N,, N is a rotation matrix 9
M,, Rot Rot Rot x y x 0 0 cos 0 sin 0 0 0 cos sin 0 0 0 cos sin 0 sin cos sin 0 cos 0 sin cos cos 0 sin 0 0 sin sin cos sin cos 0 cos sin cos sin sin cos cos 0 sin cos cos sin sin sin cos sin sin cos cos sin cos sin cos sin sin cos cos cos sin sin cos cos cos sin sin sin cos cos cos N,, Rot y Rot z Rot y cos 0 sin cos sin 0 cos 0 sin 0 0 sin cos 0 0 0 sin 0 cos 0 0 sin 0 cos cos cos cos sin sin cos 0 sin sin cos 0 0 0 sin cos sin sin cos sin 0 cos cos cos cos sin sin cos sin cos cos sin sin cos sin cos cos sin sin sin cos cos cos sin sin sin sin cos sin cos cos (latitude:, longitude: λ ) (latitude:, longitude: λ ) M,, N,, M,, Rot Rot Rot cos sin sin cos sin x sin sin cos cos sin cos sin sin cos cos cos sin y x sin cos cos sin sin cos cos sin sin cos cos cos N,, Rot y Rot z Rot y cos cos cos sin sin sin cos sin cos cos cos sin cos sin cos sin sin cos cos sin sin cos sin sin sin cos sin cos cos 0
cos 3 sin sin 0 3 sin cos sin 0 (latitude:, longitude: λ ) (latitude:, longitude: λ ) M,, N,, M cos N cos cos cos sin sin M N cos sin sin sin M N cos cos sin sin 3 cos M sin sin N sin cos cos sin cos cos cos sin tan cos cos M N sin cos cos cos ML hint: use atan to compute and. 3 sin tan sin cos,,,,,, 0, cos sin sin sin cos sin cos cos cos sin (latitude:, longitude: λ ) (latitude:, longitude: λ ),,,,,, 0, x λ λ z y cos sin cos cos cos sin tan cos cos tan sin cos cos sin sin sin cos sin cos cos cos sin ML hint: use atan to compute and. r orthodrome length r Earth radius
meridian Orthodromes are the shortest path, on the surface of sphere, between two points; Orthodromes follow great circles; Great circles share the center with the sphere; Heading changes along orthodromes; Orthodromes are straight lines in a gnomonic projection. Keeping on Course Following a rhumb line (loxodrome) 70º 70º
Cylindrical projection meridians stretching factor sec 0 R Rcos Earth S N Mercator Projection N parallels C () 70 N latitude 60 N 50 N 40 N 30 N C 0 N 0 N 0 0 S 0 S l dl ln sec tan ln tan 0 4 30 S earing: Distance: Loxodromes 40 S arctan 50 S D R sec James lexander, Loxodromes: Rhumb Way to Go, Mathematics Magazine, Vol. 77, No. 5, December 004, pp.349-356 30 W 0 0 E 70 E 00 E longitude 60 S 3