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Sistemas De Coordendas Celestes Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 1.3 Posicionamento na Terra Elipsóidica Na cartografia utiliza-se como modelo matemático para a forma da Terra o elipsóide de revolução Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 O SISTEMA GPS EFETUA MEDIÇÕES GEODÉSICAS Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Qual é a forma da Terra? Qual é a representação matemática da superfície de referência para a cartografia? A superfície topográfica da Terra apresenta uma forma muito irregular, com elevações e depressões. Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Modelos utilizados para a Terra esfera elipsóide geóide PosicionamentoTerra na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 O GEÓIDE Geóide: superfície cuja normal coincide com a vertical do lugar V V´ Superfície equipotencial O geóide é uma superfície equipotencial coincidente com o nível médio dos mares g considerados em repouso. Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Geóide tem uma superfície irregular, determinável ponto a ponto. Causas: crosta terrestre heterogenea. Isostasia |f| = k m1 m2 2 d12 Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 REPRESENTAÇÃO GEODÉSICA DA TERRA Elipsóide de revolução: elipse girando em torno do seu eixo menor (2b) Círculo máximo a= raio maior ou semi-eixo maior b= raio menor ou semi-eixo menor Prof .M A Zanetti Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Visão 3D do elipsóide de revolução Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Geometria da Elipse r1 + r2 = 2a a = semi-eixo maior Eixo de rotação f = (a-b)/a b = semi-eixo menor e = [(a2-b2)1/2]/a = 2f - f2 P r1 r2 a F F1 O 2 b r4 r3 r3+r4= 2a Q Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Geometria do elipsóide • O achatamento f é definido por: a b f a • A primeira excentricidade e2 ao quadrado é dada por: a 2 b2 e2 e2 2 f f 2 a 2 • A segunda excentricidade ao quadrado e´2 é obtida por: a 2 b2 e'2 b2 Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Normal a um ponto do elipsóide Normal = perpendicular a tangente tangente p ao elipsoide no ponto p p´ Fonte: R.E. DEAKIN and M.N. HUNTER,2010 Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 SISTEMA DE COORDENADAS GEODÉSICAS pn p normal p’ Superfície física Meridiano de Greenwich q’ q t ps = latitude geódésica = longitude geodésica h=pp’ = altitude elipsoidal Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 normal equador Latitude geodésica () é o ângulo formado entre a normal e sua projeção no plano do equador terrestre. Longitude geodésica () é o ângulo diedro fomado entre o meridiano elipsóidico de Greenwich e o meridiano elipsóidico do ponto considerado. Altitude elipsoidal ou geométrica (h) é a distância medida sobre a normal, desde o ponto no terreno até o elipsoide. Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 LONGITUDE GEODÉSICA pn p normal p’ Superfície Meridiano de física Greenwich Meridiano do ponto p’ q’ q t ps Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 RAIO DE CURVATURA SECÇÃO MERIDIANA E PRIMEIRO VERTICAL Fonte: R.E. DEAKIN and M.N. HUNTER,2010 Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 RAIOS DE CURVATURAS PRINCIPAIS Raio de curvatura da seção primeiro vertical N ou grande normal é dado por: a N 1/ 2 1 e2 sen 2 é a latitude geodésica do ponto P • O raio de curvatura da seção meridiana M é calculado por: a(1 e2 ) M (1 e2 sen 2 )3 / 2 E o raio médio de curvatura RM é dado por: RM NM Raio de curvatura de uma secção de azimute A 1 cos2 A sen2 A R M N Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 SISTEMAS GEODÉSICOS USADOS NO BRASIL Sistema Córrego Alegre SAD-69 SIRGAS Geodésico elipsóide Hayford Referencia 1967 GRS-80 a 6378388,000m 6378160,000m 6378137,000m b 6356911.946m 6356774,719m 6356752,3141m f 1/297 1/298,25 1/298,257222101 e2 0,006722670 0,0066946053 0,00669438002290 a-b 21476,054m 21385,281m 21384,6859m a- b = diferença entre o semi-eixo maior e o menor f = achatamento do elipsóide e2 = excentrecidade ao quadrado Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Definição de distância no elipsóide: geodésica Fonte: R.E. DEAKIN and M.N. HUNTER,2010 Geodésica é uma curva reversa no espaço Menor distância entre dois pontos no elipsóide Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Definição de distância no elipsóide: geodésica pn p normal T p’ Superfície física geodésica T q’ q ’ t ps Geodésica é uma curva reversa no espaço Menor distância entre dois pontos no elipsóide Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Definição de azimute geodésico entre dois pontos pn p normal T p’ Superfície física A TP APT T’ q’ q ’ t ps APT = ATP ±180+ Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 SISTEMA DE COORDENADAS CARTESIANAS GEODÉSICO (CG) Greenwich Pn =90⁰E Ps Posicionamento na Terra Elipsóidica Origem: centro de massa da Terra Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 DETALHES DO SISTEMA DE COORDENADAS CARTESIANAS GEODÉSICO Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 TRANSFORMAÇÃO DE COORDENADAS GEODÉSICAS EM CARTESIANAS TRIDIMENSIONAIS XP = (N + h) cos cos YP = (N + h) cos sen 2 ZP = [N (1 – e ) + h) sen = latitude geodésica =longitude geodésica N = grande normal h = altitude elipsoidal ou geométrica e = excentricidade do elipsóide a a b N 1/ 2 f 2 2 1e2 sen2 a e 2 f f Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 TRANSFORMAÇÃO DE COORDENADAS CARTESIANAS TRIDIMENSIONAIS EM GEODÉSICAS 2 3 ZP + e´ b sen tg = p - e 2a cos3 YP tg = X P e2 2 f f 2 p h = - N cos Z a 2 2 P p = X +Y = arctg P P '2 a b e p b b2 Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 DESVIO DA VERTICAL (i) vertical normal i Superfície p física geóide elipsóide Vertical=fio de prumo Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 N H= altitude ortométrica H=h-N h= altitude elipsoidal N= ondulação geoidal Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Sistema GNSS No ponto A HA= altitude ortométrica Nivelamento hA= altitude geométrica geométrico NA= ondulação geoidal Superfície terrestre geóide elipsóide Fonte: Move3 Manual Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 MÉTODOS DE DETERMINAÇÃO DO GEÓIDE Gravimetria – Almirante Tamandaré- Astronomia- Observação ao Paraná Sol - Atol das Rocas Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 GRAVÍMETRO LACOSTE&ROMBERG Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Estação: UFPR - RBMC Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 MAPGEO2010 – MODELO GEOIDAL DO BRASIL http://www.ibge.gov.br/home/geociencias/geodesia/modelo_geoidal.shtm Para o ponto UFPR-RBMC NA=3,67m hA =925,81 HA=925,81-3,67 H = 922,14m A Escolha do referencial Executar Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 Porta -ADR aquisição de dados a cada 6 min Nível topográfico caixa protetora Réguas ou miras do instrumento Observador (leituras diárias) Mira topográfica régua de pier maré RN bóia linimetro orifício roda dentada Definição de um datum vertical: o nível médio das águas do mar, aproximação do geóide, é usado como referência para as altitudes ortométricas (altitude zero) Posicionamento na Terra Elipsóidica Prof. DR. Carlos Aurélio Nadal - Sistemas de Referência e Tempo em Geodésia – Aula 05 DATUM VERTICAL BRASILEIRO Datum vertical do Brasil: Imbituba-SC nível médio das águas do mar observadas no marégrafo de Imbituba-SC.
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