Geod\303\251sica em uma superf\303\255cieRobert J. LopezEmeritus Professor, RHITMaple Fellow, MaplesoftLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Introdu\303\247\303\243o</Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Samir Khan, um dos engenheiros de aplica\303\247\303\243o da Maplesoft, fez uma pergunta a um usu\303\241rio que perguntou como obter curvas de comprimento m\303\255nimo em uma superf\303\255cie. Samir forneceu uma t\303\251cnica num\303\251rica interessante para encontrar o que \303\251 essencialmente uma geod\303\251sica conectando dois pontos em uma superf\303\255cie embutida em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1. Nesta planilha, implementamos uma vers\303\243o da abordagem num\303\251rica de Samir e tamb\303\251m mostramos como obter as equa\303\247\303\265es diferenciais que regem tais geod\303\251sicas. Fazemos isso com o c\303\241lculo das varia\303\247\303\265es (minimizando a integral do comprimento do arco) e com as t\303\251cnicas de c\303\241lculo tensorial implementadas no pacote Geometria Diferencial.No artigo C\303\241lculo de tensor com o pacote de geometria diferencial encontramos geod\303\251sicas no plano quando o plano foi referido \303\240s coordenadas polares. Nesta planilha, encontramos geod\303\251sicas em uma superf\303\255cie embutida em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Abordagem Num\303\251rica de Samir Khan</Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Execute a planilha inteira: Clique em "!!!" \303\255cone na barra de ferramentasExecutar passo a passo: clique no \303\255cone \303\240 direita (para inicializar) e execute cada comando
restart;
with(Student[Precalculus], Distance):
with(Optimization, Minimize): with(plots):with(VariationalCalculus):
with(DifferentialGeometry):
with(Tensor):
with(Tools):
with(VectorCalculus,diff,BasisFormat):
BasisFormat(false):
Preferences("ShowFramePrompt", false):
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=O gr\303\241fico da superf\303\255cie definida pela fun\303\247\303\243o na Tabela 1
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 1 Fun\303\247\303\243o que define uma superf\303\255cie em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=aparece na Figura 1.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
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%;Figura 1 Superf\303\255cie descrita pela fun\303\247\303\243o na Tabela 1
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=A curva de comprimento m\303\255nimo conectando os pontosLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtb0dGJDYtUSomdW1pbnVzMDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGOS8lKnN5bW1ldHJpY0dGOS8lKGxhcmdlb3BHRjkvJS5tb3ZhYmxlbGltaXRzR0Y5LyUnYWNjZW50R0Y5LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSC1JI21uR0YkNiRRIjVGJ0Y0LUYxNi1RIixGJ0Y0RjcvRjtRJXRydWVGJ0Y8Rj5GQEZCRkQvRkdRJjAuMGVtRicvRkpRLDAuMzMzMzMzM2VtRidGMC1GTDYkUSIzRidGNEY0RjRGNA== e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSI1RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMEYnRjRGNEY0RjQ= \303\251 aproximado por uma linha poligonal, ou seja, por uma spline linear, com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEjNTBGJ0Y5LUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLyUrZXhlY3V0YWJsZUdGPUY5segmentos e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIitGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= n\303\263s equidistantes. As coordenadas x dos n\303\263s inicial e terminal nesta spline s\303\243o dadas respectivamente 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JSFHe para os n\303\263s intermedi\303\241rios, 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As coordenadas y nos pontos inicial e terminal s\303\243o 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIitGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= n\303\263s 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 S\303\243o ent\303\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print(); # input placeholderJSFHonde o 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, s\303\243o desconhecidos. O comprimento da spline aproximada \303\251 dado 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%;JSFHque \303\251 simplesmente a soma dos comprimentos de cada segmento na spline. Este comprimento \303\251 minimizado 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JSFHonde as restri\303\247\303\265es de negatividade nas coordenadas y s\303\243o inspiradas pelos valores nos pontos inicial e terminal. Portanto, os n\303\263s computados s\303\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portanto, a spline resultante pode ser representada graficamente na Figura 2.
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%;Figura 2 Geod\303\251sica aproximada na superf\303\255cie na Figura 1
JSFH
<Text-field style="Heading 1" layout="Heading 1">Minimize a Integral do Comprimento do Arco</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=O comprimento do arco \303\251 dado pela integral 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, onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 o par\303\242metro ao longo da curva, e 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. Uma vez que o radicando \303\251 necessariamente diferente de zero, podemos obter uma geod\303\251sica minimizando a integral cujo integrando \303\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 pode ser feito resolvendo as equa\303\247\303\265es de Euler-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que podemos obter no Maple por meio dos c\303\241lculos na Tabela 2.
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Tabela 2 As equa\303\247\303\265es de Euler-Lagrange para uma geod\303\251sica em uma superf\303\255cie
Al\303\251m das pr\303\263prias equa\303\247\303\265es de Euler-Lagrange, o comando EulerLagrange do pacote VariationalCalculus pode fornecer as primeiras integrais nas quais as constantes de integra\303\247\303\243o est\303\243o na forma LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn. Primeiro removemos essas primeiras integrais e extra\303\255mos as duas equa\303\247\303\265es diferenciais de Euler-Lagrange restantes, que resolvemos numericamente na Tabela 3.
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Tabela 3 Solu\303\247\303\243o num\303\251rica das equa\303\247\303\265es diferenciais de Euler-Lagrange
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Na Figura 3, onde representamos graficamente a solu\303\247\303\243o das equa\303\247\303\265es de Euler-Lagrange na superf\303\255cie da Figura 1, tomamos o cuidado de adicionar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5MTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GLDYlUSJ6RicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictSSNtb0dGJDYtUSI9RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlAtSSNtbkdGJDYkUSUwLjA1RidGMkYy para cada coordenada z ao longo da curva para que possa ser vista mais claramente. A alternativa teria sido tornar a superf\303\255cie suficientemente transparente para que a curva, situada na superf\303\255cie, fosse vis\303\255vel.
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%;Figura 3 Solu\303\247\303\243o das equa\303\247\303\265es de Euler-Lagrange sobrepostas na superf\303\255cie da Figura 1
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Geod\303\251sica por meio dos pacotes de geometria diferencial</Font></Text-field>JSFHO comando GeodesicEquations no pacote DifferentialGeometry ir\303\241 gerar as equa\303\247\303\265es geod\303\251sicas para a superf\303\255cie LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= embarcado em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1. A moldura para esta superf\303\255cie \303\251 estabelecida 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%;Agora, o tensor m\303\251trico (primeira forma fundamental) para a superf\303\255cie deve ser obtido. Para este fim, definimos a superf\303\255cie na forma de vetor de raio 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JSFHem seguida, calcule os vetores de base tangente 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 de acordo com a Tabela 4.
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print(); # input 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print(); # input placeholderTabela 4 Vetores de base tangente natural na superf\303\255cie embutida 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
Os componentes do tensor m\303\251trico covariante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUYsNiUtRi82J1EiZUYnLyUlYm9sZEdGNC9GM0ZIL0Y2USVib2xkRicvJStmb250d2VpZ2h0R0Zdby1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1Rj0tRkE2LVEpJm1pZGRvdDtGJ0ZERkZGSUZLRk1GT0ZRRlMvRlZRLDAuMTY2NjY2N2VtRicvRllGaW8tRiw2JUZmbi1GIzYlLUYvNiVRImpGJ0YyRjVGMkY1Rj1GRA== est\303\243o contidos na 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print(); # input placeholdercuja forma tensorial \303\251 gerada 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print(); # input placeholderJSFHOs s\303\255mbolos de Christoffel do segundo tipo, a saber, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEnJiM5MTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRI2lqRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LUYjNiUtRi82JVEia0YnRj1GP0Y9Rj8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkgvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnRjU=, s\303\243o obtidos 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%;JSFHOs s\303\255mbolos de Christoffel individuais podem ser extra\303\255dos com os comandos mostrados na Tabela 5. As express\303\265es s\303\243o todas longas e foram suprimidas.
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JKG1zdWJzdXBHRiQ2KC1GLDYlUSgmR2FtbWE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUkjbW5HRiQ2JFEjMTFGJ0Y6RjotRiM2JC1GQDYkUSIyRidGOkY6LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZKL0krbXNlbWFudGljc0dGJFEsW25vbmUsbm9uZV1GJ0Y6RitGOg==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Tabela 5 Os s\303\255mbolos de Christoffel de segundo tipo
JSFHAs equa\303\247\303\265es geod\303\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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEoJmVxdWl2O0YnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGWEYuRj4= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEoJmVxdWl2O0YnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGWC1GLzYlUSJ5RidGMkY1Rj4=, s\303\243o ent\303\243o gerados 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JSFHe extra\303\255do 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JSFHNovamente, as sa\303\255das (muito longas) foram suprimidas. Essas duas equa\303\247\303\265es diferenciais s\303\243o ent\303\243o resolvidas numericamente como na Tabela 6, assim como suas contrapartes foram resolvidas na Tabela 3.
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%;Tabela 6 Solu\303\247\303\243o num\303\251rica de equa\303\247\303\265es geod\303\251sicas geradas pelo comando GeodesicEquations
JSFHA Figura 4, um gr\303\241fico da geod\303\251sica (levantada ligeiramente da superf\303\255cie) e da superf\303\255cie 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, \303\251 obtido exatamente como a Figura 3 foi.
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%;Figura 4 Geod\303\251sica computada numericamente desenhada na superf\303\255cie 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