Ajustando C\303\255rculos no Espa\303\247o para Dados 3-DDr. Robert J. LopezEmeritus Professor of mathematics, RHITMaple Fellow, Maplesoft
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Introdu\303\247\303\243o</Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Uma solu\303\247\303\243o alg\303\251brica foi dada em [1] para o problema de ajustar um c\303\255rculo a pontos de dados tridimensionais que n\303\243o se encontram no plano xy. Em [2], esta solu\303\247\303\243o \303\251 trabalhada como um projeto MATLAB\302\256 cujas etapas articulamos na Tabela 1.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
1.Traduzir tudo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pontos por seu centroide de modo que \316\240, o plano que cont\303\251m o c\303\255rculo de ajuste, passe atrav\303\251s (ou quase atrav\303\251s) da origem.2.Obtenha o 3 \303\227 3 fator U da decomposi\303\247\303\243o de valor singular de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, e 3 \303\227 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= matriz cujas colunas s\303\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 vetores da origem aos pontos traduzidos.3.As duas primeiras colunas em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= est\303\243o 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, uma base ortonormal para \316\240.4.A terceira coluna em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSdtc3R5bGVHRiQ2Jy1GLDYoUSJuRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lJ29wYXF1ZUdGNy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGP0Y1RjtGPUZARisvRj5RJ25vcm1hbEYn, um vetor unit\303\241rio normal para \316\240.5.As colunas no 3 \303\227 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= matriz 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 s\303\243o os componentes dos vetores 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 com respeito aos vetores de base 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. Desde a n \303\251 ortogonal a \316\240, as entradas na terceira linha de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= s\303\243o pequenos.6.Um ajuste de m\303\255nimos quadrados a um c\303\255rculo \303\251 feito no plano \316\240 tratando as duas primeiras entradas em cada coluna de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= as 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, a ponto em um plano xy.Tabela 1 Passos essenciais em uma solu\303\247\303\243o alg\303\251brica para o problema de ajustar um c\303\255rculo no espa\303\247o a dados 3-D
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Implementaremos a solu\303\247\303\243o alg\303\251brica no Maple e, em seguida, forneceremos uma solu\303\247\303\243o anal\303\255tica alternativa baseada na minimiza\303\247\303\243o direta da soma dos quadrados dos desvios. A t\303\251cnica anal\303\255tica, sugerida por [3], simultaneamente al\303\247a um plano aos dados, enquanto encontra o raio do c\303\255rculo minimizando a soma dos quadrados dos desvios de cada ponto de uma linha paralela \303\240 normal ao plano de ajuste.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Inicializa\303\247\303\265es</Font></Text-field>JSFHCarregue os pacotes e defina os dados iniciais clicando no \303\255cone abaixo.
with(LinearAlgebra):
with(Optimization):
with(plots):
with(Student[Precalculus],CenterOfMass):
UseHardwareFloats := false:
P[1] := [-6.3,1.69,1.11]:
P[2] := [-4.84,-1.46,3.08]:
P[3] := [-2.97,-2.74,5.1]:
P[4] := [2.26,-.77,9.7]:
P[5] := [4.17,7.11,10.02]:
P[6] := [3.67,8.71,9.25]:
P[7] := [2.92,9.98,8.29]:
P[8] := [1.14, 11.4, 6.33]:
P[9] := [-.82,11.68,4.42]:
P[10] := [-6.41,4.48,.47]:
JSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Solu\303\247\303\243o Alg\303\251brica</Font></Text-field>JSFHCada uma das dez colunas da matriz a seguir s\303\243o as coordenadas cartesianas dos pontos de dados retirados de [2].JSFH
JSFHPara cada ponto de dados, adicionamos ru\303\255do aleat\303\263rio de magnitude inferior a 1,5. As perturba\303\247\303\265es para cada coordenada est\303\243o na matriz
JSFHAs coordenadas do centr\303\263ide s\303\243o subtra\303\255das de cada ponto; as colunas da matriz resultante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=JSFH
JSFHs\303\243o vetores LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtSSdtc3R5bGVHRiQ2Jy1GLDYoUSJ2RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lJ29wYXF1ZUdGOi8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGQkY4Rj5GQEZDLUYsNiVRImtGJy9GPEY6L0ZBUSdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRisvRkFRJ25vcm1hbEYn,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YrLUYsNi1RIj1GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGU0YrLUkjbW5HRiQ2JFEiMUYnRi8tRiw2LVEiLEYnRi9GMi9GNkZMRjdGOUY7Rj1GP0ZBL0ZFUSwwLjMzMzMzMzNlbUYnRistRiw2LVEnJm1sZHI7RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjE2NjY2NjdlbUYnL0ZFRl1vRllGKy1GVjYkUSMxMEYnRi9GLw==, da origem a cada um dos pontos traduzidos.A decomposi\303\247\303\243o de valor singular desta matriz fornece o fator LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5 e o vetor S de valores singulares. O Matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= cont\303\251m bases ortonormais para o espa\303\247o da coluna de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e por seu complemento ortogonal.
JSFHO terceiro componente do LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== = print(); # input placeholderJSFHo vetor de valores singulares, \303\251 uma medida de qu\303\243o pr\303\263ximos os pontos transladados est\303\243o do plano \316\240. \303\200 medida que a magnitude do ru\303\255do nos dados aumenta, esse terceiro valor singular fica maior. Conseq\303\274entemente, a abordagem alg\303\251brica defendida em [1] \303\251 menos robusta do que a abordagem anal\303\255tica esbo\303\247ada na pr\303\263xima se\303\247\303\243o.JSFHSe os pontos traduzidos est\303\243o em um plano atrav\303\251s da origem, o espa\303\247o da coluna de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ser\303\241 bidimensional, ou seja, o plano \316\240. O complemento ortogonal ser\303\241 medido por um vetor ortogonal ao plano \316\240. Consequentemente, as duas primeiras colunas do LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= compreendem a base 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 o plano \316\240, e a terceira coluna de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5 nomeadamente,JSFH
JSFH\303\251 um vetor unit\303\241rio normal a \316\240.Que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 uma matriz ortogonal \303\251 verificada pelo c\303\241lculo 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, implementado no Maple via
JSFHAs colunas do produto 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 isso \303\251,JSFH
JSFHs\303\243o os componentes do LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtSSdtc3R5bGVHRiQ2Jy1GLDYoUSJ2RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lJ29wYXF1ZUdGOi8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGQkY4Rj5GQEZDLUYsNiVRImtGJy9GPEY6L0ZBUSdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRisvRkFRJ25vcm1hbEYn ao longo de 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 e n. Assim, as duas primeiras linhas de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= s\303\243o tomadas como coordenadas xey de pontos em um plano xy estendido por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtSSdtc3R5bGVHRiQ2Jy1GLDYoUSJ1RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lJ29wYXF1ZUdGOi8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGQkY4Rj5GQEZDLUkjbW5HRiQ2JFEiMUYnL0ZBUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRitGSQ== e 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 Neste plano, um ajuste de m\303\255nimos quadrados de um c\303\255rculo ao 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 pares representados por essas linhas \303\251 efetuado por escritoLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y+USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZXLUklbXN1YkdGJDYlLUYsNiVRImNGJ0Y6Rj0tSSNtbkdGJDYkUSIxRidGRC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRitGREZELUZbbzYkUSIyRidGRC8lMXN1cGVyc2NyaXB0c2hpZnRHRmBvLUZBNi9RMSZJbnZpc2libGVUaW1lcztGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdGSEZERkZGSUZLRk1GT0ZRRlNGVUZYLUZBNi1RJyZwbHVzO0YnRkRGRkZJRktGTUZPRlFGU0ZVRlhGZm8tRjA2JS1GMzYkLUYjNictRiw2JVEieUYnRjpGPUZALUZlbjYlRmduRmFvRl5vRitGREZERmFvRmRvRmZvLUZBNi1RIj1GJ0ZERkZGSUZLRk1GT0ZRRlNGVUZYRmZvLUYwNiUtRiw2JVEickYnRjpGPS1GIzYkRmFvRkRGZG9GK0ZEJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY9LUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi9RMSZJbnZpc2libGVUaW1lcztGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdRJmZhbHNlRidGLy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRjItSSVtc3ViR0YkNiUtRlA2JVEiY0YnRlNGVi1GLDYkUSIxRidGLy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjItRjM2LVEnJnBsdXM7RidGL0Y8Rj5GQEZCRkRGRkZIRkpGTUYyRitGMi1GUDYlUSJ5RidGU0ZWRjItRlk2JUZlbkYrRltvRjJGXm9GMi1GWTYlRmVuLUYsNiRRIjNGJ0YvRltvRjItRjM2LVEiPUYnRi9GPEY+RkBGQkZERkZGSEZKRk1GMi1JJW1zdXBHRiQ2JUZPRisvJTFzdXBlcnNjcmlwdHNoaWZ0R0Zdb0YyRl5vRjItRl9wNiVGYW9GK0ZhcC1GUDYjUSFGJ0YvJSFHOndeLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjNGJy9GOVEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi9RMSZJbnZpc2libGVUaW1lcztGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdRJmZhbHNlRidGPy8lJmZlbmNlR0ZNLyUqc2VwYXJhdG9yR0ZNLyUpc3RyZXRjaHlHRk0vJSpzeW1tZXRyaWNHRk0vJShsYXJnZW9wR0ZNLyUubW92YWJsZWxpbWl0c0dGTS8lJ2FjY2VudEdGTS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmhuLUZFNi1RIj1GJ0Y/Rk5GUEZSRlRGVkZYRlpGZm5GaW5GRC1JJW1zdXBHRiQ2JS1GLDYlUSJyRidGNUY4LUY8NiRRIjJGJ0Y/LyUxc3VwZXJzY3JpcHRzaGlmdEdGQ0ZELUZFNi1RKCZtaW51cztGJ0Y/Rk5GUEZSRlRGVkZYRlpGZm5GaW5GRC1JKG1zdWJzdXBHRiQ2J0YyLUY8NiRRIjFGJ0Y/RmRvRmdvRkFGREZpb0ZELUZdcDYnRjJGZG9GZG9GZ29GQUYrRj8=JSFHJSFH(como sugerido em [2]). Esta estrat\303\251gia leva aos vetores de colunaJSFH
para o raio e centro, respectivamente. Novamente, observe que o centro est\303\241 em coordenadas relativas \303\240 base ortonormal 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. As coordenadas deste centro, no espa\303\247o dos dados originais, s\303\243o ent\303\243oJSFHJSFH
JSFHNa presen\303\247a de ru\303\255do, a terceira linha da matriz LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= n\303\243o representa um componente zero ao longo da normal ao plano medido por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2KUYrLUklbXN1YkdGJDYlLUYsNidRInVGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0ZCLUkjbW5HRiQ2JFEiMUYnL0ZBUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiLEYnRkkvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zcby1GTzYvUTEmSW52aXNpYmxlVGltZXM7RicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUpcmVhZG9ubHlHRj9GSUZSRlRGVkZYRlpGZm5GaG5Gam5GXW9GNEYrRklGSS8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRictRk82LVEiLkYnRklGUkZURlZGWEZaRmZuRmhuRmpuRl1vRl9vRkk= Assim, uma condi\303\247\303\243o essencial para a validade da solu\303\247\303\243o alg\303\251brica n\303\243o \303\251 atendida. A teoria requer que os dados estejam em um plano para que o m\303\251todo alg\303\251brico ilustrado aqui seja aplic\303\241vel. No entanto, como veremos, a solu\303\247\303\243o alg\303\251brica est\303\241 pr\303\263xima da solu\303\247\303\243o anal\303\255tica obtida a seguir.JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Solu\303\247\303\243o Anal\303\255tica</Font></Text-field>JSFHUma solu\303\247\303\243o anal\303\255tica do problema de ajustar um c\303\255rculo a pontos em tr\303\252s dimens\303\265es come\303\247a com as seguintes defini\303\247\303\265es.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JSFHDefinido igual a zero, a primeira express\303\243o define o plano atrav\303\251s do ponto 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 e tendoLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjFGJy9GOVEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRj8= i + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GOVEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRj8= j + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjNGJy9GOVEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRj8= k JSFHpara seu normal de comprimento um. A segunda express\303\243o \303\251 a dist\303\242ncia do ponto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ5RidGNEY3RjotRjE2JVEiekYnRjRGN0Y+Rj5GPg== para a linha paralela ao normal, e passando por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtaUdGJDYlUSJoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJrRidGNEY3RjotRjE2JVEiakYnRjRGN0Y+Rj5GPg==. Figura 1, o uso de conceitos do c\303\241lculo multivariado \303\251 a base para a obten\303\247\303\243o desta fun\303\247\303\243o.
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o ponto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ5RidGNEY3RjpGMEY+Rj5GPg== \303\251 projetado na linha cuja dire\303\247\303\243o \303\251 dada pelo vetor unit\303\241rio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNidRIm5GJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y6LUYjNiUtSSNtb0dGJDYtUSgmc3JhcnI7RicvRjlRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS9GNkY0L0Y5USdpdGFsaWNGJy9GUkY0RkM=, um tri\303\242ngulo ret\303\242ngulo \303\251 formado. A trigonometria elementar d\303\241 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, ent\303\243o 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. Porque LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1JI21pR0YkNidRIm5GJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y6LUYjNiUtSSNtb0dGJDYtUSgmc3JhcnI7RicvRjlRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS9GNkY0L0Y5USdpdGFsaWNGJy9GUkY0LUYvNiNRIUYnRkM= tem comprimento um, portanto podemos escreverLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkobWZlbmNlZEdGJDYmLUYjNictSSNtaUdGJDYjUSFGJy1GLDYkLUYjNiZGMC1GIzYmLUkmbW92ZXJHRiQ2JS1GMTYnUSJYRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGSC1GIzYkLUkjbW9HRiQ2LVEoJnNyYXJyO0YnL0ZHUSdub3JtYWxGJy8lJmZlbmNlR0ZFLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRl1vRlEvRmpuRkItRk42LVEoJm1pbnVzO0YnRlFGU0ZVRldGWUZlbkZnbkZpbi9GXG9RLDAuMjIyMjIyMmVtRicvRl9vRmVvLUY7NiUtRjE2J1EiQ0YnRkBGQ0ZGRklGS0Zgb0ZRRjBGUUZRLUZONi1RKCZ0aW1lcztGJ0ZRRlNGVUZXRllGZW5GZ25GaW5GZG9GZm8tRjs2JS1GMTYnUSJuRidGQEZDRkZGSUZLRmBvRlFGUS8lJW9wZW5HUSYmcGFyO0YnLyUmY2xvc2VHRmZwLUZONi1RIj1GJ0ZRRlNGVUZXRllGZW5GZ25GaW4vRlxvUSwwLjI3Nzc3NzhlbUYnL0Zfb0ZdcS1GTjYtUSJ+RidGUUZTRlVGV0ZZRmVuRmduRmluRltvRl5vLUYsNiZGOEZRRmRwRmdwLUZONi1RJyZzZG90O0YnRlFGU0ZVRldGWUZlbkZnbkZpbkZbb0Zeby1JI21uR0YkNiRRIjFGJ0ZRRmRxRl9xLUYxNiVRJHNpbkYnRkNGUS1GLDYkLUYjNiQtRjE2JVEnJiM5NTI7RidGQ0ZRRlFGUUZpcC1GMTYlUSJkRicvRkRGQi9GR1EnaXRhbGljRidGUQ==O resultado desejado segue 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Figura 1 Dist\303\242ncia do ponto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ5RidGNEY3RjotRjE2JVEiekYnRjRGN0Y+Rj5GPg== para a linha com dire\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNidRIm5GJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y6LUYjNiUtSSNtb0dGJDYtUSgmc3JhcnI7RicvRjlRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS9GNkY0L0Y5USdpdGFsaWNGJy9GUkY0RkM=
Os dados ruidosos s\303\243o armazenados na matrizLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== = print(); # input placeholderJSFHPara obter uma estimativa de onde o centro da esfera pode estar, formamos listas de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-coodenadas.
JSFHJSFHJSFHA Figura 2 mostra esse plano, os pontos de dados e o centro computado.JSFH
Clique no \303\255cone abaixo para gerar a Figura 2.
p[1] := pointplot3d([convert(Transpose(M),listlist)[]], symbol=solidsphere, color=black, symbolsize=20):
p[c] := pointplot3d(cen, symbol=solidsphere, color=green, symbolsize=20):
p[2] := implicitplot3d(Pxyz, x=-8..6, y=-5..15, z=-1..12, color=red, transparency=.8, style=patchnogrid):
p[3] := display([p[1],p[2],p[c]], scaling=constrained, axes=frame, labels=[x,y,z], orientation=[-125,60], tickmarks=[3,3,3]):
p[3];
Figura 2 Os pontos de dados, o centro computado e o plano de m\303\255nimos quadrados contendo os pontos de dados
JSFHA obten\303\247\303\243o de um gr\303\241fico do c\303\255rculo de ajuste requer muito mais trabalho.JSFHEm um sistema de coordenadas esf\303\251ricas centrado em 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, e definido por
JSFHJSFHJSFHde onde obtemos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTUyO0YnRi9GMkYyRjJGMg== as
JSFHA Figura 3 mostra os pontos ajustados, o c\303\255rculo de ajuste, o plano no qual este c\303\255rculo se encontra e o centro do c\303\255rculo de ajuste.JSFH
Clique no \303\255cone abaixo para gerar a Figura 3.
p[4] := spacecurve([xt,yt,zt],theta=0..2*Pi, color=blue, axes=box,thickness=2):
display([p[3],p[4]], scaling=constrained, labels=[x,y,z], orientation=[-130,50],tickmarks=[3,3,3]);
Figura 3 O c\303\255rculo de m\303\255nimos quadrados e os dados ruidosos aos quais foi ajustado.
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