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PROCESSAMENTO GEOMéTRICOSCANNER DE CURVAS
José Eduardo de Almeida Ayres
Rio de Janeiro, 27 de março de 2018
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Introdução
# Objetivo de amostrar a borda de uma curva em R2:
◦ Pode ser realizado através da função implícita.◦ Também é possível através de uma nuvem de pontos.
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Scanner circular
# Toma-se a equação paramétrica da circunferência de raio re centro (0, 0):
xc � r cos(t)yc � r sin(t)
# Amostram-se m pontos, os quais serão as origens dos raiosdisparados contra a curva:
0 : 2πm
:(2π − 2π
m
)# Um raio iniciado no par (x , y) termina no ponto simétrico
em relação à sua origem (−x ,−y).
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2.9
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Circle scanner, rays = 180, ray intervals = 10
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5Circle scanner, rays = 180, ray intervals = 10
Tempo computacional: 0.01 segundos.
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5Circle scanner, rays = 180, ray intervals = 100
Tempo computacional: 0.03 segundos.
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0.4Circle scanner, rays = 180, ray intervals = 100
Tempo computacional: 0.03 segundos.
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2.6Circle scanner, rays = 180, ray intervals = 100
Tempo computacional: 0.03 segundos.
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Circle scanner, rays = 180, ray intervals = 100
Tempo computacional: 0.03 segundos.
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5Circle scanner, rays = 180, ray intervals = 1000
Tempo computacional: 0.27 segundos.
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5Circle scanner, rays = 180, ray intervals = 10000
Tempo computacional: 1.56 segundos.
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Tempo computacional: 0.27 segundos.
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Tempo computacional: 1.56 segundos.
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2.6Circle scanner, rays = 180, ray intervals = 1000
Tempo computacional: 0.27 segundos.
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Circle scanner, rays = 180, ray intervals = 1000
Tempo computacional: 0.27 segundos.
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5Circle scanner, rays = 720, ray intervals = 1000
Tempo computacional: 0.85 segundos.
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5Circle scanner, rays = 1440, ray intervals = 1000
Tempo computacional: 1.54 segundos.
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Tempo computacional: 0.85 segundos.
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Tempo computacional: 1.54 segundos.
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2.6Circle scanner, rays = 720, ray intervals = 1000
Tempo computacional: 0.85 segundos.
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2.6Circle scanner, rays = 1440, ray intervals = 1000
Tempo computacional: 1.54 segundos.
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0 0.2 0.4 0.6 0.8 1 1.2
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Circle scanner, rays = 720, ray intervals = 1000
Tempo computacional: 0.85 segundos.
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0 0.2 0.4 0.6 0.8 1 1.2
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Circle scanner, rays = 1440, ray intervals = 1000
Tempo computacional: 1.54 segundos.
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Scanner com anteparos
# Toma-se a equação paramétrica da circunferência de raio re centro (0, 0):
xr � r cos(t)yr � r sin(t)
# Amostram-se, novamente m pontos, os quais serão,ordenadamente, usados como extremos dos anteparos.
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Scanner com anteparos
# Dados, pontos sequenciais a e b obtidos da paratrização dacircunferência, temos os pontos iniciais dos n raios de cadaanteparo (nm raios no algoritmo):
t � 0 : 1/(n − 1) : 1
d � b − a
p0(t) � a + td
d⊥ �(−dy , dx)
|d |2d0 � point2line((0, 0), a , b)
p f (t) � p0 + 2d0d⊥
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-5 0 5
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5Aleatory rail scanner, rays = 100, rails = 4, ray intervals = 1000
Tempo computacional: 0.62 segundos.
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0.4Aleatory rail scanner, rays = 100, rails = 4, ray intervals = 1000
Tempo computacional: 0.62 segundos.
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2.35
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2.6Aleatory rail scanner, rays = 100, rails = 4, ray intervals = 1000
Tempo computacional: 0.62 segundos.
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0 0.2 0.4 0.6 0.8 1 1.2
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Aleatory rail scanner, rays = 100, rails = 4, ray intervals = 1000
Tempo computacional: 0.62 segundos.
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5Aleatory rail scanner, rays = 100, rails = 15, ray intervals = 1000
Tempo computacional: 1.44 segundos.
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Tempo computacional: 1.44 segundos.
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2.6Aleatory rail scanner, rays = 100, rails = 15, ray intervals = 1000
Tempo computacional: 1.44 segundos.
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0 0.2 0.4 0.6 0.8 1 1.2
1.2
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1.8
2
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Aleatory rail scanner, rays = 100, rails = 15, ray intervals = 1000
Tempo computacional: 1.44 segundos.
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-5 0 5
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5Aleatory rail scanner, rays = 100, rails = 100, ray intervals = 1000
Tempo computacional: 9.41 segundos.
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Scanner com anteparos aleatórios
# Toma-se a equação paramétrica da circunferência de raio re centro (0, 0), ∆r,∆θ ∼ U(0, 1) e re , θe ∈ [0, 1]:
xr � r(1 + re∆r) cos(t + 2πθr∆θ)yr � r(1 + re∆r) sin(t + 2πθr∆θ)
# Amostram-se, novamente m pontos, os quais serão,ordenadamente, usados como extremos dos anteparos.
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-5 0 5
-5
-4
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0
1
2
3
4
5Aleatory rail scanner, rays = 100, rails = 15, ray intervals = 1000
Tempo computacional: 1.69 segundos.
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-1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6
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0
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0.2
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0.4Aleatory rail scanner, rays = 100, rails = 15, ray intervals = 1000
Tempo computacional: 1.69 segundos.
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2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6Aleatory rail scanner, rays = 100, rails = 15, ray intervals = 1000
Tempo computacional: 1.69 segundos.
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0 0.2 0.4 0.6 0.8 1 1.2
1.2
1.4
1.6
1.8
2
2.2
Aleatory rail scanner, rays = 100, rails = 15, ray intervals = 1000
Tempo computacional: 1.69 segundos.
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-5 0 5
-5
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0
1
2
3
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5Aleatory rail scanner, rays = 100, rails = 100, ray intervals = 1000
Tempo computacional: 8.37 segundos.
![Page 41: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/41.jpg)
-1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6
-0.2
-0.1
0
0.1
0.2
0.3
0.4Aleatory rail scanner, rays = 100, rails = 100, ray intervals = 1000
Tempo computacional: 8.37 segundos.
![Page 42: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/42.jpg)
-0.2 -0.1 0 0.1 0.2 0.3
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6Aleatory rail scanner, rays = 100, rails = 100, ray intervals = 1000
Tempo computacional: 8.37 segundos.
![Page 43: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/43.jpg)
0 0.2 0.4 0.6 0.8 1 1.2
1.2
1.4
1.6
1.8
2
2.2
Aleatory rail scanner, rays = 100, rails = 100, ray intervals = 1000
Tempo computacional: 8.37 segundos.
![Page 44: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/44.jpg)
Scanner para pontos
# Tentar manter as mesmas abordagens anteriores, porémcom novos problemas a se resolver.
# não temos mais disponível o sinal da f (x , y), e é quaseimpossível um raio passar exatamente em um ponto.
# Abordar com distâncias, dados dois pontos p j e p j+1 de umsegmento de raio, e dado um ponto qualquer do conjundode dados ql , a partir da desigualdade triangular:
|p j+1 − ql |1 + |ql − p j |1 ≥ |p j+1 − p j |1
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Scanner para pontos
# O ideal seria possuir a igualdade estrita, o que não épossível na prática, então façamos o uso de uma tolerânciaε:
|p j+1 − ql |1 + |ql − p j |1 − |p j+1 − p j |1 ≤ ε# Coleta-se todos os pontos que satisfaçam a desigualdade e
tomamos a média deles. Em cada raio, no primeirosegmento que possuir algum ponto.
# A escolha do ε pode gerar problemas, como o raioatravessando os primeiros pontos, por se usar um ε muitopequeno. Um valor bom, constatado para o parâmentro foi:
ε �rns
1000
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-5 0 5
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-1
0
1
2
3
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5Points to scan
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0 0.2 0.4 0.6 0.8 1 1.2
1.2
1.4
1.6
1.8
2
2.2
Points to scan
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-0.2 -0.1 0 0.1 0.2 0.3
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6Points to scan
![Page 49: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/49.jpg)
-1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6
-0.2
-0.1
0
0.1
0.2
0.3
0.4Points to scan
![Page 50: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/50.jpg)
-5 0 5
-5
-4
-3
-2
-1
0
1
2
3
4
5Circle scanner points, rays = 180, ray intervals = 100
Tempo computacional: 4.37 segundos.
![Page 51: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/51.jpg)
-1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6
-0.2
-0.1
0
0.1
0.2
0.3
0.4Circle scanner points, rays = 180, ray intervals = 100
Tempo computacional: 4.37 segundos.
![Page 52: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/52.jpg)
-0.2 -0.1 0 0.1 0.2 0.3
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6Circle scanner points, rays = 180, ray intervals = 100
Tempo computacional: 4.37 segundos.
![Page 53: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/53.jpg)
-5 0 5
-5
-4
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-2
-1
0
1
2
3
4
5Circle scanner points, rays = 180, ray intervals = 100
Tempo computacional: 5.12 segundos.
![Page 54: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/54.jpg)
-1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6
-0.2
-0.1
0
0.1
0.2
0.3
0.4Circle scanner points, rays = 180, ray intervals = 100
Tempo computacional: 5.12 segundos.
![Page 55: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/55.jpg)
-0.2 -0.1 0 0.1 0.2 0.3
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6Circle scanner points, rays = 180, ray intervals = 100
Tempo computacional: 5.12 segundos.
![Page 56: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/56.jpg)
0 0.2 0.4 0.6 0.8 1 1.2
1.2
1.4
1.6
1.8
2
2.2
Circle scanner points, rays = 180, ray intervals = 100
Tempo computacional: 5.12 segundos.
![Page 57: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/57.jpg)
-5 0 5
-5
-4
-3
-2
-1
0
1
2
3
4
5Circle scanner points, rays = 180, ray intervals = 1000
Tempo computacional: 42.88 segundos.
![Page 58: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/58.jpg)
-5 0 5
-5
-4
-3
-2
-1
0
1
2
3
4
5Circle scanner points, rays = 180, ray intervals = 1000
Tempo computacional: 41.77 segundos.
![Page 59: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/59.jpg)
-1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6
-0.2
-0.1
0
0.1
0.2
0.3
0.4Circle scanner points, rays = 180, ray intervals = 1000
Tempo computacional: 41.77 segundos.
![Page 60: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/60.jpg)
-0.2 -0.1 0 0.1 0.2 0.3
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6Circle scanner points, rays = 180, ray intervals = 1000
Tempo computacional: 41.77 segundos.
![Page 61: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/61.jpg)
0 0.2 0.4 0.6 0.8 1 1.2
1.2
1.4
1.6
1.8
2
2.2
Circle scanner points, rays = 180, ray intervals = 1000
Tempo computacional: 41.77 segundos.
![Page 62: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/62.jpg)
-5 0 5
-5
-4
-3
-2
-1
0
1
2
3
4
5Rail scanner points, rays = 50, rails = 4, ray intervals = 1000
Tempo computacional: 74 segundos.
![Page 63: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/63.jpg)
-1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6
-0.2
-0.1
0
0.1
0.2
0.3
0.4Rail scanner points, rays = 50, rails = 4, ray intervals = 1000
Tempo computacional: 74 segundos.
![Page 64: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/64.jpg)
-0.2 -0.1 0 0.1 0.2 0.3
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6Rail scanner points, rays = 50, rails = 4, ray intervals = 1000
Tempo computacional: 74 segundos.
![Page 65: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/65.jpg)
0 0.2 0.4 0.6 0.8 1 1.2
1.2
1.4
1.6
1.8
2
2.2
Rail scanner points, rays = 50, rails = 4, ray intervals = 1000
Tempo computacional: 74 segundos.
![Page 66: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/66.jpg)
-5 0 5
-5
-4
-3
-2
-1
0
1
2
3
4
5Rail scanner points, rays = 12, rails = 15, ray intervals = 1000
Tempo computacional: 38 segundos.
![Page 67: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/67.jpg)
-1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6
-0.2
-0.1
0
0.1
0.2
0.3
0.4Rail scanner points, rays = 12, rails = 15, ray intervals = 1000
Tempo computacional: 38 segundos.
![Page 68: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/68.jpg)
-0.2 -0.1 0 0.1 0.2 0.3
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6Rail scanner points, rays = 12, rails = 15, ray intervals = 1000
Tempo computacional: 38 segundos.
![Page 69: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/69.jpg)
0 0.2 0.4 0.6 0.8 1 1.2
1.2
1.4
1.6
1.8
2
2.2
Rail scanner points, rays = 12, rails = 15, ray intervals = 1000
Tempo computacional: 38 segundos.
![Page 70: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/70.jpg)
-5 0 5
-5
-4
-3
-2
-1
0
1
2
3
4
5Rail scanner points, rays = 20, rails = 15, ray intervals = 1000
Tempo computacional: 79.2 segundos.
![Page 71: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/71.jpg)
-1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6
-0.2
-0.1
0
0.1
0.2
0.3
0.4Rail scanner points, rays = 20, rails = 15, ray intervals = 1000
Tempo computacional: 79.2 segundos.
![Page 72: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/72.jpg)
-0.2 -0.1 0 0.1 0.2 0.3
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6Rail scanner points, rays = 20, rails = 15, ray intervals = 1000
Tempo computacional: 79.2 segundos.
![Page 73: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/73.jpg)
0 0.2 0.4 0.6 0.8 1 1.2
1.2
1.4
1.6
1.8
2
2.2
Rail scanner points, rays = 20, rails = 15, ray intervals = 1000
Tempo computacional: 79.2 segundos.
![Page 74: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/74.jpg)
-5 0 5
-5
-4
-3
-2
-1
0
1
2
3
4
5Points to scan
![Page 75: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/75.jpg)
-5 0 5
-5
-4
-3
-2
-1
0
1
2
3
4
5Circle scanner points, rays = 180, ray intervals = 1000
Tempo computacional: 43.2 segundos.
![Page 76: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/76.jpg)
-5 0 5
-5
-4
-3
-2
-1
0
1
2
3
4
5Rail scanner points, rays = 20, rails = 15, ray intervals = 1000
Tempo computacional: 63.8 segundos.
![Page 77: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/77.jpg)
-5 0 5
-5
-4
-3
-2
-1
0
1
2
3
4
5Rail scanner points, rays = 20, rails = 4, ray intervals = 1000
Tempo computacional: 27.7 segundos.
![Page 78: Processamento Geométrico - Scanner de curvasw3.impa.br/~jeaayres/resources/pdf/GP3.pdf · Processamento Geométrico Scannerdecurvas José Eduardo de Almeida Ayres Rio de Janeiro,](https://reader035.vdocuments.com.br/reader035/viewer/2022063018/5fda7b2f33450073786dbc07/html5/thumbnails/78.jpg)