如何通过多边形找到与最大和弦垂直的最长和弦
我有一组由点定义的不规则多边形。我可以找到最大的和弦位置和长度，但是我不确定如何分析这些点以找到垂直于最大和弦的最长和弦位置和长度。
这是我到目前为止所掌握的，一些定义多边形的点的示例数据：
points_ex <- structure(list(V1 = c(68L, 67L, 66L, 66L, 65L, 65L, 64L, 63L,
62L, 61L, 61L, 60L, 59L, 58L, 57L, 56L, 56L, 55L, 55L, 55L, 55L,
54L, 54L, 54L, 54L, 54L, 54L, 54L, 53L, 53L, 53L, 52L, 52L, 52L,
51L, 51L, 50L, 50L, 49L, 49L, 49L, 48L, 48L, 47L, 47L, 46L, 46L,
45L, 45L, 45L, 44L, 44L, 44L, 44L, 44L, 44L, 44L, 43L, 43L, 42L,
42L, 41L, 41L, 41L, 41L, 40L, 40L, 40L, 40L, 40L, 40L, 39L, 39L,
38L, 38L, 38L, 38L, 37L, 37L, 36L, 36L, 36L, 35L, 35L, 35L, 35L,
35L, 34L, 34L, 34L, 33L, 33L, 33L, 32L, 32L, 32L, 32L, 32L, 32L,
32L, 32L, 32L, 32L, 32L, 32L, 32L, 32L, 32L, 32L, 32L, 32L, 32L,
32L, 32L, 32L, 32L, 32L, 32L, 32L, 32L, 32L, 32L, 32L, 33L, 33L,
33L, 33L, 33L, 33L, 33L, 33L, 33L, 33L, 33L, 33L, 33L, 33L, 33L,
33L, 33L, 32L, 32L, 32L, 32L, 32L, 32L, 32L, 31L, 31L, 31L, 31L,
31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L,
30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L,
30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 30L, 31L, 31L,
31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L,
31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L, 31L, 32L, 32L, 33L, 33L,
34L, 34L, 35L, 36L, 36L, 37L, 38L, 39L, 40L, 41L, 42L, 43L, 43L,
44L, 44L, 45L, 46L, 46L, 47L, 47L, 48L, 48L, 49L, 49L, 50L, 51L,
51L, 51L, 52L, 52L, 52L, 52L, 53L, 53L, 53L, 53L, 54L, 54L, 54L,
54L, 54L, 54L, 54L, 53L, 53L, 53L, 53L, 53L, 53L, 52L, 52L, 52L,
51L, 51L, 51L, 51L, 51L, 51L, 51L, 51L, 52L, 52L, 52L, 52L, 52L,
53L, 53L, 54L, 54L, 55L, 55L, 56L, 57L, 58L, 59L, 60L, 61L, 62L,
63L, 64L, 65L, 66L, 67L, 68L, 69L, 70L, 71L, 72L, 73L, 74L, 75L,
76L, 77L, 78L, 79L, 80L, 81L, 82L, 83L, 84L, 84L, 85L, 85L, 86L,
86L, 86L, 87L, 87L, 88L, 88L, 88L, 89L, 89L, 90L, 90L, 91L, 91L,
92L, 93L, 94L, 95L, 96L, 97L, 98L, 99L, 100L, 101L, 102L, 103L,
104L, 105L, 105L, 106L, 106L, 107L, 108L, 109L, 109L, 110L, 111L,
112L, 113L, 114L, 115L, 116L, 117L, 118L, 118L, 119L, 120L, 121L,
122L, 123L, 123L, 124L, 125L, 126L, 126L, 127L, 127L, 127L, 127L,
127L, 128L, 128L, 128L, 128L, 128L, 128L, 128L, 128L, 128L, 129L,
129L, 129L, 129L, 129L, 129L, 130L, 130L, 130L, 130L, 131L, 131L,
131L, 131L, 131L, 132L, 132L, 132L, 132L, 132L, 133L, 133L, 133L,
133L, 133L, 133L, 133L, 132L, 132L, 132L, 131L, 131L, 130L, 130L,
129L, 129L, 128L, 128L, 128L, 128L, 128L, 128L, 128L, 128L, 128L,
129L, 129L, 129L, 129L, 129L, 129L, 129L, 129L, 129L, 129L, 128L,
127L, 126L, 125L, 125L, 124L, 123L, 123L, 122L, 121L, 120L, 120L,
119L, 119L, 118L, 117L, 117L, 116L, 115L, 115L, 115L, 114L, 114L,
113L, 113L, 112L, 111L, 111L, 110L, 110L, 109L, 109L, 109L, 109L,
108L, 108L, 108L, 108L, 107L, 107L, 107L, 107L, 106L, 106L, 106L,
106L, 105L, 105L, 105L, 105L, 104L, 104L, 104L, 104L, 103L, 103L,
103L, 103L, 103L, 102L, 102L, 102L, 102L, 102L, 102L, 102L, 102L,
102L, 102L, 102L, 102L, 101L, 101L, 101L, 101L, 101L, 101L, 101L,
100L, 100L, 100L, 100L, 100L, 100L, 100L, 99L, 99L, 99L, 99L,
99L, 98L, 98L, 98L, 98L, 98L, 98L, 98L, 98L, 98L, 98L, 97L, 97L,
97L, 97L, 97L, 97L, 96L, 96L, 96L, 96L, 96L, 96L, 96L, 96L, 96L,
96L, 95L, 95L, 95L, 95L, 95L, 95L, 95L, 95L, 95L, 95L, 95L, 95L,
94L, 94L, 94L, 94L, 93L, 93L, 92L, 92L, 91L, 90L, 90L, 89L, 89L,
89L, 88L, 88L, 88L, 88L, 88L, 87L, 87L, 87L, 86L, 86L, 86L, 85L,
84L, 83L, 82L, 81L, 80L, 79L, 78L, 77L, 76L, 75L, 74L, 73L, 72L,
71L, 70L, 69L), V2 = c(20L, 21L, 22L, 23L, 24L, 25L, 26L, 27L,
28L, 29L, 30L, 31L, 32L, 33L, 34L, 35L, 36L, 37L, 38L, 39L, 40L,
41L, 42L, 43L, 44L, 45L, 46L, 47L, 48L, 49L, 50L, 51L, 52L, 53L,
54L, 55L, 56L, 57L, 58L, 59L, 60L, 61L, 62L, 63L, 64L, 65L, 66L,
67L, 68L, 69L, 70L, 71L, 72L, 73L, 74L, 75L, 76L, 77L, 78L, 79L,
80L, 81L, 82L, 83L, 84L, 85L, 86L, 87L, 88L, 89L, 90L, 91L, 92L,
93L, 94L, 95L, 96L, 97L, 98L, 99L, 100L, 101L, 102L, 103L, 104L,
105L, 106L, 107L, 108L, 109L, 110L, 111L, 112L, 113L, 114L, 115L,
116L, 117L, 118L, 119L, 120L, 121L, 122L, 123L, 124L, 125L, 126L,
127L, 128L, 129L, 130L, 131L, 132L, 133L, 134L, 135L, 136L, 137L,
138L, 139L, 140L, 141L, 142L, 143L, 144L, 145L, 146L, 147L, 148L,
149L, 150L, 151L, 152L, 153L, 154L, 155L, 156L, 157L, 158L, 159L,
160L, 161L, 162L, 163L, 164L, 165L, 166L, 167L, 168L, 169L, 170L,
171L, 172L, 173L, 174L, 175L, 176L, 177L, 178L, 179L, 180L, 181L,
182L, 183L, 184L, 185L, 186L, 187L, 188L, 189L, 190L, 191L, 192L,
193L, 194L, 195L, 196L, 197L, 198L, 199L, 200L, 201L, 202L, 203L,
204L, 205L, 206L, 207L, 208L, 209L, 210L, 211L, 212L, 213L, 214L,
215L, 216L, 217L, 218L, 219L, 220L, 221L, 222L, 223L, 224L, 225L,
226L, 227L, 228L, 229L, 230L, 231L, 232L, 233L, 234L, 235L, 236L,
237L, 238L, 239L, 240L, 241L, 242L, 243L, 244L, 245L, 246L, 247L,
248L, 249L, 250L, 251L, 252L, 253L, 254L, 255L, 256L, 257L, 258L,
259L, 260L, 261L, 262L, 263L, 264L, 265L, 266L, 267L, 268L, 269L,
270L, 271L, 272L, 273L, 274L, 275L, 276L, 277L, 278L, 279L, 280L,
281L, 282L, 283L, 284L, 285L, 286L, 287L, 288L, 289L, 290L, 291L,
292L, 293L, 294L, 295L, 296L, 297L, 298L, 299L, 300L, 301L, 302L,
303L, 304L, 305L, 306L, 307L, 308L, 308L, 308L, 309L, 309L, 309L,
310L, 310L, 310L, 311L, 311L, 312L, 312L, 312L, 312L, 312L, 312L,
312L, 311L, 311L, 310L, 310L, 310L, 310L, 309L, 308L, 307L, 306L,
305L, 304L, 303L, 302L, 301L, 300L, 299L, 298L, 297L, 296L, 295L,
294L, 293L, 292L, 291L, 290L, 289L, 288L, 287L, 286L, 285L, 284L,
283L, 282L, 281L, 280L, 279L, 279L, 278L, 277L, 276L, 275L, 274L,
273L, 272L, 271L, 270L, 269L, 268L, 267L, 266L, 265L, 264L, 263L,
262L, 261L, 260L, 259L, 258L, 257L, 256L, 255L, 254L, 253L, 252L,
252L, 251L, 250L, 249L, 248L, 247L, 246L, 245L, 244L, 243L, 242L,
241L, 240L, 239L, 238L, 237L, 236L, 235L, 234L, 233L, 232L, 231L,
230L, 229L, 228L, 227L, 226L, 225L, 224L, 223L, 222L, 221L, 220L,
219L, 218L, 217L, 216L, 215L, 214L, 213L, 212L, 211L, 210L, 209L,
208L, 207L, 206L, 205L, 204L, 203L, 202L, 201L, 200L, 199L, 198L,
197L, 196L, 195L, 194L, 193L, 192L, 191L, 190L, 189L, 188L, 187L,
186L, 185L, 184L, 183L, 182L, 181L, 180L, 179L, 178L, 177L, 176L,
175L, 174L, 173L, 172L, 171L, 170L, 169L, 168L, 167L, 166L, 165L,
164L, 163L, 162L, 161L, 160L, 159L, 158L, 157L, 156L, 155L, 154L,
153L, 152L, 151L, 150L, 149L, 148L, 147L, 146L, 145L, 144L, 143L,
142L, 141L, 140L, 139L, 138L, 137L, 136L, 135L, 134L, 133L, 132L,
131L, 130L, 129L, 128L, 127L, 126L, 125L, 124L, 123L, 122L, 121L,
120L, 119L, 118L, 117L, 116L, 115L, 114L, 113L, 112L, 111L, 110L,
109L, 108L, 107L, 106L, 105L, 104L, 103L, 102L, 101L, 100L, 99L,
98L, 97L, 96L, 95L, 94L, 93L, 92L, 91L, 90L, 89L, 88L, 87L, 86L,
85L, 84L, 83L, 82L, 81L, 80L, 79L, 78L, 77L, 76L, 75L, 74L, 73L,
72L, 71L, 70L, 69L, 68L, 67L, 66L, 65L, 64L, 63L, 62L, 61L, 60L,
59L, 58L, 57L, 56L, 55L, 54L, 53L, 52L, 51L, 50L, 49L, 48L, 47L,
46L, 45L, 44L, 43L, 42L, 41L, 40L, 39L, 38L, 37L, 36L, 35L, 34L,
33L, 32L, 31L, 30L, 29L, 28L, 27L, 26L, 26L, 26L, 25L, 25L, 24L,
24L, 23L, 23L, 22L, 22L, 21L, 21L, 20L, 20L, 20L)), class = "data.frame", row.names = c(NA,
-613L))
看起来像这样：
我可以找到最大的和弦并绘制：
# draw max dimension points and line
suppressPackageStartupMessages(library(tidyverse))
df_dist = data.frame(as.matrix(dist(cbind(points_ex$V1,points_ex$V2))))
df_dist_x = df_dist %>%
mutate(row.1 = 1:nrow(df_dist)) %>%
mutate(Y = paste0("Y", row_number())) %>%
gather(X, distance, X1:nrow(.)) %>%
select(X, Y, distance) %>%
mutate_at(vars(X, Y), parse_number)
df_dist_x_max <-
df_dist_x %>%
dplyr::filter(distance == max(distance))
points(points_ex[df_dist_x_max$X[1],], col = "red", cex = 2)
points(points_ex[df_dist_x_max$X[2],], col = "red", cex = 2.5)
segments(points_ex[df_dist_x_max$X[1], 'V1'],
points_ex[df_dist_x_max$X[1], 'V2'],
points_ex[df_dist_x_max$X[2], 'V1'],
points_ex[df_dist_x_max$X[2], 'V2'],
col = "green")
这就是我一直试图使最长的和弦垂直于最大长度的和弦的方法：
# transform the points and lines into spatial objects
library(sf)
library(sp)
library(rgeos)
points_sf <- st_as_sf(points_ex, coords = c("V1", "V2"))
newline = matrix(c(points_ex[df_dist_x_max$X[1], 'V1'],
points_ex[df_dist_x_max$X[1], 'V2'],
points_ex[df_dist_x_max$X[2], 'V1'],
points_ex[df_dist_x_max$X[2], 'V2']), byrow = T, nrow = 2)
spline <- as(st_as_sfc(st_as_text(st_linestring(newline))), "Spatial") # there is probably a more straighforward solution...
position <- gProject(spline, as(points_sf, "Spatial"))
position <- coordinates(gInterpolate(spline, position))
colnames(position) <- c("X2", "Y2")
segments <-
data.frame(st_coordinates(points_sf), position)
segments$dist <- NULL
for(i in 1:nrow(segments)){
segments$dist[i] <-
proxy::dist(data.frame(segments$X[i], segments$Y[i]),
data.frame(segments$X2[i], segments$Y2[i]))
}
# max width perpendicular to length axis
max_segment <- segments[which.max(segments$dist), ]
max_segment <- segments[segments$Y == max_segment$Y, ]
segments(max_segment$X[1], max_segment$Y[1],
max_segment$X2[1], max_segment$Y2[1],
col = "purple")
segments(max_segment$X[2], max_segment$Y[2],
max_segment$X2[2], max_segment$Y2[2],
col = "purple")
看起来不错，但我的问题是，这种用于查找与最大和弦垂直的最长和弦的方法仅在最大和弦的一侧查找与多边形边缘的最长距离。
我不知道如何测量垂直于最大弦的多边形的边缘到边缘的每个距离。
这意味着我的方法根本无法很好地泛化，此处将其应用于另一个多边形（此处的数据为https://pastebin.com/XpiB6UnX，因为dput的输出使该帖子过长了）
显然，这很不好，因为两个紫色段不应位于绿色段的同一侧，而且对于垂直于最大和弦的最长和弦而言，它看起来根本不正确。
如何稳健地找到与最大和弦垂直的最长和弦？
2 个答案:
答案 0 :(得分：4)
这是一个完整的解决方案。首先是一些功能：
## this returns the i,j of the largest elements in matrix `m`
findmax <- function(m){
v = which.max(m) - 1
c(v %% nrow(m)+1, v %/% nrow(m)+1)
}
## Return an sf line through a point at an angle of a given length
pline <- function(pt, angle, length){
st_linestring(
cbind(
pt[1] + c(length,-length) * sin(angle),
pt[2] + c(length,-length) * cos(angle)
)
)
}
## return the line that is the chord at angle perp.angle of length through any of the polygon vertices
max_perp_chord <- function(polygon, perp.angle, length){
## get polygon vertices
pts = st_coordinates(polygon)[,c(1,2)]
## return the perpendicular lines
perplines = lapply(1:nrow(pts), function(i){
## through the i-th vertex
xy = pts[i,,drop=FALSE]
perpline = pline(xy, perp.angle, length)
## intersect it with the polygon
inters = st_intersection(polygon, perpline)
inters
}
)
## get the vector of intersection lengths, find the largest
perplengths = unlist(lapply(perplines, st_length))
longest = which.max(perplengths)
## return the longest line
perplines[[longest]]
}
### find the max length chord across all pairs of vertex points
max_chord <- function(polygon){
## get polygon coordinates
xy = st_coordinates(polygon)[,1:2]
## compute the distance matrix and find largest element
df_dist = as.matrix(dist(xy))
maxij = findmax(df_dist)
## those elements define the largest chord
chord = rbind(
xy[maxij[1],],
xy[maxij[2],]
)
chord
}
find_max_chord <- function(spolygon, chord=max_chord(spolygon)){
## Now compute the length and angle of the longest chord
chord.length = sqrt(diff(chord[,1])^2 + diff(chord[,2])^2)
chord.theta = atan2(diff(chord[,1]), diff(chord[,2]))
## The perpendicular is at this angle plus pi/2 radians
perp = chord.theta + pi/2
max_perp_chord(spolygon, perp, chord.length)
}
这是一起进行的方式。唯一的依存关系是sf-除了使用基数R之外没有其他用途：
library(sf)
接下来，将数据压缩到sf多边形对象中：
## construct an sf polygon from points:
polygon = st_polygon(list(as.matrix(rbind(points_ex, points_ex[1,]))))
获取顶点之间的最大弦长：
chord = max_chord(polygon)
绘制多边形，和弦以及和弦点：
plot(polygon)
points(chord, col="red",cex=2)
lines(chord,col="green",lwd=2)
现在吃肉。通过一个多边形顶点获取与最大和弦垂直的和弦：
## get the max perpendicular chord
pchord = find_max_chord(polygon, chord)
绘制它。
plot(pchord,add=TRUE)
我已经在另一个示例上对此进行了测试，但不是您发布的第二个示例...应该可以，但是...
它确实...
答案 1 :(得分：1)
如果多边形是单调的，则可以应用下一种方法：
我假设旋转多边形以使最大长度的和弦垂直（如您的两个图像所示）。
从最高点开始。沿CCW方向遍历左侧部分，沿顺时针方向遍历右侧部分。
关键时刻：最长的弦必须接触某个多边形顶点（因为在边缘的两个内部点之间的线段上无法达到最大值）。
因此，获得Y坐标最高的顶点（在左侧或右侧）。计算相对侧边上的点。得到长度。
获取具有下一个Y坐标的顶点。计算相对侧边上的点。得到长度。继续。
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