根据R中的顶点坐标计算多边形的内角

Question

给定多个顶点的2D坐标，我想计算相应多边形的内角。

这是一个数据框，其中包含一些多边形及其顶点的坐标。问题是顶点没有排序。

structure(list(x = c(173L, 173L, 185L, 190L, 231L, 267L, 185L, 
190L, 233L, 260L, 190L, 231L, 260L, 230L, 230L, 172L, 233L, 230L, 
231L, 267L, 185L, 172L, 233L, 231L, 231L), y = c(299L, 299L, 
321L, 360L, 361L, 377L, 321L, 360L, 363L, 309L, 360L, 361L, 309L, 
322L, 322L, 378L, 363L, 322L, 391L, 377L, 321L, 378L, 363L, 391L, 
361L), polygonID = c(2L, 4L, 5L, 3L, 6L, 7L, 2L, 5L, 7L, 6L, 2L, 
3L, 4L, 5L, 6L, 2L, 3L, 4L, 3L, 6L, 4L, 3L, 6L, 7L, 5L)), row.names = c(1L, 
2L, 3L, 4L, 5L, 6L, 7L, 9L, 10L, 12L, 13L, 14L, 15L, 16L, 17L, 
19L, 20L, 21L, 24L, 25L, 27L, 29L, 30L, 31L, 34L), class = "data.frame")

从本质上讲，我会从一个顶点到另一个顶点来计算边缘/矢量，并使用它们来计算内角。但是我不清楚如何以算法方式进行此操作。任何提示都非常感谢。

Answer 1

这是一个镜头：

shp <- structure(list(x = c(173L, 173L, 185L, 190L, 231L, 267L, 185L, 
190L, 233L, 260L, 190L, 231L, 260L, 230L, 230L, 172L, 233L, 230L, 
231L, 267L, 185L, 172L, 233L, 231L, 231L), y = c(299L, 299L, 
321L, 360L, 361L, 377L, 321L, 360L, 363L, 309L, 360L, 361L, 309L, 
322L, 322L, 378L, 363L, 322L, 391L, 377L, 321L, 378L, 363L, 391L, 
361L), polygonID = c(2L, 4L, 5L, 3L, 6L, 7L, 2L, 5L, 7L, 6L, 
                     2L, 3L, 4L, 5L, 6L, 2L, 3L, 4L, 3L, 6L, 4L, 3L, 6L, 7L, 5L)), class = "data.frame",
row.names = c(NA, -25L))

aa <- shp[ shp$polygonID == 2, ]
aa <- aa[ chull(aa[,1:2]), ]
aa
#      x   y polygonID
# 7  185 321         2
# 1  173 299         2
# 16 172 378         2
# 11 190 360         2

现在aa是一个4点多边形，按顺时针方向排序。现在让我们对一些计算进行硬编码：

indm1 <- c(ind[-1], ind[1])
indp1 <- c(ind[length(ind)], ind[-length(ind)])
angles <- ((atan2(aa$y[indm1] - aa$y[ind], aa$x[indm1] - aa$x[ind]) -
              atan2(aa$y[indp1] - aa$y[ind], aa$x[indp1] - aa$x[ind])) * 180 / pi) %% 360
cbind(indm1,ind,indp1)
#      indm1 ind indp1
# [1,]     2   1     4
# [2,]     3   2     1
# [3,]     4   3     2
# [4,]     1   4     3
angles
# [1] 158.69530  29.33568  44.27478 127.69424

让我们看看（最初，我感到困惑的是，它在视觉上是不相关的，直到我意识到长宽比已经关闭，{ergo asp=1）。

plot(y~x, data=aa, type='l', asp=1)
with(aa, text(x-5, y, seq_len(nrow(aa)), col="red"))
with(aa, text(x+5, y, round(angles, 0)))

好的，让我们尝试将其正式化：

getangles <- function(aa) {
  aa <- aa[chull(aa[,1:2]),]
  ind <- seq_len(nrow(aa))
  indm1 <- c(ind[-1], ind[1])
  indp1 <- c(ind[length(ind)], ind[-length(ind)])
  ((atan2(aa$y[indm1] - aa$y[ind], aa$x[indm1] - aa$x[ind]) -
      atan2(aa$y[indp1] - aa$y[ind], aa$x[indp1] - aa$x[ind])) * 180 / pi) %% 360
}
by(shp, shp$polygonID, getangles)
# shp$polygonID: 2
# [1] 158.69530  29.33568  44.27478 127.69424
# ------------------------------------------------------------ 
# shp$polygonID: 3
# [1] 130.91438 136.39718 133.60282  57.42594  81.65967
# ------------------------------------------------------------ 
# shp$polygonID: 4
# [1]  29.98564  54.83259 119.88349 155.29828
# ------------------------------------------------------------ 
# shp$polygonID: 5
# [1] 92.74183 81.42121 98.70294 87.13402
# ------------------------------------------------------------ 
# shp$polygonID: 6
# [1]  72.44870 111.95989 136.46880 157.38014  61.74247
# ------------------------------------------------------------ 
# shp$polygonID: 7
# [1] 71.70548 64.66388 43.63064

（舍入/模数可能存在一些问题，我留给您来美化和验证其他问题。）