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## getMinCirc() requires convex hull without collinear points
## https://stackoverflow.com/q/64866942/484139
## identify collinear points on convex hull
## assumes that points in xy are already oredered
idCollinear <-
function(xy) {
if(!is.matrix(xy)) { stop("xy must be a matrix") }
if(!is.numeric(xy)) { stop("xy must be numeric") }
if(nrow(xy) < 3L) { stop("xy must have at least 3 points") }
if(ncol(xy) != 2L) { stop("xy must have 2 columns") }
n <- nrow(xy)
idx <- seq_len(n)
post <- (idx %% n) + 1 # next point in S
prev <- idx[order(post)] # previous point in S
del <- integer(0)
## check all sets of 3 consecutive points if they lie in 1D sub-space
for(i in idx) {
pts <- rbind(xy[prev[i], ],
xy[i, ],
xy[post[i], ])
pts_rank <- qr(scale(pts, center=TRUE, scale=FALSE))$rank
if(pts_rank < 2L) {
del[length(del) + 1] <- i
}
}
del
}
## circle defined by three points
getCircleFrom3 <-
function(xy) {
if(!is.matrix(xy)) { stop("xy must be a matrix") }
if(!is.numeric(xy)) { stop("xy must be numeric") }
if(any(dim(xy) != c(3L, 2L))) { stop("xy must be (3x2)-matrix") }
aa <- xy[1, ]
bb <- xy[2, ]
cc <- xy[3, ]
y <- xy[ , 2]
xDeltaA <- bb[1] - aa[1]
yDeltaA <- bb[2] - aa[2]
xDeltaB <- cc[1] - bb[1]
yDeltaB <- cc[2] - bb[2]
xDeltaC <- cc[1] - aa[1]
yDeltaC <- cc[2] - aa[2]
## check if the points are collinear: qr(xy)$rank == 1, or:
## determinant of difference matrix = 0, no need to use det()
dMat <- rbind(c(xDeltaA, yDeltaA), c(xDeltaB, yDeltaB))
if(isTRUE(all.equal(dMat[1,1]*dMat[2,2] - dMat[1,2]*dMat[2,1], 0, check.attributes=FALSE))) {
## define the circle as the one that's centered between the points
rangeX <- range(c(aa[1], bb[1], cc[1]))
rangeY <- range(c(aa[2], bb[2], cc[2]))
ctr <- c(rangeX[1] + 0.5*diff(rangeX), rangeY[1] + 0.5*diff(rangeY))
rad <- sqrt((0.5*diff(rangeX))^2 + (0.5*diff(rangeY))^2)
} else {
rad <- prod(dist(xy)) / (2 * abs(det(cbind(xy, 1)))) # circle radius
v1 <- rowSums(xy^2) # first vector in the numerator
v2x <- c( xDeltaB, -xDeltaC, xDeltaA) # 2nd vector numerator for Mx
v2y <- c(-yDeltaB, yDeltaC, -yDeltaA) # 2nd vector numerator for My
ctr <- c(t(v1) %*% v2y, t(v1) %*% v2x) / c(2 * (t(y) %*% v2x)) # center
}
return(list(ctr=ctr, rad=rad))
}
## alternative formulas
## detABC <- (cx-ax)*(cy+ay) + (bx-cx)*(by+cy) + (ax-bx)*(ay+by)
## Mx <- 0.5 * (((ax^2 + ay^2)*(by-cy) + (bx^2+by^2)*(cy-ay) + (cx^2+cy^2)*(ay-by)) /
## (ay*(cx-bx) + by*(ax-cx) + cy*(bx-ax)))
## My <- 0.5 * (((ax^2 + ay^2)*(cx-bx) + (bx^2+by^2)*(ax-cx) + (cx^2+cy^2)*(bx-ax)) /
## (ay*(cx-bx) + by*(ax-cx) + cy*(bx-ax)))
## rad <- dist(rbind(xy, ctr))[3]
## vertex that produces the circle with the maximum radius
## used in getMinCircle()
getMaxRad <-
function(xy, S) {
if(!is.matrix(xy)) { stop("xy must be a matrix") }
if(!is.numeric(xy)) { stop("xy must be numeric") }
if(ncol(xy) != 2L) { stop("xy must have two columns") }
if(!is.numeric(S)) { stop("S must be numeric") }
if((length(S) < 2L) || (nrow(xy) < 2L)) { stop("There must be at least two points") }
if(length(S) > nrow(xy)) { stop("There can only be as many indices in S as points in xy") }
n <- length(S) # number of points
Sidx <- seq(along=S) # index for points
rads <- numeric(n) # radii for all circles
post <- (Sidx %% n) + 1 # next point in S
prev <- Sidx[order(post)] # previous point in S
for(i in Sidx) {
pts <- rbind(xy[S[prev[i]], ], xy[S[i], ], xy[S[post[i]], ])
rads[i] <- getCircleFrom3(pts)$rad # circle radius
}
return(which.max(rads))
}
## angle (in degrees) at B in triangle ABC
getAngleTri <-
function(xy, deg=TRUE) {
if(!is.matrix(xy)) { stop("xy must be a matrix") }
if(!is.numeric(xy)) { stop("xy must be numeric") }
if(any(dim(xy) != c(3L, 2L))) { stop("xy must be (3x2)-matrix") }
d <- dist(xy)
dAB <- d[1]
dAC <- d[2]
dBC <- d[3]
## dAB*dAC should not be 0
if(isTRUE(all.equal(dAB*dAC, 0, check.attributes=FALSE))) {
stop("Some edges have zero length")
}
Wabc <- (dAB^2 + dBC^2 - dAC^2)
arc <- acos(Wabc / (2*dAB*dBC)) # angle in radians
ang <- ifelse(deg, arc*180/pi, arc) # return angle in degree or radians
return(ang)
}
## checks if the angle at B in triangle ABC is bigger than 90 degrees
isBiggerThan90 <-
function(xy) {
if(!is.matrix(xy)) { stop("xy must be a matrix") }
if(!is.numeric(xy)) { stop("xy must be numeric") }
if(any(dim(xy) != c(3L, 2L))) { stop("xy must be (3x2)-matrix") }
d <- dist(xy)
dAB <- d[1]
dAC <- d[2]
dBC <- d[3]
return((dAB^2 + dBC^2 - dAC^2) < 0)
}
## minimal enclosing circle
getMinCircle <-
function(xy) {
UseMethod("getMinCircle")
}
getMinCircle.data.frame <-
function(xy) {
xy <- getXYmat(xy, xyTopLeft=FALSE, relPOA=FALSE)
NextMethod("getMinCircle")
}
getMinCircle.default <-
function(xy) {
if(!is.matrix(xy)) { stop("xy must be a matrix") }
if(!is.numeric(xy)) { stop("xy must be numeric") }
if(nrow(xy) < 2L) { stop("There must be at least two points") }
if(ncol(xy) != 2L) { stop("xy must have two columns") }
H <- chull(xy) # convex hull indices (vertices ordered clockwise)
hPts <- xy[H, ] # points that make up the convex hull
## remove collinear points on convex hull, if any
del <- idCollinear(hPts)
if(length(del) >= 1L) {
H <- H[-del]
hPts <- hPts[-del, ]
if(length(H) < 2L) {
stop("less than 2 points left after removing collinear points on convex hull")
}
}
## min circle may touch convex hull in only two points
## if so, it is centered between the hull points with max distance
maxPD <- getMaxPairDist(hPts)
idx <- maxPD$idx # index of points with max distance
rad <- maxPD$d / 2 # half the distance -> radius
rangeXY <- hPts[idx, ]
ctr <- rangeXY[1, ] + 0.5 * diff(rangeXY)
## check if circle centered between hPts[pt1Idx, ] and hPts[pt2Idx, ]
## contains all points (all distances <= rad)
dst2ctr <- dist(rbind(ctr, hPts[-idx, ])) # distances to center
if(all(as.matrix(dst2ctr)[-1, 1] <= rad)) { # if all <= rad, we're done
tri <- rbind(hPts[idx, ], ctr)
return(getCircleFrom3(tri))
}
## min circle touches hull in three points - Skyum algorithm
S <- H # copy of hull indices that will be changed
while(length(S) >= 2L) {
n <- length(S) # number of remaining hull vertices
Sidx <- seq(along=S) # index for vertices
post <- (Sidx %% n) + 1 # next vertex in S
prev <- Sidx[order(post)] # previous vertex in S
mIdx <- getMaxRad(xy, S) # idx for maximum radius
## triangle where mIdx is vertex B in ABC
Smax <- rbind(xy[S[prev[mIdx]], ],
xy[S[mIdx], ],
xy[S[post[mIdx]], ])
## if there's only two hull vertices, we're done
if(n <= 2L) { break }
## check if angle(ABC) is > 90
## if so, eliminate B - if not, we're done
if(isBiggerThan90(Smax)) { S <- S[-mIdx] } else { break }
}
return(getCircleFrom3(Smax))
}
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