#' Calculate shape indices listed in Borg 1998
#'
#' @description used as internal function. see \code{?geometryVariables}
#' and Borg 1998: Vergleichende Analyse von Formindizes zur Charakterisierung
#' von Landschaftsobjekten unter oekologischen Aspekten.
#' Zeitschrift fuer Photogrammetrie und Fernerkundung 4: 108-119.
#'
#' @param Ar Area of the entities
#' @param Ur Perimeter of the entities
#' @param De Diameter of the inner circle
#' @param Du Diameter of the outer circle
#'
#' @return A RasterStack
#'
#' @author Hanna Meyer
#'
#' @seealso \code{\link{geometryVariables}}.
borgIndices<-function(Ar,Ur,De,Du){
Dk <- sqrt((Ar/pi))*2 # Diameter of the circle of area Ar
Uk <- Dk*pi # perimeter of the circle of area Ar
Ue <- De*pi #perimeter of the inner circle
Uu <- Du*pi #perimeter of the outer circle
Au <- pi*((Du/2)^2) #area of outer circle
Ae <- pi*((De/2)^2) #area of the inner circle
CI1 <- Ue/Uk
CO1 <- Uk/Uu
CI2 <- (Uk-Ue)/Uk
CO2 <- (Uu-Uk)/Uu
CCI1 <- (CI1+CO1)/2
CCI2 <- 1-sqrt(CI2*CO2)
### orientation at re circle of same area
CO <- (2*sqrt(pi*Ar))/Ur
SHD <- Ur/(2*sqrt(pi*Ar))
C1 <- (4*Ar)/(Ur^2)
E <- Ur/(sqrt(Ar))
TR <- (4*pi*Ar)/(Ur^2)
###orientation at the smallest outer circle
CR <- Ar/Au
C2 <- sqrt(Ar/Au)
FR <- Ar/Du^2
EI <- (pi*Du*(0.5*Du))/Ar
SF1 <- Uu/Ur
GSI <- Ur/Du
###orientation at the largest inner circle
SF2 <- Ae/Ar
###orientation on inner and outer circle
C3 <- sqrt(De/Du)
SF3 <- (Au-Ae)/Ar
result <- stack(CI1,CO1,CI2,CO2,CCI1,CCI2,CO,SHD,C1,E,TR,CR,
C2,FR,EI,SF1,GSI,SF2,C3,SF3)
names(result) <- c("CI1","CO1","CI2","CO2","CCI1","CCI2","CO","SHD","C1","E",
"TR","CR","C2","FR","EI","SF1","GSI","SF2","C3","SF3")
return(result)
}
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