This article shows how to draw a geometry from given line elements. We provide three examples:

Note that we use spherical coordinate (r,theta,phi), but plot on cartesian coordinate (x,y,z).

0. Main Program

For each line element, the main program composes of:

Calling functions
Initialization (we use the following initialization for every example)

init <- c(1., 2.001, pi/2., 0., 0.0001, 10, 5, 10) # M, r > 2M, theta, phi, distance, n_grid, r_grid, phi_grid
view_angle <- c(45.,0.) # theta, phi in degree

Constructing coordinates from the line element in spherical coordinate
Transforming from spherical to cartesian coordinate
Visualization

1. Circle

source("Circle.R")
source("Delta_Phi_Circle.R")
source("Sphere_2_XYZ.R")
sphere_coor <- Circle(init)
xyz_coor <- Sphere_2_XYZ(sphere_coor)
plot(x = xyz_coor[,1],y = xyz_coor[,2], type="l")

2. Sphere

## Warning: package 'plot3D' was built under R version 3.2.2

3. Wormhole

source("WormHole.R")
source("Delta_R_WormHole.R")
source("Show3D_WormHole.R")
sphere_coor <- WormHole(init)
length <- length(sphere_coor)
xyz_coor <- vector(mode = "list", length = length)
for(i in 1:length){
     ll <- length(sphere_coor[[i]])
     xyz_coor[[i]] <- vector(mode = "list",length = ll)
     for(j in 1:ll)
     xyz_coor[[i]][[j]] <- Sphere_2_XYZ(sphere_coor[[i]][[j]])
}
Show3D_WormHole(xyz_coor, view_angle)

4. Codes

4.1. Circle.R

Circle<-function(init){
     r <- init[2]
     theta <- init[3]
     #phi <- init[4]
     distance <- init[5]
     #n_grid <- init[6]
     delta_phi <- Delta_Phi_Circle(r,theta,distance)
     n_grid <- as.integer((2*pi/delta_phi) + 1)
     length <- n_grid+1
     sphere_coor <- array(data = 0,dim = c(length,3))
     for (i in 1:length){
          sphere_coor[i,1] <- r
          sphere_coor[i,2] <- theta
          sphere_coor[i,3] <- (i-1.) * delta_phi
     }
     sphere_coor
}

4.2. Delta_Phi_Circle.R

Delta_Phi_Circle<-function(r,theta,distance){
     delta_phi <- abs(distance / ( r*sin(theta) ))
     delta_phi
}

4.3. Sphere_2_XYZ.R

Sphere_2_XYZ<-function(sphere_coor){
     length <- dim(sphere_coor)[1]
     xyz <- array(data = 0,dim = c(length,3))
     for(i in 1:length){
          r <- sphere_coor[i,1]
          theta <- sphere_coor[i,2]
          phi <- sphere_coor[i,3]
          xyz[i,1] <- r*sin(theta)*cos(phi)
          xyz[i,2] <- r*sin(theta)*sin(phi)
          xyz[i,3] <- r*cos(theta)
     }
     xyz
}

4.4. Sphere.R

Sphere<-function(init){
     r <- init[2]
     #theta <- init[3]
     #phi <- init[4]
     distance <- init[5]
     n_grid <- init[6]
     phi_grid <- init[8]
     delta_theta <- pi/n_grid
     length_theta <- n_grid+1
     sphere_coor <- vector(mode = "list",length = length_theta)
     delta_phi <- 2*pi/phi_grid
     n_grid_phi <- phi_grid
     length_phi <- n_grid_phi+1
     for(i in 1:length_theta){
          theta <- (i-1.) * delta_theta
          sphere_coor[[i]] <- array(data = 0,dim = c(length_phi,3))
          for (j in 1:length_phi){
               sphere_coor[[i]][j,1] <- r
               sphere_coor[[i]][j,2] <- theta
               sphere_coor[[i]][j,3] <- (j-1.) * delta_phi
          }
     }
     sphere_coor
}

4.5. Show3D.R

Show3D<-function(xyz_coor,view_angle){
     length <- length(xyz_coor)
     total <- 0
     for(i in 1:length){
          total <- total + dim(xyz_coor[[i]])[1]
     }
     x <- array(data = 0,dim = c(total))
     y <- array(data = 0,dim = c(total))
     z <- array(data = 0,dim = c(total))
     k <- 1
     for(i in 1:length){
          ll <- dim(xyz_coor[[i]])[1]
          for(j in 1:ll){
               x[k] <- xyz_coor[[i]][j,1]
               y[k] <- xyz_coor[[i]][j,2]
               z[k] <- xyz_coor[[i]][j,3]
               k <- k+1
          }
     }
     library(plot3D)
     lines3D(x = x,y = y,z = z,theta=view_angle[1], phi=view_angle[2])
}

4.6. WormHole.R

WormHole<-function(init){
     m <- init[1]
     r <- init[2]
     #theta <- init[3]
     #phi <- init[4]
     distance <- init[5]
     n_grid <- init[6]
     r_grid <- init[7]
     phi_grid <- init[8]
     delta_theta <- (pi/4)/n_grid
     length_theta <- n_grid+1
     sphere_coor <- vector(mode = "list",length = length_theta)
     delta_phi <- 2*pi/phi_grid
     n_grid_phi <- phi_grid
     length_phi <- n_grid_phi+1
     for(i in 1:length_theta){
          theta <- (i-1.) * delta_theta
          sphere_coor[[i]] <- vector(mode = "list",length = length_phi)
          for (j in 1:length_phi){
               phi <- (j-1.) * delta_phi
               sphere_coor[[i]][[j]] <- array(data = 0,dim = c(r_grid,3))
               for(k in 1:r_grid){
                    sphere_coor[[i]][[j]][k,1] <- r + Delta_R_WormHole(r,theta,distance,m)
                    sphere_coor[[i]][[j]][k,2] <- theta
                    sphere_coor[[i]][[j]][k,3] <- phi
                    r <- sphere_coor[[i]][[j]][k,1]
               }
          }
     }
     sphere_coor
}

4.7. Delta_R_WormHole.R

Delta_R_WormHole<-function(r,theta,distance,m){
     delta_r =  sqrt(abs(distance**2 / ( 1- (2*m/r) )))
     delta_r
}

4.8. Show3D_WormHole.R

Show3D_WormHole<-function(xyz_coor, view_angle){
     length <- length(xyz_coor)
     total <- 0
     for(i in 1:length){
          ll <- length(xyz_coor[[i]])
          for(j in 1:ll){
               total <- total + dim(xyz_coor[[i]][[j]])[1]
          }
     }
     x <- array(data = 0,dim = c(total))
     y <- array(data = 0,dim = c(total))
     z <- array(data = 0,dim = c(total))
     k <- 1
     k <- 1
     for(i in 1:length){
          ll <- length(xyz_coor[[i]])
          for(j in 1:ll){
               lll <- dim(xyz_coor[[i]][[j]])[1]
               for(p in 1:lll){
                    x[k] <- xyz_coor[[i]][[j]][p,1]
                    y[k] <- xyz_coor[[i]][[j]][p,2]
                    z[k] <- xyz_coor[[i]][[j]][p,3]
                    k <- k+1
               }
          }
     }
     library(plot3D)
     lines3D(x = x,y = y,z = z,theta=view_angle[1], phi=view_angle[2])
}

4.9. Main.R

source("Circle.R")
source("Delta_Phi_Circle.R")
source("Sphere_2_XYZ.R")
source("Sphere.R")
source("Show3D.R")
source("WormHole.R")
source("Delta_R_WormHole.R")
source("Show3D_WormHole.R")
################################################################################
init <- c(1., 2.001, pi/2., 0., 0.0001, 10, 5, 10) # M, r > 2M, theta, phi, distance, n_grid, r_grid, phi_grid
view_angle <- c(45.,0.) # theta, phi in degree
################################################################################
print("Hello World!")

if(init[2]>2*init[1]){
     sphere_coor <- Circle(init)
     xyz_coor <- Sphere_2_XYZ(sphere_coor)
     plot(x = xyz_coor[,1],y = xyz_coor[,2], type="l")
     readline()
     
     sphere_coor <- Sphere(init)
     length <- length(sphere_coor)
     xyz_coor <- vector(mode = "list", length = (length-2))
     for(i in 1:(length-2)){
          xyz_coor[[i]] <- Sphere_2_XYZ(sphere_coor[[i+1]])
     }
     Show3D(xyz_coor, view_angle)
     readline()

     sphere_coor <- WormHole(init)
     length <- length(sphere_coor)
     xyz_coor <- vector(mode = "list", length = length)
     for(i in 1:length){
          ll <- length(sphere_coor[[i]])
          xyz_coor[[i]] <- vector(mode = "list",length = ll)
          for(j in 1:ll)
          xyz_coor[[i]][[j]] <- Sphere_2_XYZ(sphere_coor[[i]][[j]])
     }
     Show3D_WormHole(xyz_coor, view_angle)
     
} else {
     print("Error: r>2M")
}
print("End program.")

Drawing Geometry from Line Element

Kornpob Bhirombhakdi

March 6, 2016