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A schematic of the 5D spacekime manifold, two cones representing space (\(\mathbb{R}^3\)) - kime (\(\mathbb{R}^2\))
Complex-time (kime) can be defined using polar coordinates, \(\kappa = t\ e^{i \varphi}\).
library(plotly)
library(dplyr)
# parameter space sweep for the spherical coordinates
<- seq(from = 0, to = 2*pi, by = ((2*pi - 0)/(200 - 1)))
phi <- seq(from = 0, to = pi, by = ((pi - 0)/(200 - 1)))
psi
# shape=="cone1")
# rendering (u,v) parametric surfaces requires x,y,z arguments to be 2D arrays
# In out case, the three coordinates have to be 200*200 parameterized tensors/arrays
= 10 # cone height
h1= seq(from = 0, to = h1, by = ((h1 - 0)/(200 - 1))) # r = radius
r1 = 20* ((h1 - r1)/h1 ) %o% rep(1, 200) # x = 3*r
x1 = 3* ((h1 - r1)/h1 ) %o% sin(phi) # y = r*sin(phi)
y1 = 3* ((h1 - r1)/h1 ) %o% cos(phi) # z = r*cos(phi)
z1
# circle1 boundary
= rep(20, 200) %o% rep(1, 200) # x = 20
x11 = 3* ((h1 - r1)/h1 ) %o% sin(phi) # y = r*sin(phi)
y11 = 3* ((h1 - r1)/h1 ) %o% cos(phi) # z = r*cos(phi)
z11
# shape=="cone2")
= 10 # cone height
h2= seq(from = 0, to = h2, by = ((h2 - 0)/(200 - 1))) # r = radius
r2 = 20* ((h2 - r2)/h2 ) %o% rep(1, 200) # x = 3*r
x2 = 2* ((h2 - r2)/h2 ) %o% sin(phi) # y = r*sin(phi)
y2 = 2* ((h2 - r2)/h2 ) %o% cos(phi) # z = r*cos(phi)
z2
# circle2 boundary
= rep(20, 200) %o% rep(1, 200) # x = 20
x21 = 2* ((h2 - r2)/h2 ) %o% sin(phi) # y = r*sin(phi)
y21 = 2* ((h2 - r2)/h2 ) %o% cos(phi) # z = r*cos(phi)
z21
# shape=="cone3")
= 10 # cone height
h3= seq(from = 0, to = h3, by = ((h3 - 0)/(200 - 1))) # r = radius
r3 = 15* ((h3 - r3)/h3 ) %o% rep(1, 200) # x = 3*r
x3 = 3* ((h3 - r3)/h3 ) %o% sin(phi) # y = r*sin(phi)
y3 = 3* ((h3 - r3)/h3 ) %o% cos(phi) # z = r*cos(phi)
z3
# circle3 boundary
= rep(15, 200) %o% rep(1, 200) # x = 15
x31 = 3* ((h3 - r3)/h3) %o% sin(phi) # y = r*sin(phi)
y31 = 3* ((h3 - r3)/h3) %o% cos(phi) # z = r*cos(phi)
z31
<- c("cone1", "cone2", "cone3")
shape_names
# https://plot.ly/r/custom-buttons/
# updatemenus component
<- list(
updatemenus list(
active = -1,
type = 'buttons',
buttons = list(
list(
label = shape_names[1],
method = "update",
args = list(list(visible = c(TRUE, FALSE, FALSE)),
list(title = shape_names[1]))),
list(
label = shape_names[2],
method = "update",
args = list(list(visible = c(FALSE, TRUE, FALSE)),
list(title = shape_names[2]))),
list(
label = shape_names[3],
method = "update",
args = list(list(visible = c(FALSE, FALSE, TRUE)),
list(title = shape_names[3])))
)
)
)
<-
p plot_ly(hoverinfo="none", legendshow=FALSE, showscale = FALSE) %>%
# add cone1
add_trace(x = ~x1, y = ~y1, z = ~z1, type = 'surface', opacity=0.3, visible=T,
contour=list(show=F, color="#000", width=15, lwd=10,
opacity=0.5, hoverinfo="none", legendshow=F)) %>%
# add cone2
add_trace(x = ~x2, y = ~y2, z = ~z2, type='surface', opacity=0.4,visible=T,
contour=list(show=F, color="#000", width=15, lwd=10,
opacity=0.5, hoverinfo="none", legendshow=F)) %>%
# add cone3
add_trace(x = ~x3, y = ~y3, z = ~z3, type='surface', opacity=0.5,visible=T,
contour=list(show=F, color="#000", width=15, lwd=10,
opacity=0.5, hoverinfo="none", legendshow=F)) %>%
#
#
# trace the x-axis
add_trace(x=~1.1*x1[,1], y=0, z=0, type="scatter3d", mode="lines",
line = list(width = 10, color="light blue"), name="Z",
hoverinfo="none", legendshow=F) %>%
#
#
# trace the boundary of cone1 surface
add_trace(x=~x1[1,], y=~y11[1,], z=~z11[1,], type="scatter3d", mode="lines",
line = list(width = 10, color="red"), name="Surface Boundary",
hoverinfo="none", legendshow=F) %>%
# add center for kime circle1 at location x1
add_trace(x=~x1[1,1], y=0, z=0, type="scatter3d", mode="markers",
line = list(width = 10, color="navy blue"), name="Z",
hoverinfo="none", legendshow=F) %>%
# trace the boundary of cone2 surface
add_trace(x=~x2[1,], y=~y21[1,], z=~z21[1,], type="scatter3d", mode="lines",
line = list(width = 10, color="green"), name="Surface Boundary",
hoverinfo="none", legendshow=F) %>%
# trace the boundary of cone3 surface
add_trace(x=~x3[1,], y=~y31[1,], z=~z31[1,], type="scatter3d", mode="lines",
line = list(width = 10, color="blue"), name="Surface Boundary",
hoverinfo="none", legendshow=F) %>%
# add center for cime circle3 at location x3
add_trace(x=~x3[1,1], y=0, z=0, type="scatter3d", mode="markers",
line = list(width = 10, color="navy blue"), name="Z",
hoverinfo="none", legendshow=F) %>%
layout(title = "Choose a Cime Cone", showlegend = FALSE,
updatemenus = updatemenus)
p