---
title: "Definition of Kime"
subtitle: "[Back To TCIU Contents](https://tciu.predictive.space/)"
author: "SOCR Team "
date: "`r format(Sys.time(),'%m/%d/%Y')`"
output: 
  html_document:
    theme: spacelab
    highlight: tango
    includes:
      before_body: TCIU_header.html
    toc: true
    number_sections: true
    toc_depth: 2
    toc_float:
      collapsed: false
      smooth_scroll: true
    code_folding: hide
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, warings = FALSE)
```

# Figure 1.8
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](https://en.wikipedia.org/wiki/Polar_coordinate_system), $\kappa = t\ e^{i \varphi}$.

```{r message=F, warning=F}
library(plotly)
library(dplyr)
```

```{r message=F, warning=F}

# parameter space sweep for the spherical coordinates
phi <- seq(from = 0, to = 2*pi, by = ((2*pi - 0)/(200 - 1)))
psi <- seq(from = 0, to = pi, by = ((pi - 0)/(200 - 1)))

# 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
    h1= 10   # cone height
    r1 = seq(from = 0, to = h1, by = ((h1 - 0)/(200 - 1)))  # r = radius
    x1 = 20* ((h1 - r1)/h1 ) %o% rep(1, 200)             # x = 3*r
    y1 = 3* ((h1 - r1)/h1 ) %o% sin(phi)   # y = r*sin(phi)
    z1 = 3* ((h1 - r1)/h1 ) %o% cos(phi)   # z = r*cos(phi)

    # circle1 boundary
    x11 = rep(20, 200) %o% rep(1, 200)             # x = 20
    y11 = 3* ((h1 - r1)/h1 ) %o% sin(phi)   # y = r*sin(phi)
    z11 = 3* ((h1 - r1)/h1 ) %o% cos(phi)   # z = r*cos(phi)

# shape=="cone2")
    h2= 10   # cone height
    r2 = seq(from = 0, to = h2, by = ((h2 - 0)/(200 - 1)))  # r = radius
    x2 = 20* ((h2 - r2)/h2 ) %o% rep(1, 200)             # x = 3*r
    y2 = 2* ((h2 - r2)/h2 ) %o% sin(phi)   # y = r*sin(phi)
    z2 = 2* ((h2 - r2)/h2 ) %o% cos(phi)   # z = r*cos(phi)

    # circle2 boundary
    x21 = rep(20, 200) %o% rep(1, 200)             # x = 20
    y21 = 2* ((h2 - r2)/h2 ) %o% sin(phi)   # y = r*sin(phi)
    z21 = 2* ((h2 - r2)/h2 ) %o% cos(phi)   # z = r*cos(phi)
    
# shape=="cone3")
    h3= 10   # cone height
    r3 = seq(from = 0, to = h3, by = ((h3 - 0)/(200 - 1)))  # r = radius
    x3 = 15* ((h3 - r3)/h3 ) %o% rep(1, 200)             # x = 3*r
    y3 = 3* ((h3 - r3)/h3 ) %o% sin(phi)   # y = r*sin(phi)
    z3 = 3* ((h3 - r3)/h3 ) %o% cos(phi)   # z = r*cos(phi)

    # circle3 boundary
    x31 = rep(15, 200) %o% rep(1, 200)             # x = 15
    y31 = 3* ((h3 - r3)/h3) %o% sin(phi)   # y = r*sin(phi)
    z31 = 3* ((h3 - r3)/h3) %o% cos(phi)   # z = r*cos(phi)

shape_names <- c("cone1", "cone2", "cone3")

# https://plot.ly/r/custom-buttons/

# updatemenus component
updatemenus <- list(
  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
```

<!--html_preserve-->
<div>
    	<footer><center>
			<a href="https://www.socr.umich.edu/">SOCR Resource</a>
				Visitor number <img class="statcounter" src="https://c.statcounter.com/5714596/0/038e9ac4/0/" alt="Web Analytics" align="middle" border="0">
				<script type="text/javascript">
					var d = new Date();
					document.write(" | " + d.getFullYear() + " | ");
				</script> 
				<a href="https://socr.umich.edu/img/SOCR_Email.png"><img alt="SOCR Email"
	 			title="SOCR Email" src="https://socr.umich.edu/img/SOCR_Email.png"
	 			style="border: 0px solid ;"></a>
	 		 </center>
	 	</footer>

	<!-- Start of StatCounter Code -->
		<script type="text/javascript">
			var sc_project=5714596; 
			var sc_invisible=1; 
			var sc_partition=71; 
			var sc_click_stat=1; 
			var sc_security="038e9ac4"; 
		</script>
		
		<script type="text/javascript" src="https://www.statcounter.com/counter/counter.js"></script>
	<!-- End of StatCounter Code -->
	
	<!-- GoogleAnalytics -->
		<script src="https://www.google-analytics.com/urchin.js" type="text/javascript"> </script>
		<script type="text/javascript"> _uacct = "UA-676559-1"; urchinTracker(); </script>
	<!-- End of GoogleAnalytics Code -->
</div>
<!--/html_preserve-->

