---
title: "torus_demo"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{torus_demo}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup}
library(ashapesampler)
library(alphashape3d)
library(alphahull)
library(doParallel)
library(parallel)
cores <- min(2L, detectCores())
library(rgl)
options(rgl.useNULL = TRUE)
```

In this document, we demonstrate the $\alpha$-shape sampler pipeline by simulating
the process of learning a set of three-dimensional shapes (in this case, tori) and
simulating a new shape from that. This vignette requires the packages `alphashape3d`,
`alphahull`, `rgl`, `parallel`, and `doParallel` in addition to `ashapesampler`.

Within vignettes, we are unable to set the number of cores above 2L. However, we highly 
recommend using as many cores on the machine as possible to parallelize the code and 
expedite computation, especially with 3D calculations.

We begin by setting the parameters for our simulation. We will fix $\alpha=0.15$ 
and $n=1000$, and draw 20 shapes for our data set. Our true underlying manifold
will be the torus with major radius 0.5 and minor radius 0.15.

```{r}
N=10
tau_vec <- vector("numeric", N)
my_alpha <- 0.15
n <- 1000
r_maj <- 0.5
r_min <- 0.15
```

Next we will draw the shapes themselves.

```{r}
torus_list <- list()
complex_torus_list <- list()

for (k in 1:N){
  torus_pts <- rtorus(n, r_min, r_maj)
  temp_torus <- ashape3d(torus_pts, my_alpha)
  torus_list[[k]] <- temp_torus
  complex_torus_list[[k]] <- get_alpha_complex(torus_pts, my_alpha)
  tau_vec[k] <- tau_bound(torus_list[[k]]$x, complex_torus_list[[k]])
}
```

Now that we have the shapes generated and imported, we can learn the underlying
manifold. First we will put together the entire point cloud into one two column 
matrix.

```{r}
choose_2 <- sample(N,2)
point_cloud = rbind(torus_list[[choose_2[1]]]$x, torus_list[[choose_2[[2]]]]$x)
```

Then we will have our $\tau$ bound be a summary statistic of the $\tau$ found for 
each input shape. Here, we will use mean, but one can tweak this to see different
results. Note that if $\tau$ is too small, then the random walk won't be able to 
execute around the point cloud, but if $\tau$ is too big, then we risk losing 
geometric and topological information in the reconstruction.

```{r}
tau = min(tau_vec[choose_2[1]], tau_vec[choose_2[2]])
```

Now we can take the parameters and generate a new shape and plot it. Note we assume `k_min=3` as we are in three dimensions.

```{r}
new_torus <- generate_ashape3d(point_cloud, J=2, tau=tau, cores=cores)
plot(new_torus, indexAlpha="all")
rglwidget()
```