# Versailles's Floor generator

# Introduction

Versailles's floor is a puzzle: many pieces to assemble together.
I wanted to create a simple simulator for the most common pattern.



# Simulator

This is a simple javascript simulator, made with [Bokeh](https://docs.bokeh.org/en/latest/).


You can play with using two sliders:

- The `R` slider modifies the trade-off between the square and the lines (of lenght `a` and `b` respectively).
- The `length` slider doesn't change anything visual, this is only if you don't want to compute the size of the pieces yourself. The length represents the total length (borders included).

{% include versailles.html %}


# Formulas

We will first forget about the total length. The reference unit will be `R`, the ratio between the large squares, and the spacing rectangles.

![](/assets/images/Versailles/versailles_whiteB.png)

## Main Components

Edge of `A`:

$$a = R$$

Width of `B`:

$$b = 1 - R$$

from which you get the diagonal:

$$D_a = \sqrt{2 \times a^2}$$

$$D_b = \sqrt{2 \times b^2}$$

For the rectangle `B`, $$D_b$$ is not the diagonal: it is the length of the smallest edge cut with an angle of `45°`.

## Triangles

Diagonal of the angle's triangle `C`:

$$x = 2 \times a + b$$

From which you get the side of the triangle:

$$y = \sqrt{\frac{x^2}{2}}$$

For the small triangle `D`, the hypotenuse is:

$$z = D_a - D_b$$

## Internal Length

The internal length (not the total one) is:

$$L = 2 \times y + 4 \times D_b + z$$

This is the length of a border `E` (and $$b$$ is the width).

## Converting Units

If we want to move from our unit defined as the length of two main elements glued together to the total size of a pannel, this is easy.

- The total length in our unit is $$L + 2 \times b$$
- The "true" total length is $$L_0$$

To convert all the previous numbers into the target unit, we just need to compute:

$$w' = w \times \frac{L_0}{L + 2 \times b}$$

where $$w$$ is the length to convert and $$w'$$ the resulting value.

# How to Generate that ?

Well, you need to find the coordinate of `B` rectangles.

Next, the simplest is to perform translation (horizontal and vertical) of $$D_a + D_b$$.

There are rectangles that are cut at `45°`.
To avoid finding the coordinates, the simplest thing to do (from a computer science perspective) is to add above these `B` rectangles the borders `E`. So they hide the edges.