How to work with triangular shaped products

It is common among retailers to have at least some products that are
shaped as triangles. Examples may include:

Protein powder bags
Furniture parts designed to work well in duos

How are triangles handled
Our Optimization first looks for matching pairs of triangles that can be efficiently combine together, since
this always tends to be the optimal outcome.
For any non-matched triangular products, we will instead try to use the remaining empty space in the triangle's bounding box
to place other products.

To determine the empty space to use within a triangle, we use some simple geometry that is not the most readable code,
which is why we have this documentation page.

Only works for right-angled triangles at the moment

The solution is based around expressing the intersection of two lines in the form of two equations:

These form simultaneous equations we can solve by substitution. So, what are these lines?
Line 1
The first one is quite simply the hypotenuse of the triangle, as seen below.

So if we have length  and width  for the triangle it means we know that the hypotenuse is expressed by:

Line 2
Finding the second line is more tricky. To do this, let us assume that we have a ratio  that will
determine the relative length and width of the rectangle we find in the empty space.
Let's consider the example of  as it will help illustrate the point.
To find the second line, we simply place a box with the desired ratio starting at the furthest side
of our triangle. Given the size ratio of the box, we know that along the hypotenuse of the box, this ratio will always hold.
Therefore, this hypotenuse will become our  - our second line.

We can then simply find the intersection between these lines and take the remaining distance to the
edges of the bounding box to find the empty space we can use.

Importantly, this will always also give two new triangles of empty space that
we could repeat the process for, which can allow more efficient usage of empty space.

In 3D it is actually not much more complicated for right-angled triangles, as illustrated by this visualization:

Advanced

There are some weird-seeming cases when for example a ratio is used that makes the second box
extend outside the original box, but the solution is still valid.
It can be a little hard to follow the code that implements this, since there are not any great ways of working with expressions and equations, but this describes the logic behind what is happening



Triangles

How are triangles handled