Rolling Cube along a path in C4D using math and Xpresso

A couple of days ago I came across this video of Maxon Training Team (channel that I recommend you follow):

This tutorial shows techniques for rolling a cube along a path using XPresso and the new Nodes system. The result is good and in certain situations it can be fine, but it is not precise, it is an approximation. I tried to look if there was a mathematical solution to the problem but in my short search I found only these other methods of solving the problem:

This is another fast and effective approach that leverages the internal physics of C4D.

Here, on the other hand, a solution is offered to obtain a mathematically precise rolling, but it would probably be difficult, to deny me if I am wrong, to adapt it to make the cube follow a path other than the straight line.

So I thought I’d solve the problem mathematically so that the cube can follow a path along a spline. But first I want to show you a short example video where I used my own preset.

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Realmagazine preset for Cinema 4D – released

Finally my new Realmagazine preset for Cinema 4D is available. 
With Realmagazine you can create a personalized and animatable 3D magazine, newspaper or exercise book in just a few steps. You can choose the size, the number of pages, his thickness, add creases, crumbling and much more.
Watch the video below to quickly see what you can do with it. And if you want to understand more download the user guide here:

Realmagazine – preview

After creating the Realbook preset for C4D, perfect for creating and animating a book, I received several messages asking me to create a preset for newspapers, magazines and notebooks.
The substantial difference lies in the form of the sheets. While in the Realbook each sheet starts from the spine and contains two pages, in the Realmagazine (this is the name of the new preset) the sheets are folded and each of them can contain other sheets inside, as in reality. This is the substantial difference that appears to the eye, but there are others that make this preset more powerful. Powerful because it allows much more control in terms of configurations and possible animations. To manage this power and be able to express it as much as possible, it will be necessary to make a greater effort to understand how it works.
Below you can see some sample videos that partially illustrate its potential.
Also I could not resist the temptation to use a C4D content browser scene, created by the extraordinary Raphael Rau, to get the renderings you find at the end of the post.