#3DxMathematics – Calculating the volume of a mesh @ Nervous System #3DThursday #3DPrinting
Leaving introductory geometry behind, here is an illuminating explanation for how to draw on differential geometry and other helpful tools to calculate the volume of a mesh, from Jesse of nervous system — one of Adafruit’s favorite design teams as well as an excellent resource to learn more about leveraging good math to produce good art:
In order to generate the price of a custom design on the fly, we need to calculate the volume of the piece for 3d printing. By constantly updating the volume, the customer gets instant feedback on how their changes are affecting price. Calculating the volume of a mesh is a relatively simple and well-known problem, and I’ll go over the straight forward case as well as an optimization we’ve incorporated into our latest project.
…Now, onto the good stuff. What happens if you have an object that is made of (at least in part) an aggregate of a bunch of identical but complex parts. I don’t mean a booleaning together primitives, but you could imagine something like a buckyball where each face is articulated with some kind of intricate shape. The brute force approach would be to move and rotate the shape to the proper position then go through each triangle and calculate the volume. This means you have to calculate a transform on each of the points of your shape, and then go through each triangle. If your shape has 1000 triangles and you have 100 shapes, that ends up being a lot calculation. We can drastically increase the efficiency of this by computing a “general volume” for the shape once, and applying our transforms only to that simplified representation. But what does this general volume look like?
The key idea behind this general volume is the fact that volume is rotation invariant. This is one of the basic results of differential geometry. It is intuitively obvious; no matter how I orient an object in space its volume does not change. What is less intuitive is that the same thing holds true for the signed volume of open shapes. Mathematically this can be seen easily by noting that the volume is the determinant of a matrix, and the rotation matrix has a determinant of 1. The determinant of one matrix multiplied by another is the multiplication of their individual determinants. So, I can rotate my primitive element however I want, and the volume stays the same. If I was only rotating my shape, then I could calculate the volume of my shape once and multiply it by the number of shapes I have.
We are angry, frustrated, and in pain because of the violence and murder of Black people by the police because of racism. We are in the fight AGAINST RACISM. George Floyd was murdered, his life stolen. The Adafruit teams have specific actions we’ve done, are doing, and will do together as a company and culture. We are asking the Adafruit community to get involved and share what you are doing. The Adafruit teams will not settle for a hash tag, a Tweet, or an icon change. We will work on real change, and that requires real action and real work together. That is what we will do each day, each month, each year – we will hold ourselves accountable and publish our collective efforts, partnerships, activism, donations, openly and publicly. Our blog and social media platforms will be utilized in actionable ways. Join us and the anti-racist efforts working to end police brutality, reform the criminal justice system, and dismantle the many other forms of systemic racism at work in this country, read more @ adafruit.com/blacklivesmatter
Stop breadboarding and soldering – start making immediately! Adafruit’s Circuit Playground is jam-packed with LEDs, sensors, buttons, alligator clip pads and more. Build projects with Circuit Playground in a few minutes with the drag-and-drop MakeCode programming site, learn computer science using the CS Discoveries class on code.org, jump into CircuitPython to learn Python and hardware together, TinyGO, or even use the Arduino IDE. Circuit Playground Express is the newest and best Circuit Playground board, with support for CircuitPython, MakeCode, and Arduino. It has a powerful processor, 10 NeoPixels, mini speaker, InfraRed receive and transmit, two buttons, a switch, 14 alligator clip pads, and lots of sensors: capacitive touch, IR proximity, temperature, light, motion and sound. A whole wide world of electronics and coding is waiting for you, and it fits in the palm of your hand.