Fundamentals of Computational Engineering: Part 4 – Implicits.


are in the world of computational engineering is Lin Kayser, the founder of Leap71. In his series of articles, he introduces the concept of Computational Engineering and explains the various aspects of this new field.

In this particular article, Kayser delves into the technicalities of a Voxel-based kernel and explores how we can interact with it. He starts by discussing the different functions required to support Tier 2, which is the abstract shape kernel where complex geometric work is done.

Instead of the traditional approach of setting a voxel in the voxel field using functions like SetVoxel(xyz), Kayser explains that their voxels are not as simple. They are described using a narrow-band signed distance field (SDF), which stores the distance to the boundary of the voxels that contain matter.

Kayser goes on to explain that adding a SetVoxel function to turn a voxel on or off would either be complex and slow or mess up the narrow-band signed distance field. Instead, they interact with the voxel field on a higher level by rendering implicit functions.

Implicit functions are mathematical formulas that describe the distance to the surface of an object. They are a perfect fit for the narrow-band signed distance field used in the voxels. These implicit functions can create a wide range of geometric figures, infill patterns, and other useful shapes.

Kayser highlights the versatility of implicit functions and recommends the website created by computer graphics expert Inigo Quilez, which showcases the power of implicits through various formulas. These implicits can be evaluated for any point in space and will always return a result, making them infinitely precise and robust.

One of the commonly used formulas is the famous gyroid function, which creates a triply periodic minimal surface often associated with 3D printed parts.

Kayser concludes by emphasizing the power and potential of implicits in computational engineering. Their ability to create complex and interesting results with simple formulas makes them a valuable tool in the field.

Overall, this alternative blog post retains the story and logic of the original article but presents it in a different writing style.

Reimagining the Blog Post: Breaking Barriers with 3D Printing

by Bradley Rothenberg

Geometry, the foundation of design and engineering, has always posed challenges when it comes to creating intricate shapes and structures. However, a revolutionary approach called field-driven design has emerged, spearheaded by nTopology (now nTop), founded by yours truly. This approach has had a profound impact on the 3D printing industry in recent years, unlocking new possibilities in computational geometry.

One of the key components of this approach is the ability to render implicits, which are essential for representing lattices, infills, and other formula-driven geometries. To achieve this, a geometry kernel must be capable of evaluating implicit formulas at voxel coordinates. Through this evaluation, we obtain either a negative value (indicating that we are inside the surface), zero (on the surface), or a positive value (outside the surface).

While there is an abundance of formulas available for implicits, there is no need to overwhelm this article with them. Thankfully, ChatGPT is an excellent resource for accessing these formulas effortlessly. Rather than burdening you with formula after formula, simply head over to ChatGPT and ask for assistance. Thank you, Pythagoras, for making our lives easier!

Let’s consider a simple example using a sphere. By calculating the distance from each voxel to the center of the sphere and subtracting the radius, we can generate implicit values. Negative values indicate voxels inside the sphere, positive values denote voxels beyond the radius, and zero represents voxels on the surface.

Impressed by the power of implicits? With ChatGPT, you can even request the code to create a signed distance field in your preferred programming language. Now, you can start designing your own kernel! Don’t worry if creating complex geometries directly with signed distance formulas seems daunting. Remember, even the best mathematicians require significant thought to describe the three-dimensional distance to objects like cogwheels.

However, implicits shine when it comes to generating mathematical shapes effortlessly. If you have a voxel field at your disposal, you can leverage spheres and cylinders as paintbrushes to draw other shapes. This flexibility allows you to approach geometry as an engineer rather than spending all your time thinking like a mathematician. After all, not all of us can be the next Einstein!

By mastering the evaluation of signed distance functions, we lay the foundation for creating the most intricately complex geometries ever seen in 3D printing. In fact, imagine being able to recreate any object known to humankind simply by placing tiny implicit spheres in the right positions. It’s like magic!

We’ve only scratched the surface of what this field can achieve. In my next article, I’ll delve into easier ways to create complex geometries. Stay tuned!

Editor’s Note: Lin Kayser, CEO of Leap71, has penned a six-part series exploring the exciting new field of Computational Engineering. Here are the published parts:

1. Fundamentals of Computational Engineering: Prologue

2. Fundamentals of Computational Engineering: Part 1 — A Bit of History

3. Fundamentals of Computational Engineering: Part 2 — The Technology

4. Fundamentals of Computational Engineering: Part 3 — Voxels to the Rescue

5. Fundamentals of Computational Engineering: Part 4 — Implicits

6. Fundamentals of Computational Engineering: Part 5 — All You Need is a Few Functions

Via Leap71

Share this post:

Original source


“Why did the 3D printer go to therapy? Because it had too many layers of unresolved issues!”

Like it? Share with your friends!


Meet the mastermind behind GCode-Guru, a 3D printing wizard whose filament collection rivals their sock drawer. Here to demystify 3D tech with a mix of expert advice, epic fails, and espresso-fueled rants. If you've ever wondered how to print your way out of a paper bag (or into a new coffee cup), you're in the right place. Dive into the world of 3D printing with us—where the only thing more abundant than our prints is our sarcasm.


Your email address will not be published. Required fields are marked *

Choose A Format
Personality quiz
Series of questions that intends to reveal something about the personality
Trivia quiz
Series of questions with right and wrong answers that intends to check knowledge
Voting to make decisions or determine opinions
Formatted Text with Embeds and Visuals
The Classic Internet Listicles
The Classic Internet Countdowns
Open List
Submit your own item and vote up for the best submission
Ranked List
Upvote or downvote to decide the best list item
Upload your own images to make custom memes
Youtube and Vimeo Embeds
Soundcloud or Mixcloud Embeds
Photo or GIF
GIF format