moDelnik Reviews: Arctan Calculator by RC87

Neither only an expensive 3D printer nor a premium 3D model guarantees the printing of smooth, flat surfaces. The proper object orientation is a key factor in ensuring optimal print quality and smooth surfaces.

Why Does the Printing Angle Matter?

Placing a model with a flat side parallel to the build plate is not always the best approach. Flat surfaces often benefit from being printed at an angle to improve surface quality. Reduce the need for excessive supports, compared to the parallel oriantation, and prevent deformation.

There is no perfect angle, but there is a better angle, and we’ll discuss that soon. First, let’s go over some key theory. But before we continue with it, allow me to say that if you’re printing miniature figures, curved objects, etc. You don’t need to be as picky about the printing angle! When we refer to a flat side, we mean a side that lies on a plane.

Also, keep in mind that the quality of the print for flat sides is not only determined by the orientation. There are various factors such as settings, support structure, and the overall quality or condition of the printer.

Promised Theory: What is a voxel?

A voxel (short for “volume element”) is a 3D “pixel”. Or in other words the smallest unit of a 3D space in a digital model. Similar to how a pixel represents the smallest unit of a 2D image.

In more detail:

• 2D Pixels: In digital images, a pixel is the basic unit of color. Representing a single point on a flat plane (height x width).
• 3D Voxels: In our 3D objects, voxels represent the smallest unit of volume. In a three-dimensional grid ofcourse (height x width x depth). You can think of it like a tiny cube within a 3D space.

In short, voxels are a fundamental part of working with 3D objects. They represent the 3D equivalent of pixels in the digital world. In resin printing, the model can be represented as a grid of voxels. Each voxel corresponds to a small part of the object that is cured layer by layer.

Why and how to use the RC87 Angle Calculator?

The optimal angle for the best results when resin printing flat surfaces is often not 45°. This optimal angle (or “better angle”, as I referred to it earlier) varies for each printer. And depends on factors like layer size and the pixel width of the printer.

Some resin 3D printer screens, such as the one on the Elegoo Mars 3, have square pixels. While others, like one of mine which is Anycubic Photon M5s produce rectangular pixels. Keep this in mind for now, we will discuss it in more detail later.

Earlier we said that the voxel have height, width and depth. The width and depth of the voxels you print, come from your printer’s screen pixel size. While the height come from the layer height you set.

When 45° is just fine?

Now, let’s assume we are printing a flat detail on the Elegoo Mars 3. It has a pixel size of 35 µm x 35 µm (which is 0.035 mm x 0.035 mm). In order to achieve a perfect voxel (cube), we would need to set the layer height to 35 µm, right?

In this case. To ensure an equal diagonal offset between every printed layer, which results in a flat and uniform surface. We can set the orientation of the flat surface to 45° along either the X or Y axis. And we will achieve the desired result.

In other words when we have equal pixel size sides and layer height we always go for 45°. And this is what we exactly see on Figure 1.

When do we need a different angle?

Ok, now lets say that with the same printer we want to print the same object. But this time with layer height of 20 µm (0.020 mm). Then our voxel is not a cube anymore. And If we keep the same angle of 45°, we may won’t have the same surface quality as before. Why?

Because, for layer height of 0.020 mm the angle giving us equal layer offset is not 45° anymore. Our optimal angle will be 60.2551°, from the vertical axis. Or 29.7449° from the build plate.

Well, I know it with the help of the RC87 calculator. As it’s shown on Figure 2 , which comes from its interface. Since I was a kid, I’ve never been very good at or involved in mathematics. Thats why I love this calculator. xD

Being lazy, I preffer to choose my printer’s brand and model. Than to fill in the desired by me layer height. And on the end simply see what angles I must set when orient the model. So thats why I use and recommend you the RC87 Angle calculator. The tool is completely FREE and web-based.

What if my Pixels are NOT ^Square!?

As already said, I am a proud owner of Anycubic Photon Mono M5s. For sure is not as modern nowdays, but always had and have great print quality. It has a 10.1-inch diagonal screen with a 2:1 aspect ratio (width:depth). Where the pixel size sides are different by X and Y, and they look’s like this:

• X-axis Pixel Size: Width / Resolution Width = 218.88 mm / 11,520 pixels ≈ 0.019 mm per pixel
• Y-axis Pixel Size: Depth / Resolution Depth = 122.88 mm / 5,120 pixels ≈ 0.024 mm per pixel

In this case, I simply open the RC87 Angle calculator, enter the layer height, and adjust the model’s orientation accordingly.

midArticle TIPs:

In theory, a resin 3D printer with the smallest and perfectly equal X and Y pixel sizes, along with a minimal layer height matching those pixel sizes, should produce the highest-quality flat surfaces, and tiny details ofcourse.

However, in reality, if the printer lacks a well-built Z-axis and overall good mechanical quality, nothing is guaranteed. That’s why I’m really happy and proud of my Photon M5s, even if it doesn’t have the smallest and most uniform pixel size out there.

More Theory & Formulas

Formula for Optimal Printing Angle

In case youre not lazy, not as I am. You can use formula to find the optimal angle for orientation and your pixel size:

Your optimal angle in degrees = arctan⁡(Layer Height / Pixel Width)

And here is the breakdown of the formula:

• arctan: This is the inverse tangent function (also known as atan or tan⁻¹). It is used to find the angle from the ratio of two sides of a right triangle. The arctan function is the inverse of the tangent function. Which means that if you know the tangent (ratio of opposite to adjacent side), arctan will give you the angle.

• Layer Height: This represents the height of each individual layer of the model being printed. In resin printing, the layer height is typically measured in microns (μm) or millimeters (mm). Smaller layer heights create finer details, but the print process may take longer.

• Pixel Width: This refers to the width of a pixel in the context of the printer’s resolution. Or the size of the smallest printable unit in a single layer. The pixel width represents how wide each pixel of resin exposure is during a single pass or exposure cycle.

What the Formula Does?

• The Layer Height is divided by the Pixel Width. It gets the ratio of the vertical dimension (height) to the horizontal dimension (width) of the printed surface.
• The arctan function then converts this ratio into an angle.

Video Recap: “Printing” the Arctan

This video was one of the first sources I came across on the topic a long time ago. Highly informative, packed with great examples, and brought to us by True Elite Geek.

Formula for Finding the Pixel Size

To determine the pixel size accurately, you need the screen’s resolution and the physical screen dimensions. Once you have that, next you can calculate the pixel size. And ofcourse better understand the printer’s capability to reproduce fine details in 3D prints.

With the RC87 Angle Calculator it’s easy. After selection of the brand and the model of your resin printer, you will see the pixel size. But if you wish to calculate the pixel size alone, you can use the following formula:

Pixel Size = Screen Size / Resolution

Where:

• Screen Size is the physical size of the display (usually the diagonal size of the screen in inches or millimeters).
• Resolution is the number of pixels along each axis (X and Y) that the printer’s display can produce.

Step-by-Step Process to Calculate Pixel Size:

1. Determine the Screen Size:
   • This is typically provided in the printer’s specifications. It is usually given as the diagonal size of the display (in inches or millimeters).
   Example: A common resin 3D printer might have a screen size of 5.5 inches.
2. Determine the Printer’s Resolution:
   • The resolution refers to the number of pixels along the X and Y axes of the display. Most resin printers have a resolution given by the number of pixels in width and height.
   Example: A common resolution for a resin printer might be 2560 x 1440 pixels. Typically, a screen with a 5.5-inch display).
3. Convert Screen Size into Dimensions (if needed):
   • If your screen size is provided in inches (e.g., 5.5 inches), you might need to calculate the width and height of the screen.
   You can use the Pythagorean theorem to calculate the width and height of the screen from the diagonal size. For example, if the resolution is 2560 x 1440 pixels and the screen is 5.5 inches diagonal. The width and height of the screen can be calculated using the ratio of the resolution. Ths calculation will give us in milimmeters 120 x 67.5 mm screen sides.
4. To Calculate the Pixel Size:
   • Divide the width or height of the screen by the number of pixels along that axis:
   • Pixel Width = 120 mm / 2560 ≈ 0.0469 mm per pixel
   • Pixel Height = 67.5 mm / 1440 ≈ 0.0469 mm per pixel

Why Pixel Size Matters:

• Smaller pixel sizes generally mean better resolution, which can result in finer details in your prints.
• If your printer has a larger pixel size, the print may have less detail or appear more “blocky.”
• In some cases, the pixel size affects the minimum feature size your printer can resolve. Smaller pixels help create finer details, which is especially important for miniatures or models with intricate designs.

Final Thoughts:

If you’re new to resin 3D printing, it’s just like being new to scale modeling or any other skill. It’s something you start researching and mastering over time. Make sure to give yourself enough time to learn without unproductive frustration if something goes wrong. Or doesn’t turn out exactly as you imagined. These situations are not failures but valuable experiences.

The more you print, the more advanced you’ll become, and over time, it will require less effort.

Stay positive and I wish you Happy Printing & Modeling!

Nick

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