The key to the Loomis Head drawing method is to understand how the circles, both the outer and inner, work in tandem. Especially, what they mean and how they contract or expand as the head rotates or tilts.
The outer circle represents the cranium, excluding the chin, while the inner circles represent the sides of the face. You have one outer circle and two inner circles since there are two sides to the face.
However, you need to understand that the outer circle is not actually a circle, but a sphere, since your head is a 3D object. But the inner circle is a circle, since it's a 2D representation of the side. Remember that even 2D objects expand or contract as per the perspective: foreshortening in this case.
The outer circle never changes because it's a sphere, but the inner circle does when heads rotate or tilt. The confusion arises from the fact that its rotation axis does not align with the center of the outer circle. Rather, it rotates with the spine of the head.
Another point of confusion is all those lines drawn on the circle. Again, it's not a circle; it's a sphere. The lines on a sphere also contract or expand as the sphere rotates or tilts. Essentially, you need to draw any lines on this sphere considering its surface, not a flat circle. The only lines that are straight are the horizontal and vertical center lines. If the head rotates, the vertical center line becomes a curve, but the horizontal center line does not. If the head tilts, the horizontal center line becomes a curve, but the vertical center line does not. If the head both rotates and tilts, both center lines become curves.
The slope of these curves, mathematically speaking, is the same as the slope of the head's rotation or tilt. Differential calculus is your friend here. The tangent line of the curve at any point is the slope of the curve at that point. But we don't need to go that far. Just remember the slope of the curve is the same as the slope of the head's rotation or tilt. At the very least, remember the outer circle is a 3D sphere, but the inner circle is a 2D circle, which is susceptible to foreshortening.
The inner circle becomes an ellipse when the head rotates or tilts. The slope of the ellipse is the same as the slope of the head's rotation or tilt.
Don't get confused. You are viewing the head from the front. The camera is not moving. The head is moving. The head is rotating or tilting, not the camera. The camera is always at the front. The head is moving around the camera, hence the perspective or foreshortening.
The fixed horizontal or vertical center lines (diameters of the inner circle) never become curves. Only those tilted or rotated ones do. This is crucial to remember. The inner circles are used to cut off the sides of the sphere, the outer circle. Even the center lines of the outer circle or the sphere suddenly turn into straight lines when they meet the inner circle, or now the ellipse. If you draw all these lines just as curves, you get incorrect placements of the features.
The slope of every curvature on the face matters. Do note where the lines are straight and where they are curved.
Back to the rotation logic. We're only considering the horizontal rotation here in order to exclude the tilt. The tilt is a different beast and makes the understanding more complicated.
It's not precise math, but an approximation. However, if you want to, you can make it a precise math problem. And indeed, you can draw perfectly accurate heads at any angle. That's how Blender or any 3D software does it. But that's not the goal here.
Start with non-tilted heads excluding lower or higher angles.
The default profile head's inner circle is just a tad smaller than the 7/8 of the outer circle. Even at this angle, the inner circles cut off both sides of the face. You just don't realize it because the head is not rotated. The perfect front view shows this clearly.
Remember the rotation axis: if the head rotates towards the front, the inner circle contracts or expands depending on which curvature you're looking at. The face-side curvature of the inner circle contracts while the back-side curvature expands. And the back-side curvature is way closer to the rotation axis than the face-side curvature. Thus, there's a clear limit to how much the back-side curvature can expand. It almost stops expanding just about at the 3/4 view.
The focal length of the camera lens comes into play if you use photo references. The longer the focal length, the less distortion you get. The shorter the focal length, the more distortion you get. The human eye is about 50mm, which is considered a normal focal length. However, considering the field of view, it's about 24-35mm. Essentially, this means there can be no right or wrong in the distortion. It's all about the perspective and the focal length. The crucial determinant is what makes your character look more visually appealing. Exaggeration is the key to making your drawing more visually appealing. The facial features closer to the center of the focus should be more exaggerated than the ones farther away. When the head rotates or tilts, the facial features closer to the center of the focus become more exaggerated and look larger. Just imagine you're inspecting this head with a magnifying glass. You'll get the idea.
See those anime big eyes and nose when the head rotates or tilts? That's the exaggeration. With a wider lens, the exaggeration is more pronounced, which is why they call it a fish-eye view. Fish have their eyes on the sides of their heads, which is why they see everything in a fish-eye view. Imagine their field of view! The larger the field of view, the more visual information you get at a glance. Humans have a field of view of about 60 degrees, while fish have about 180 degrees. You get motion sickness in those games usually due to misaligned field of view. Just a fun fact.
Anyway, just go ahead and make every feature a bit exaggerated towards the center of the focus. This means drawing the nose wing or the eye towards the focus way larger than the other side. And they occupy way more space than the other side. That's the key to how you place these elusive facial features in rotated or tilted heads.
The 7/8th semi-profile view means less than 1/8th length rotation, which means the diameter of the outer circle, in terms of the horizontal center line, cuts through right between the eyes, nose, philtrum, and the chin. The face-side curvature of the inner circle or ellipse now contracts about 3 times this length, but the back-side curvature expands about the same length due to its rotation constraints. That is the key to understanding this peculiar foreshortening.
It makes perfect sense then, in the 3/4 view, the face-side curvature of the inner ellipse contracts to about three times the rotation length of the horizontal center line, while the back-side curvature reaches its maximum expansion and almost stops expanding at this view. It even contracts a bit due to the perspective, resulting in just about 1/16th of the outer circle's diameter. Again, the inner circle is a 2D surface, hence the sudden contraction or expansion.
This is to help understand the logic behind the Loomis Head rotation and the placement of the facial features. Once you get the hang of it and can view the outer circle as a sphere and the inner circle as a 2D ellipse, you can draw any head at any angle without much trouble.
However, if your inner ellipse looks way off, your final drawing will look awful because it would mean your facial features are misplaced and disproportionate.
The perspective and foreshortening are what make your drawing visually appealing.
Drawing itself is not a math problem, but a visual problem. You need to see the head as a 3D object and then draw it as a 2D object. Think in 3D but draw in 2D. That's the key. The real problem is, if you're not trained enough, you get lost in the 2D world and forget the 3D world.
Keep practicing while shouting out loud: "I'm drawing a 3D object, not a 2D object!"
It might sound silly, but it helps. Really.