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8:20 PM, Sunday November 22nd 2020
edited at 12:32 AM, Nov 23rd 2020

I am not really sure what your point is, but I can answer your question regarding the “mathematically reliable way of” finding the minor axis: Look for the largest and shortest diameters of each ellips and you will find the major and minor axes.

I took your picture, rotated it in PowerPoint to add perfect ellipses and the major and minor axes (first image), and rotated it back to how you drew it (second image):

https://drawaboxchallenger.wordpress.com/circles/

The major axis of an ellips divides it symmetrically. It has nothing to do with the center of the circle. I think your point is that they are concentric circles in perspective, not concentric ellipses, which is how Uncomfortable describes them.

These are concentric ellipses (having a common central axis):

https://etc.usf.edu/clipart/42600/42661/conellipses_42661.htm

So now, I do think you have a point, but it might depend on how one defines the word concentric , that is what is the center point: an object’s axis disregarding perspective (like in the second example) or with context in mind such as perspective (your/my example).

edited at 12:32 AM, Nov 23rd 2020
10:31 PM, Sunday November 22nd 2020

Hey man, thanks for taking the time to respond. My point is that doing concentric elipses on a wheel is wrong, since the elipses should represent circles on a wheel. It is however a very minor thing, but the lack of foreshortening flattens the image out.

The second purpose of this post is to find learning resources which describe this topic in great detail. I want to understand how one would go about drawing two concentric circles in perspective with complete mathematical accuracy.

12:30 AM, Monday November 23rd 2020
edited at 9:12 AM, Nov 23rd 2020

If you look back at my edit of your wheel in perspective, as far as I know, the major axis can be found if you look at the square in perspective, and take the absolute width of the drawn square (without considering perspective). The major axis shouls lie there, because it divides the left and right part of the foreshortened square evenly. The minor axis is perpendicular to the major axis and you're done.

The only thing is that you need to draw a square in perspective in the first place. As of yet, I do not know a mathematical way for that.

The opposite, however—creating a square in perspective from an ellips—is an option. Currently, I can only imagine the steps, but I think they are correct:

  1. Draw an ellips (narrow horizontally, wide vertically) and draw a vertical line A that is larger than the height of the ellips, and lean it on either side (it should hit either the left or right part of the ellips).

  2. Create the square’s horizontal lines by drawing them from both ends of line A to a vanishing point to the opposite side while hitting the top and bottom of the ellips.

  3. Create the square’s other vertical line by drawing it next to the other part of the ellips.

  4. You’ve drawn a perfect square in perspective. This is for one-perspective. I think that more steps are needed for two-point (or three-point) perspective.

I will see what I can do later today. I could create examples for each mentioned step and see if my theory is correct. I do know that a circle (which with context is a sphere, obviously) should always fit perfectly inside a 3D cube (drawn on a 2D plane). That logic made me think that an ellips should always fit inside a square edge.

EDIT: I think this tutorial will help you a lot!:

https://youtu.be/h0HrmywKzFk

And this is the geometry behind it that you’re looking for:

https://www.handprint.com/HP/WCL/perspect3.html

edited at 9:12 AM, Nov 23rd 2020
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